Physics Reports 396 (2004) 1 – 39 www.elsevier.com/locate/physrep
Double decimation and sliding vacua in the nuclear many-body system G.E. Browna;∗ , Mannque Rhob; c a b
Department of Physics and Astronomy, State University of New York, Stony Brook, NY 11794, USA Service de Physique Th#eorique, CEA/DSM/SPhT, Unit#e de recherche associ#ee au CNRS, CEA/Saclay, 91191 Gif-sur-Yvette c#edex, France c Department of Physics, Hanyang University, Seoul 133-791, South Korea Accepted 13 February 2004 editor: G.E. Brown
Abstract We propose that e/ective 0eld theories for nuclei and nuclear matter comprise of “double decimation”: (1) the chiral symmetry decimation (CSD) and (2) Fermi liquid decimation (FLD). The Brown–Rho scaling recently identi0ed as the parametric dependence intrinsic in the “vector manifestation” of hidden local symmetry theory of Harada and Yamawaki results from the 0rst decimation. This scaling governs dynamics down to the scale at which the Fermi surface is formed as a quantum critical phenomenon. The next decimation to the top of the Fermi sea where standard nuclear physics is operative makes up the FLD. Thus, nuclear dynamics are dictated by two 0xed points, namely, the vector manifestation 0xed point and the Fermi liquid 0xed point. It has been a prevalent practice in nuclear physics community to proceed with the second decimation only, assuming density-independent masses, without implementing the 0rst, CSD. We show why most nuclear phenomena can be reproduced by theories using either density-independent, or density-dependent masses, a grand conspiracy of nature that is an aspect that could be tied to the Cheshire Cat phenomenon in hadron physics. We identify what is left out in the FLD that does not incorporate the CSD. Experiments such as the dilepton production in relativistic heavy ion reactions, which are speci0cally designed to observe e/ects of dropping masses, could exhibit large e/ects from the reduced masses. However, they are compounded with e/ects that are not directly tied to chiral symmetry. We discuss a recent STAR/RHIC observation where BR scaling can be singled out in a pristine environment. c 2004 Elsevier B.V. All rights reserved. PACS: 11.30.Rd; 21.30.Fe; 21.65.+f; 24.10.Cn; 24.85.+p Keywords: Double decimation; MEEFT; Vector manifestation; BR scaling; Sobar con0guration; Cheshire Cat
∗
Corresponding author. E-mail address:
[email protected] (G.E. Brown).
c 2004 Elsevier B.V. All rights reserved. 0370-1573/$ - see front matter doi:10.1016/j.physrep.2004.02.002
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Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. The double decimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. The “intrinsic dependence (ID)” from chiral symmetry decimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1. Sliding vacua . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2. “Soft glue” and quark condensate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3. Nuclear matter from chiral symmetry decimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Fermi-liquid decimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. The vector mass and gauge coupling scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. MEEFT or more e/ective e/ective 0eld theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Doing e/ective 0eld theory in nuclear physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. E/ective interactions in nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Electroweak processes in nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Saturation and Dirac phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. In-medium pion decay constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. The VM and the photon coupling to the nucleon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. “Sobar” con0gurations and CERES dileptons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. The role of the “-sobar” in the 3 He (; 0 )ppn reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. The CERES: fusion of B/R and R/W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. RHIC resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. E/ective forces in nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. Conclusion: return of the Cheshire Cat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Uncited references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 4 5 5 7 9 10 11 12 12 13 15 16 20 24 25 26 27 28 31 34 35 36 36
1. Introduction Brown and Rho adduced empirical evidence for BR scaling in terms of nuclear interactions and nuclear structure more than a decade ago [1–3]. One of the most convincing evidences was the decrease in the tensor interaction in nuclei: The and exchange enter into the tensor interaction with opposite signs. The -coupling to the nucleon is about twice that of vector dominance, so its square is four times greater. Brown and Machleidt [4] showed this to follow unambiguously from the phase shifts from nucleon–nucleon scattering. The large tensor coupling is important in free space, in that it cancels the otherwise extremely strong pionic coupling, which goes as r −3 at short distances, at a distance of ∼ 0:6 fm. Inside of this, the net tensor interaction is repulsive, but strongly cut down in net e/ect by the large repulsion from !-exchange which keeps the two nucleons apart. Because of the exchange, the nucleon–nucleon interaction never gets very strong, explaining why the e/ective nucleon–nucleon interaction obtained in the Sussex work [5], which neglected o/-shell e/ects, worked well in reproducing the e/ective nucleon–nucleon interaction in nuclei. The very smooth behavior of the nuclear interactions in going o/ shell minimizes the di/erence between the on-shell interaction and the half o/-shell one that is commonly calculated as e/ective interactions. In medium, with the decrease in m? to ∼ 0:8m by nuclear matter density n0 , the tensor interaction becomes even weaker, and this was the early evidence adduced by Brown and Rho for BR scaling.
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Our work is prompted by two recent developments that are both profound and powerful for nuclear structure. One is the notion of “vector-manifestation (VM) 0xed point” in e/ective 0eld theory of hadrons discovered by Harada and Yamawaki [6,7] and the other is the identi0cation of nuclear matter as a Fermi liquid at its 0xed point [8–10]. The 0rst 0xed point is dictated by the matching of e/ective 0eld theories to QCD and the second accounts for the stability of strongly interacting many-body systems that possess a Fermi surface arising from a quantum critical phenomenon. We discuss how one could combine these two 0xed point structures into an e/ective 0eld theory of nuclei and nuclear matter. The plan of this review is as follows. In Section 2, we develop the concept of “double decimation” for describing nuclear phenomena based on the general strategy of e/ective 0eld theories that can represent low-energy nonperturbative QCD. The 0rst decimation deals with chiral symmetry scale and we suggest that the “parametric” density dependence encoded in BR scaling [3] is the consequence of the “chiral scale decimation” (CSD) tied to the “VM” aM la Harada and Yamawaki [6,7] in hidden local symmetry theory that is matched to QCD. The second decimation deals with the Fermi momentum scale and corresponds to (Landau) Fermi liquid description of nuclear matter. We shall refer to this as “Fermi liquid decimation (FLD).” In Section 3, one way of doing e/ective 0eld theory in nuclear physics is presented with a focus on how to incorporate the standard nuclear physics approach (SNPA) that has had a spectacular success, into the framework of modern e/ective 0eld theory that we shall refer to as more e;ective e/ective 0eld theory (MEEFT in short). Both the e/ective nucleon interaction Vlow-k and the electroweak matrix elements of few-nucleon systems are described in this formalism. In Section 4, we discuss in a general way why the need for density-dependent hadron masses was missed for so many years in nuclear phenomena; namely, the description of most phenomena with density-independent masses could 0t experiments. We show that with a common scaling with density of hadron masses, the kinematics of the nuclear many-body problem are not changed very much in the region of densities up to nuclear matter density and we illustrate this with the highly successful Dirac phenomenology. As cases where BR scaling can be singled out in a speci0c way in nuclear processes, we discuss in Section 5 how the scaling enters in what is known as Warburton’s jMEC factor in axial charge transitions in heavy nuclei as well as in deeply bound pionic atoms which have been recently studied experimentally. In Section 6, we suggest that the VM 0xed point a=1 of Harada and Yamawaki is “precociously” reached when baryons are present and hence in nuclear medium, as a consequence of which the photon couples half and half to the vector meson and the nucleon core (or skyrmion). This picture may be relevant to the electromagnetic form factors of the proton recently measured at the J-Lab as well as to the longitudinal and transverse form factors of nuclei with the meson and nucleon masses dropping aM la BR scaling. While it is now generally accepted that the CERES dileptons can be explained by modi0ed properties of hadrons in dense and hot matter, it is diQcult to single out BR scaling as the principal mechanism for the shifted spectral distribution as temperature and density e/ects are compounded in the process. We sketch in Section 7 what we believe to be the correct way of interpreting the dilepton data.
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It is discussed in Section 8 how the situation can be a lot clearer in the RHIC experiments. We discuss in particular how a direct measurement of medium-dependent vector-meson mass m∗V could be made in a pristine environment where temperature e/ects are small and where the density can be well reconstructed. A discussion of the (absence of) need for “double decimation” in nuclear structure makes up Section 9 where we discuss e/ective interaction between nucleons in nuclei. The Kuo–Brown interactions [11] have undergone a renaissance with the 0nding from the renormalization-group (RG) formulation that all nucleon–nucleon interactions which 0t the two-body phase shifts give the same e/ective interactions Vlow-k in nuclei. The RG cuto/ is chosen to be at the upper scale to which phase shifts have been obtained from nucleon–nucleon scattering, i.e., ∼ 400 MeV. That Vlow-k provides a unique e/ective interaction which works well in reproducing e/ective forces in nuclei is unquestionable, given the excellent 0ts to all sorts of data. Nonetheless, there are questions: (1) Where does the modi0cation in the tensor interaction, which is lowered by the decreasing mass of the -meson which contributes with opposite sign to the pion (whose mass is presumably unchanged in medium at low density), come in? It is well known that certain states, such as the ground state of 14 N, depend sensitively on the tensor interaction. Also in the nuclear many-body problem, the decrease in the tensor force will give less binding energy and less help with saturation, the second-order tensor force giving a large contribution to the binding energy of nuclear matter. In fact, this second-order term which has the form 3 + ˜1 · ˜2 Ve/ V (r) (1) 4 is large as we discuss. In an old paper, Feshbach and Schwinger [12] describe the di/erence between 1 S and 3 S potentials in terms of the above Ve/ which acts only in the latter state. (2) In addition to the modi0cations in both tensor and spin–orbit interactions, which have not been put into the FLD, there is the question of o/-shell energies of intermediate particles 1 which are taken to be plane waves. The conclusion following from the Bethe reference spectrum [13] was that there was a small amount of binding in the intermediate states with energies just above the Fermi surface, referred to as “dispersion” correction. Generally, taking the intermediate states to be free, as in the Schwinger interaction representation, seemed to be a good approximation. This is what is done in the RG calculations, in the FLD. Concluding remarks are given in Section 10 with a reference to the Cheshire Cat Principle revisited. We postulate that the missing “smoking gun” for BR scaling in nuclear structure physics is an aspect of the Cheshire Cat Principle discovered in the baryon structure as reTected in many-nucleon systems. 2. The double decimation Our ultimate objective is to describe in a uni0ed way 0nite nuclei, nuclear matter and dense matter up to chiral restoration. For this, we introduce the approximation—double decimation—by which the phase structure in the hadronic sector can be drastically simpli0ed. The e/ective 0eld theory 1
The holes are on-shell in the Brueckner–Bethe theory.
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(EFT) approach to few-nucleon systems described below does not require both decimations but if one wants to correctly describe phase transitions of hadronic systems under extreme conditions of density and/or temperature, one needs at least two decimations which we now describe. We propose that the procedure consists of what we call CSD and FLD. 2.1. The “intrinsic dependence (ID)” from chiral symmetry decimation As we have explained in a series of recent papers [14–17], a candidate e/ective 0eld theory relevant in the hadronic sector matched to QCD at a scale near the chiral scale is hidden local symmetry (HLS) theory with the VM found by Harada and Yamawaki [6,7]. This theory, denoted HLS/VM in short, with the vector mesons, 2 , !, etc. treated as light on the same footing as the (pseudo) Goldstone pions as originally suggested by Georgi [18], turns out to yield results consistent with chiral perturbation theory [19]. Given this theory valid at low energy in matter-free space, one can then construct the chiral phase transition in both hot [20] and dense [21] media recovering BR scaling [3] near chiral restoration. This theory makes unambiguous predictions, at least at one-loop order, on the vector and axial-vector susceptibilities and the pion velocity at the transition point that seem to be di/erent from the standard scenario and that can ultimately be tested by QCD lattice measurements [22]. In the rest of this section, we will con0ne out consideration on 0nite density at zero temperature. 3 An early, seminal attempt to arrive at nuclei and nuclear matter in chiral Lagrangian 0eld theory that models QCD was made by Lynn [23]. What we are interested in here is to exploit the recent development of HLS/VM and arrive at a 0eld theoretic description anchored on QCD. In studying nuclear systems, the EFT and QCD have to be matched via current correlators at a suitable scale M in the background of matter characterized by density n. The matching de0nes the “bare” HLS Lagrangian, giving to the parameters of the Lagrangian the “intrinsic” density dependence (IDD). This means that the parameters of the Lagrangian such as hidden gauge coupling constant g∗ , gauge boson mass m∗ , etc. will depend intricately not only on M but also on density n. This IDD is generally missing in the calculations that do not make the matching to QCD. Next to account for quantum e/ects, we need to decimate the degrees of freedom and excitations from the matching scale M to a scale commensurate with the presence of a Fermi sea, FS . The latter is the scale relevant to nuclear physics, typically FS . 1:5 fm−1 . What governs this 0rst decimation is chiral symmetry. 2.1.1. Sliding vacua The HLS/VM at the present stage of development can tell us what happens only at n ≈ 0 and nc . It does not tell us how the IDD interpolates from n = 0 to nc . What we are interested in, however, involves a wide range of densities, from zero to nuclear matter and ultimately to chiral restoration. How to do this is at present poorly known. There are, however, two approximate ways to address this issue. One is based on extending NJL model so as to simulate the VM e/ect associated with vector-meson degrees of freedom. This is the approach we have proposed before [17] which we shall follow in the rest of the paper. The other which is more ambitious and perhaps closer to the 2 3
We are assuming U (2) Tavor symmetry in the two-Tavor case. Implementing temperature e/ects is straightforward and hence will be ignored in what follows in this section.
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spirit of QCD, hence harder to quantify, gives a clearer qualitative picture of “sliding vacua.” It is based on a skyrmion description of dense hadronic matter. The skyrmion approach to dense matter developed recently [24–26] makes transparent the crucial role of the IDD in nuclear processes. The power of the skyrmion model is that a single e/ective Lagrangian provides a uni0ed description of mesons and baryons and treats single- and multi-baryon interactions on the same footing, thereby describing 0nite nuclei as well as in0nite nuclear system —nuclear matter. Although the notion of skyrmions as modelling of QCD is fairly well established [34], one does not yet know at present how to write down a fully realistic skyrmion Lagrangian, and hence cannot do a quantitative calculation but one can gain a valuable insight into the structure of the “sliding vacua” that we are interested in. Following [24–26], one considers a skyrmion-type Lagrangian with spontaneously broken chiral symmetry and scale symmetry, associated, respectively, with nearly massless quarks and trace anomaly of QCD. The e/ective 0elds involved are the chiral 0eld U = ei=f with the (pseudo)Goldstone bosons for the former and the scalar “dilaton 0eld” for the latter. In the next subsection, we will specify more precisely what physics is represented by this “dilaton 0eld.” We will see that this 0eld interpolates the “soft component” of the trace anomaly that “melts” across the critical temperature Tc intimately locked to the order parameter of chiral symmetry qq. V We will assume that it is this component that 0gures in the skyrmion matter as was done in [24]. A theory so constructed which may be considered to be an Nc → ∞ approximation to QCD can, albeit approximately, describe not only the lowest excitation, i.e., pionic, sector but also the baryonic sector and massive vector-meson sector all lying below the chiral scale . What is even more signi0cant, it can treat on the same footing both single- and multi-baryon systems including in0nite nuclear matter [25]. We are interested in how low-energy degrees of freedom in many-body systems behave in dense matter, in particular, as the density reaches a density at which QCD predicts a phase transition from the broken chiral symmetry to the unbroken chiral symmetry, i.e., chiral restoration. The simplest possible Lagrangian with the given symmetry requirements consistent with QCD is of the form [24] f2 L= 4
2
1 f2 m2 † † 2 Tr(9 U 9 U ) + Tr([U 9 U; U 9 U ]) + 32e2 4 1 1 m2 4 1 1 + : ln(=f ) − + 9 9 − 2 2 4 f 4 4 f
†
f
3
Tr(U + U † − 2) (2)
We have denoted the vacuum expectation value of as f , a constant which describes the decay of the scalar into pions in matter-free space. The QCD trace anomaly can be reproduced by the last term of (2), i.e., the potential energy V () for the scalar 0eld, which is adjusted so that V =dV=d=0 and d 2 V=d2 = m2 at = f . The vacuum state of the Lagrangian at zero baryon number density is de0ned by U =1 and =f . The Tuctuations of the pion and the scalar 0elds about this vacuum, de0ned through ˜ ) and = f + ˜ U = exp(i˜! · "=f
(3)
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give physical meaning to the model parameters: f as the pion decay constant, m as the pion mass, f as the scalar decay constant, and m as the scalar mass. The coupled classical equations of motion for the U and 0elds of (2) give rise to solitons for given winding numbers corresponding to given baryon numbers with their structure constrained by the classical scalar 0eld. Nuclear matter is then described classically by an FCC crystal and possibly a Fermi liquid when quantized. When the system is strongly squeezed, there is a phase transition from the FCC state to a half-skyrmion state representing chiral restoration in dense system [24]. It is assumed that this transition remains intact when the initial state is in Fermi liquid. Fluctuations on top of the background crystal structure describe excitations of the pionic and scalar quantum numbers, the properties of which are then governed by the mean 0eld values—denoted ∗ —of at a given density 0xed by the unit cell size L of the crystal. What is found in [24] is that the mean 0eld of the scalar mostly—though not entirely—governs the scaling behavior of the parameters— such as the pion decay constant—of the Lagrangian indicative of the sliding vacuum structure. The results are found to be in qualitative agreement with what is described below in NJL. The skyrmion matter of [24] cannot address directly the property of light-quark vector mesons although the latter can be probed through correlation functions. In order to see how the vector mesons and ! behave in matter, one must introduce the vector 0elds explicitly. This can be done in the HLS formalism. A recent work along this line [27] with the dilaton suitably incorporated has indeed con0rmed that BR scaling is satis0ed by the skyrmion matter at least for low density n . n0 : 2 ∗ ∗ ∗ m m! f : (4) = = = m m! f f 2.1.2. “Soft glue” and quark condensate In discussing the notion of “sliding vacua,” we used the dilaton 0eld to interpolate the soft component of the trace anomaly. We can interpret the as interpolating “soft glue.” In this subsection, we give a precise meaning to this interpretation in terms of lattice data in hot matter. In the literature [28–30] the QCD condensed glue that 0gures in chiral restoration has been discussed as “soft glue,” that glue that gets condensed at temperatures below Tc . The condensed glue in the vacuum implied by the trace anomaly is thought to be composed of “soft glue” and “hard glue” associated, respectively, with the spontaneous breaking and explicit breaking of conformal symmetry. It is this soft glue that melts across Tc . Now conformal symmetry can be broken spontaneously only if it is also explicitly broken [31]. The restoration of the spontaneously broken scale invariance takes place together with chiral restoration. Precisely this restoration provides Brown–Rho scaling in temperature. The melting of soft glue is easily discussed in terms of NJL [28], connected, as it is, with the change in mass of the constituent quark with scale (temperature or density). The binding energy of the soft glue can be interpreted simply as the gain in energy by putting massive rather than massless quarks into negative energy states, i.e.,
3 d k 2 2 ˜k + m − |˜k| B:E: (soft glue) = 12 (5) Q 3 0 (2)
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4.05
G [GeV-2]
4 3.95 3.9 3.85 3.8 0
0.05
0.1
0.15 T [GeV]
0.2
0.25
0
0.05
0.1
0.15 T [GeV]
0.2
0.25
(a)
B.E. [GeV4]
0.015
0.01
0.005
0 (b)
Fig. 1. (a) Temperature dependence of the coupling G and (b) B.E. (soft glue) with varying G. In this calculation = 695 MeV and mcurrent quark = 0 are used.
with ∼ 700 MeV in order to give Tc = 170 MeV. This binding energy is given up by the quarks as they go massless, going into the melting of the soft glue [28]. Now the information about the glue is carried in the (dimensionful) coupling constant G in NJL, which has only the parameters G and . Lattice gauge calculations show that the soft glue melts rapidly with increasing temperature just before Tc . We show in Fig. 1 calculation [28] compared with the lattice results of David Miller [32]. In order to bring about the rapid drop, the coupling constant G must be decreased, but only by ∼ 6%. This small drop shows that chiral symmetry breaking (in temperature) is fragile, chiral restoration being reached by such as small change in G. The hard glue, which has nothing direct to do with either chiral restoration or BR scaling is melted only slowly as T goes from Tc to about 2Tc [33]. The explicit conformal symmetry breaking is thus unscathed with chiral invariance (and with BR scaling); it is therefore natural that it takes over at Tc .
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Unfortunately, no such simple demonstration exists for increase of scale with density but we saw above that the behavior of dense skyrmion matter is consistent with what we have seen in temperature. Furthermore, in Section 8, we give direct empirical evidence constructed in a pristine environment that BR scaling is realized also in density, at least in dilute systems. 2.1.3. Nuclear matter from chiral symmetry decimation Given an HLS/VM theory decimated to FS , how does one go about obtaining nuclei and nuclear matter? The 0rst thing we need is a Fermi surface characterized by the Fermi momentum kF ∼ FS and the Fermi surface arises in e/ective 0eld theories as a quantum critical phenomenon. As sketched in [34] and developed in [25,26], in the skyrmion picture, a nucleus of mass number A arises as a topological soliton of winding number W = A. Nuclear matter is then given by W = ∞. When quantized, the extended soliton system will naturally possess a Fermi surface characterizing the 0lled Fermi system. In confronting nuclei and nuclear matter in nature, however, it is more advantageous to work with explicit nucleon 0elds rather than with multi-winding number skyrmions. When nucleons are explicitly present in the theory, a nucleus will no longer emerge as a topological object. Instead, it must arise as a nontopological soliton in a way conjectured by Lynn [23]. Nuclear matter will then be more like a chiral liquid in Lynn’s language. We would like this soliton to emerge in a simple way from an e/ective Lagrangian endowed with the parameters of the Lagrangian intrinsically density dependent. A systematic derivation of such a soliton structure that is realistic enough is lacking at the moment. However, there is a short-cut approach to this and it relies on Walecka’s mean 0eld theory of nuclear matter [35]. The principal point we put forward is that the Walecka mean-0eld solution of certain e/ective Lagrangian (speci0ed below) can be identi0ed as the soliton—topological or nontopological—solution described above. Now, with the nucleon 0elds explicitly incorporated, there are two (equivalent) ways to write down such an e/ective Lagrangian of Walecka type that results from decimating down to FS [14]. One is the type-I approach in which the heavy-meson degrees of freedom of the HLS Lagrangian are integrated out and the other is the type-II one in which relevant heavy-meson degrees of freedom are retained. The two versions give equivalent descriptions of the same physics for the ground state and low-frequency Tuctuations. The type-I Lagrangian has the form LI = NV [i (9 + iv + gA? 5 a ) − M ? ]N − Ci? (NV +i N )2 + · · · ; (6) i
where the ellipsis stands for higher dimension and/or higher derivative operators and the +i ’s Dirac and Tavor matrices as well as derivatives consistent with chiral symmetry. The star aQxed on the masses and coupling constants represents the intrinsic parametric dependence on density relevant at the scale FS . The induced vector and axial vector “0elds” are given by v = −(i=2)(.† 9 . + .9 .† ) and a = −(i=2)(.† 9 . − .9 .† ). In (6) only the pion () and nucleon (N ) 0elds appear explicitly: all other 0elds have been integrated out. The e/ect of massive degrees of freedom that are integrated out and that of the decimated “shells” will be lodged in higher-dimension and/or higher-derivative interactions. The external electro-weak 0elds if needed are straightforwardly incorporated by suitable gauging.
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To write the Lagrangian for the type-II approach, we need to pick the appropriate heavy degrees of freedom we want to consider explicitly. This Lagrangian will be essentially the HLS Lagrangian implemented with nucleon 0elds and chiral scalar heavy mesons that are not taken explicitly into account. The relevant heavy mesons for nuclear physics are a vector meson in the ! channel which is a chiral scalar and a Tavor scalar that plays an important role in Walecka-type model for nuclear matter. Assuming that U (Nf ) (Nf = 2 for low-energy nuclear physics) symmetry holds in medium, we can put the and ! in the U (Nf ) multiplet and write a Harada–Yamawaki HLS Lagrangian with the parametric dependence suitably taken into account. To make the discussion simple, let us consider symmetric nuclear matter in which case the type-II Lagrangian can be written in the form of Walecka linear theory [35] with the parametric density dependence represented by the star, LII = NV (i (9 + igv? ! ) − M ? + h? )N −
m2 ? m2 ? 1 1 2 F + (9 )2 + ! !2 − 2 + · · · ; 4 2 2 2
(7)
where the ellipsis denotes higher-dimension operators. We should stress that (7) is consistent with chiral symmetry since here both the ! and 0elds are chiral singlets. In fact, the here has nothing to do with the chiral fourth-component scalar 0eld of the linear sigma model except near the chiral phase transition density; it is a “dilaton” connected with the trace anomaly of QCD. The possibility that this dilaton turns in-medium into the fourth component of the chiral four-vector in the “mended symmetry” way aM la Weinberg [36] near chiral restoration has been discussed in [37]. Since we are far from the critical density at which chiral restoration takes place, the vector mesons are massive and the would-be Goldstone scalars (the longitudinal components of the vector mesons) are absent. In (7), the pion and -meson 0elds are dropped since they do not enter in the mean 0eld approximation but they can be put back if needed (as for Tuctuations in the pionic channel) in HLS symmetric way. 2.2. Fermi-liquid decimation Given the type-I (6) or type-II (7) Lagrangians, the next step is to do the FLD. In [38], Schwenk et al. build on the e/ective interactions Vlow-k a decimation scheme to arrive at Fermi-liquid parameters, namely the Landau e/ective mass for the quasi-nucleon m? N and the quasiparticle interactions F. In doing this, the intrinsic density/temperature dependence coming from the chiral symmetry decimation was not included. It also makes the implicit assumption that n-body interactions for n ¿ 2 are negligible. In any event, it has been shown that both m? N and F are 0xed-point quantities. This approach purports to “calculate” the parameters of the “penultimate” e/ective Lagrangian from 0rst principles. An alternative method which we follow in this article is based on the mapping of the e/ective 0eld theory Lagrangians (6) and (7) to the Fermi liquid 0xed point theory as beautifully explained in a review by Shankar [8]. The mapping relies on the work done by Matsui [39] who showed that Walecka linear mean ?eld theory is equivalent to Landau Fermi-liquid theory. Using this argument, it is then immediate to map the mean-0eld solution of (6) or (7) to Landau EFT: in the mean 0eld
G.E. Brown, M. Rho / Physics Reports 396 (2004) 1 – 39 * < qq >
?? ~
n0
< q q >*
g* /g
< qq >
mV */mv
~
11
?? ~
< qq >
nc Tc
n0
< q q >* < qq >
nc Tc
? Fig. 2. Schematic scaling behavior for m? V and g versus density. The cross-over region which is not understood is marked by ??.
approximation, the two theories, (6) and (7), yield the same results [40,41]. This chain of reasoning, developed in [10,42,43], has shown that the Tuctuations on top of the Fermi liquid 0xed point— represented by the mean 0eld of (6) or (7)—in response to EW external 0elds can be related to the BR scaling parameter given by the ratio 1 = m? =m.
2.3. The vector mass and gauge coupling scaling ? In [14], we discussed on how the vector-meson mass m? in HLS V and the gauge coupling g theory with the VM—both of which are the essential elements of our EFT—drop in medium as one goes from below to above nuclear matter density. At very low density, chiral perturbation theory provides relevant information and near the critical density, the VM tells us how they scale. In between, little could be said at the moment. Here we extrapolate the scaling behavior from zero density (or temperature modulo Dey–Eletsky–Io/e low-temperature theorem [44]) to the critical density (or temperature). Our picture is summarized in Fig. 2. Up to near nuclear matter density
relevant to the chiral symmetry decimation, the BR scaling of the mass is expected to go like qq V ? and the vector coupling (more precisely the HLS gauge coupling) to stay more or less unchanged. The former is deduced from the GMOR relation for in-medium pion mass which should hold in HLS theory for small density, with the assumption that the pion mass does not change in density as indicated, e.g., by recent SU(2)c lattice results [45]. The latter generically follows from chiral models and is consistent with the weakening tensor forces in nuclei. We do not know precisely how the scaling changes as we go above the matter density. However, as we approach the critical density, we learn from HLS theory matched to QCD that the VM takes place, which means that both the hadronic mass and the gauge coupling constant should scale linearly in qq V ? [21]. Thus, near the chiral transition point, g? =m? V should go to a constant of density as required for quark number susceptibilities [46].
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3. MEEFT or more eective eective eld theory 3.1. Doing e;ective ?eld theory in nuclear physics In this section, we consider the situation where one does not need to go through the double decimation procedure described above. Suppose one is interested only in describing low-energy nucleon–nucleon scattering. For this, there are a variety of equivalent ways to proceed [47]. First, one writes down an e/ective Lagrangian using the 0elds describing the degrees of freedom relevant for the process in question. For instance, if one is looking at the S-wave two-nucleon scattering at low energy, say much less than the pion mass, one can simply take only the nucleon 0eld as an explicit degree of freedom, integrating out all others, including the pion. The pions which are important due to the broken chiral symmetry may be introduced perturbatively. A systematic power counting can then be developed and used, in conjunction with an appropriate regularization (e.g., the power divergence substraction (PDS) scheme [47]), to compute the scattering amplitude by summing to all orders a particular set of diagrams. In principle, the parameters of the pionless Lagrangian could be determined for a given scale by lattice QCD. In this sense, such a systematic higher-order calculation in this EFT can be considered to be equivalent to doing QCD. In practice, however, the parameters are obtained from experiments. For two nucleon systems, two classes of parameters are to be determined from experiments. One is single-particle vertex and this is given by on-shell information. The other is intrinsically two-particle in nature requiring data on two-nucleon processes. When the parameters are fully determined, this procedure with the pionic contributions taken into account perturbatively does lead to sensible results. For instance, it correctly reproduces such standard nuclear physics results as the e/ective-range formula. The EFT which adheres strictly to order-by-order consistency in the power counting (that we shall refer to as “purist’s EFT”), however, su/ers from the lack of predictivity. The number of unknown parameters increases rapidly as the number of nucleons involved in the process increases. Thus, even if two-nucleon systems are well described by the EFT in question, treating systems involving more than two or three nucleons becomes prohibitively diQcult, if not impossible. Furthermore, treating pions as perturbative misses the power of the “chiral 0lter mechanism” [48] that plays an important role in predictive calculations [49]. At present, going to nuclear matter is out of reach by this EFT approach. Even if such a method were available so that we could write down an e/ective Lagrangian on top of a Fermi sea, the parameters of the penultimate e/ective theory for, say, nuclear matter, would be very far from the 0rst principles, QCD: calculating them would be somewhat like calculating the boiling point of the water starting from QED. We propose here how to circumvent this impasse. Our approach proposed here is admittedly indirect and drastically simpli0ed. Broadly speaking, it involves a two-step decimation starting from the chiral Lagrangian that is matched to QCD at the chiral scale ∼ 1 GeV. The key ingredient that allows this feat is the highly re0ned “standard nuclear physics approach (SNPA)” and the objective is to marry the SNPA to an EFT. As an illustration of our strategy, we brieTy discuss here how the marriage can be e/ectuated. As summarized recently [14,50], a thesis developed since some time posits that by combining the SNPA based on potentials 0t to experiments with modern e/ective 0eld theory, one can achieve a more predictive power than the purist’s EFT alone can. The idea was recently given a test in a variety of electroweak processes in nuclei, in particular, the solar pp fusion and hep processes where highly
G.E. Brown, M. Rho / Physics Reports 396 (2004) 1 – 39
13
accurate predictions could be made free of parameters [51]. As alluded above, a closely related mechanism has been found at work in nuclear e/ective interactions that 0gure in nuclear structure calculations [52]. We brieTy sketch these two recent developments which are closely connected to the issue at hand. 3.2. E;ective interactions in nuclei The calculation of the EW response functions that will be the subject of the next subsection relied on an implicit assumption on the e/ective forces that enter in the calculation of the wave functions. A recent development provides a support to this assumption. Consider two nucleon scattering at very low energy, a process very well understood in SNPA. The relevant T matrix for the scattering is a solution of the Lippmann–Schwinger equation with a “bare” potential Vbare 0guring as the driving term. The long-range part of the bare potential Vbare is governed by chiral symmetry, namely, by a pion exchange and hence is unambiguous. But the short-range parts are not unique. Even if the potentials are determined accurately by 0tting scattering data up to say Elab ∼ 350 MeV, those potentials that give the equivalent phase shifts can di/er appreciably, in particular in the short-distance parts. In terms of e/ective 0eld theory, what this means is that while the long-range parts given by “low-order” expansion are the same for all the realistic potentials, shorter-range terms that are given by higher-order terms can di/er depending upon how they are computed. They will depend upon how the power counting is organized, what regularization is used, etc. In practice, those “higher-order” terms are 0xed by 0tting to experimental data. The examples for such realistic potentials are the Paris potential, the Bonn potential, the chiral potential, etc. Suppose in the integral equation satis0ed by the scattering T -matrix, one integrates out the momentum scales above a given cuto/ suQciently high to accommodate the relevant degrees of freedom and probe momentum but low enough to exclude the massive scales that do not explicitly 0gure in the theory, with the requirement that the resulting e/ective potential reproduce the phase shifts while preserving the long-range wave function tails as given by the half-on-shell (HOS), T (k ; k; k 2 ). To the extent that the HOS T matrix is a physical quantity, it should be independent of where the cuto/ is set d T (k ; k; k 2 ) = 0 : d
(8)
This condition leads to a renormalization group equation (RGE) for the e/ective potential, denoted Vlow-k . It is important to note that the 0t to experiments de?nes the complete theory for the probe momentum k. Now since the HOS T is 0t to experiments, integrating out the momentum component p ¿ transfers the physics operative above the cuto/ into the counter terms that are to be added to the bare potential to give the e/ective one, Vlow-k . Bogner et al. [52] have shown that the resulting Vlow-k is independent of the bare potential one starts with as long as it is consistent with the chiral structure, i.e., the long-range tails of the wave functions, and the T is 0t to experiments. It is important to recognize that this strategy is none other than that of the SNPA. The results of Ref. [52] are reproduced in Fig. 3. The 0gure on LHS shows the variety of bare potentials that are 0t to experiments and the one on the RHS the e/ective potentials collapsing into a universal curve at = 2 fm−1 after the integrating-out procedure. How the collapse occurs from di/erent Vlow-k ’s for di/erent potentials is shown in Fig. 4. For ¿ 2 fm−1 , Vlow-k ’s can be di/erent for di/erent potentials but they “collapse” to one universal curve for = 2:1 fm−1 . The reason for
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G.E. Brown, M. Rho / Physics Reports 396 (2004) 1 – 39
Vlow k (k,k) [fm]
4 VNN (k,k) [fm]
1
Paris Bonn-A Chiral V-18
2
0
0
-1
Paris Bonn-A Chiral V-18
-2 -2
0
5
10
0
0.5
k [fm-1]
1
1.5
2
k [fm-1]
Fig. 3. Diagonal matrix elements of several “realistic potentials” Vbare and the e/ective potentials Vlow-k as a function of relative momentum in the 1 S0 partial wave. The e/ective potentials are calculated with the cuto/ = 2 fm−1 , from [52].
Fig. 4. The collapse of the diagonal momentum-space matrix elements of Vlow-k as the cuto/ is lowered to = 2:1 fm−1 in the 3 S1 partial wave.
this is easy to understand: that the phase shift analyses from which the two-body interactions in Fig. 4 were 0t were carried out for experiments up to laboratory energies of ∼ 350 MeV, which corresponds to a c.m. momentum of 2:1 fm−1 . 4 Although the phase shifts were determined on-shell, as we noted in Section 1 the interaction is not far o/-shell when used in nuclei. This is how one 4
At about this energy substantial inelasticity sets in, complicating analysis.
G.E. Brown, M. Rho / Physics Reports 396 (2004) 1 – 39
15
can understand that the half-o/-shell Vlow-k reproduced the same diagonal matrix elements for the various potentials 0t to experiments to the given cut-o/ scale. The point is that o/-shell e/ects are unimportant in considerations of the smooth parts of the shell-model wave functions. Of course, these e/ects can be large if the wave functions are forced far o/ shell by short-range two-body interactions, but these will involve a scale higher than our 2:1 fm−1 not constrained by the experimental data. That while the bare potentials are widely di/erent for all momenta, all realistic potentials give the identical e/ective potential illustrates the power of the strategy that combines the accuracy achieved by the SNPA and chiral e;ective ?eld theory of QCD into a “more e;ective e;ective ?eld theory (MEEFT).” The key lesson from this result is that the short-distance part of the potential which represents higher orders in EFT power counting schemes which di/er for di/erent counting schemes may be di/erent from one “realistic potential” to another but when suitably regularized taking into account the constraints by experimental data, the resulting e/ective potential comes out unique. For instance, the “chiral potential” which is consistent with the chiral counting, and hence presumed to be more in line with the tenet of EFT and the v18 potential which, apart from the long-range part, is not, can di/er at “higher orders” but gives the same Vlow-k . This reTects how the MEEFT works. An identical mechanism is at work for the weak matrix elements relevant for the solar neutrino processes [51] as described below. The Vlow-k is the basis of nuclear structure calculations replacing the role of G-matrix. For low-energy processes, it is insensitive to short-distance physics properly taking into account the standard short-range correlations. It is also the input for 0eld theoretic calculation of the Landau parameters for nuclear matter [38] which are the Fermi liquid 0xed point quantities. 3.3. Electroweak processes in nuclei The next question we raise is: How can one do a uni0ed e/ective 0eld theory calculation which is truly predictive for the following processes? p + p → d + e + + ve ;
(9)
H → 3 He + e− + vVe ;
(10)
3
p + 3 He → 4 He + e+ + ve ;
(11)
n + 3 He → 4 He + :
(12)
Processes (9) and (11) take place in the Sun, playing an essential role for the solar neutrino problem but have not been measured in the laboratories while processes (10) and (12) are accurately measured in the laboratories. Given the triton beta decay data, the primary objective then is to make an accurate prediction for the rates for (9) and (11). This highly nontrivial feat has been accomplished recently by Park et al. [51] by an MEEFT. This MEEFT method used for the prediction for the hep process (11) has been recently tested by Song and Park [53] with the hen process (12). The theoretical uncertainty is veri0ed to be ∼10% in both processes. To the best of our knowledge, such a predictive calculation (for both (11) and (12)) is not yet feasible in other EFT strategies developed so far (e.g., the purist’s EFT).
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The MEEFT strategy of [50] employed in the publications by Park et al. [51] goes as follows. 5 First one picks a potential 0t to on-shell NN scattering up to typically Elab ∼ 300 MeV plus many-body (typically three-body) potentials, the combination of which is to describe accurately many-nucleon scattering data. As stated, this potential is required to be consistent with chiral symmetry, the symmetry of QCD, which means that the long-range part of the potential is dictated by the pion exchange. An example for such a potential is the v18 potential which has been used in the numerical calculation. Next one computes the weak current in a chiral perturbation theory, e.g., heavy-baryon chiral perturbation theory, to an appropriate order in the power counting, say, Qn where Q is the characteristic momentum scale. The current matrix element is then computed with this current and the “realistic” wave functions computed with the given potential. In doing this, one integrates out all momentum components above the cuto/ , the e/ect of which is a set of counter terms appearing in the e/ective current. Working to the next-to-next-to-next order (i.e., Q3 ) relative to the leading matrix element given by the single-particle Gamow–Teller matrix element in (9)–(11), one encounters a number of counter terms but symmetry consideration reduces them to a single combination, denoted dˆ R in [51], entering in all three processes. The counter term coeQcient dˆ R is cuto/-dependent and so is the matrix element of the current with the momentum component p ¿ integrated out. The numerical values of these individual terms di/er from one potential to another but the sum of the two does not. The strategy then is to exploit that the short-range component of the interactions must be universal, that is, independent of the mass number. This means that the coeQcient dˆ R can be determined for any given cuto/ from the accurately measured process (10). Since there are no other unknowns to order Q3 , we have a parameter-free theory to calculate all the matrix elements that 0gure in (9) and (11). Indeed, this is what was done in [51]: The S factor for the pp fusion process was calculated within the accuracy of 0.4% and that for the hep process within the accuracy of 15%. Both results, particularly the latter, are of unprecedent accuracy unmatched by other calculations. 6 The con0dence in this prediction is justi0ed by the similar accuracy obtained by the same method for the hen process in the work of Song and Park [53]. 4. Saturation and Dirac phenomenology In detail, the situation with dropping masses is complicated because the pion, both in lowest order and in two-pion exchange, contributes to the energy, but m is most likely unscaled, whereas the scalar and vector mesons, two most important e/ective 0elds in nuclear medium, are. Rapp et al. [54] showed that given these scalings, saturation does come at the right density. The chief contributions to the binding energy come from the and ! exchange. Here we show schematically how these behave in the “swelled” world. Consider a Hamiltonian 1 e−m! rij e−m rij 1 2 H= g! (13) ∇ + − g 2mN i 2 ij rij rij i 5 The procedure employed for the hen process [53] is identical to that used for the hep process, except that there is a subtle di/erence in the “chiral 0lter property” which however does not a/ect the analysis involved. 6 We remind the readers that up to date, the calculated values for the hep S factor varied by orders of magnitude, hence completely unknown.
G.E. Brown, M. Rho / Physics Reports 396 (2004) 1 – 39
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with ground state energy H = E0
:
(14)
Now scale all m’s m? N = ;mN ;
m? ; ! = ;m; ! :
(15)
Then H ? (;;˜x) = ;H (1;˜r) ;
(16)
˜x = ;˜r :
(17)
H ? (;; r) = E0 (;)
(18)
where Now or ;H (1; x) = E0 (;)
:
(19)
But H (1; x) is just a relabelling of H via r → x, etc. so the solution of H (1; x) = E
(20)
for the lowest eigenvalue is E0 , or E0 = E0 (;)=; :
(21)
E0 (;) = ;E0
(22)
Thus, and the scaled system (with ;) is less bound than the original unscaled system since ; = 0:8 at nuclear matter density. The replacement of the correlated two exchange, with the two ’s in a relative S-state by a with scaling mass is, however, not a good approximation as shown by Rapp et al. [54]. Even with the loss in binding energy, because of the scaling, saturation does not occur in the correct region of densities, as can be seen from Fig. 2 of the quoted paper. Although crossed channel exchange of a with decreased mass increases the attraction, such a decreased mass increases the (repulsive) part of the Lorentz–Lorenz correction of the pion coupling to NN −1 and ZN −1 bubbles. In addition, the form factor ∗ of the t-channel exchange must be scaled. Also repulsive contact interactions in the scattering needed to preserve chiral invariance are proportional to f−∗2 ; they balance the increase in attraction from t-channel -exchange. When all of the constraints from chiral invariance and BR scaling are enforced, the 2 exchange potential— which at low densities behaves approximately like a scalar meson with BR-scaling mass—gives an e/ective scalar interaction corresponding to a in which the decrease in mass is slowed down. This decrease in the rate of dropping enables saturation at the correct density. From the references in [54] the interested reader will be able to construct the unphysical artifacts that arise from neglecting the chiral constraints in the two-pion exchange interactions and convince oneself that they are necessary on physical as well as on formal grounds.
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G.E. Brown, M. Rho / Physics Reports 396 (2004) 1 – 39
In conventional calculations of nuclear matter, it is well known that the chief agent of saturation is the second-order tensor force (1). The lower intermediate states are progressively cut out by Pauli exclusion principle as the density is increased, slowing down the gain in binding energy. In our scheme with dropping -meson mass (with the pion mass left unchanged), the tensor force decreases rapidly with increasing density. As discussed, this is due to the fact that with the decreasing mass, the cancellation between the tensor and the tensor becomes stronger. Numerically with m∗ (n0 ) ≈ 0:8m , the tensor force at n = n0 is found to be only about half of that at n = 0. Thus, the square of the tensor force is severely cut down and this was the main cause of saturation in Rapp et al. [54] where m∗ ≈ 0:85m was used. 7 One might think that with BR scaling, and the above modi0cations of it, the mechanism of saturation is quite di/erent from that of introduction of three-body forces, as in Pieper et al. [55], who carry out an essentially exact calculation of binding energy of light nuclei in the conventional approach with bare hadron masses (that does not explicitly implement the intrinsic density dependence required by matching to QCD). However this thinking is too hasty. The simplest way to see that this is so is to implement the dropping of the m∗ by attaching a qq V ∗ tadpole (see [97] later in connection with RHIC physics) which incorporates a three-body force. More generally, as explained in [42], in order to assure thermodynamic consistency, one has to consider the IDD of parameters, i.e., masses and coupling constants, introduced as BR scaling in Walecka-type e/ective Lagrangians as a parametric dependence on the ground-state expectation value of a chirally invariant fermion bilinear which in the rest frame can be identi0ed as the density of the system. This means that with the correct implementation of BR scaling, the theory is basically nonlinear, with 2n-fermion interactions (for n ¿ 2) entering in such a way as to preserve chiral invariance, energy-momentum conservation, etc. The six-fermion interactions which represent three-body interactions, for instance, are thus automatically included. As shown in [42], this “linear” Walecka-type theory with BR scaling does provide a correct saturation of nuclear matter. From the foregoing, we see that the dynamics in the “sliding vacuum” substantially changed the binding energy, but in many other respects is only slightly di/erent from that in the perturbative vacuum at low densities. Indeed, now it is clear why Brown and Rho found in their early work the change in tensor force as indicative of dropping masses; there the -meson, which does drop in mass, beats against the -meson, which does not. It is quite remarkable that in much of nuclear structure, it does not seem to matter whether BR scaling is operative or not. For instance, consider one of the most successful predictions in nuclear physics, the Dirac phenomenology. The most complete theoretical paper on this was by Clark et al. [56]. Predictions of this theory as compared with experiment were incredibly successful, the rapid oscillation in the polarization and spin rotation as function of angle being reproduced in great detail. Brown et al. [57] persuaded John Tjon and Steve Wallace to put a linear scaling of meson masses with density into their relativistic impulse approximation (RIA) calculation. We reproduce Figs. 12 and 13 from Brown, Sethi and Hinz as Figs. 5 and 6 here.
7 The compression modulus obtained in [54] was K = 356 MeV, a bit too high. However, the scales of the meson form factors have in the meantime come down substantially, as a consequence of lattice calculations, from those in the DBHF used there. Use of these updated form factors and the slightly more rapid dropping m∗ with density could smooth out the curve of binding energy versus density somewhat, thereby giving a lower compression modulus.
G.E. Brown, M. Rho / Physics Reports 396 (2004) 1 – 39
Fig. 5. Results of Tjon and Wallace (see [57]) for the di/erential cross-section of 500 MeV protons from 40 Ca. The dotted line gives the IA2 results. For the solid curve a linear scaling with density ? ? m? (n0 )=m = m! (n0 )=m! = m (n0 )=m = 0:85. Note that the two are essentially on top of each other. ? ? m and m! scaled in this way but m was held constant at its free-space value. The experimental solid round dots.
19
scattered elastically was assumed, with For the dashed line points are given as
Fig. 6. Results of Tjon and Wallace for the spin observables for 500 MeV protons scattered elastically from 40 Ca. The lines represent the same as in Fig. 5. Note that the dashed line starts deviating strongly from the others beyond about 30◦ .
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G.E. Brown, M. Rho / Physics Reports 396 (2004) 1 – 39
Looking at these 0gures with a magnifying glass, one can see that the 0t to data is not worsened by putting in density-dependent masses. 8 We now outline the calculations that Tjon and Wallace have carried out, scaling meson masses other than the pion as m? M =mM ≈ 1 − 0:15n=n0
(23)
and nucleon mass as m? N =mN ≈ 1 − 0:2n=n0 :
(24)
? This scaling is somewhat milder than what we now advocate m? M =mM ≈ 1 − 0:2n=n0 and mN =mN ≈ 1 − 0:3n=n0 . The Tjon–Wallace calculation, putting in the local density, was incredibly complicated, with 128 components of Dirac wave functions. Tjon ran part of the program, Wallace the rest, and they had to be together to complete it. Such a daunting calculation is highly unlikely to be repeated in the foreseeable future. So we will draw what conclusion we can draw from their results. Before proceeding, we must point out several caveats here in re-interpreting the Tjon–Wallace results in terms of our language. Whereas the IA2 used by Tjon and Wallace to check the scaling masses in Dirac phenomenology, with results given in Brown, Sethi and Hintz did not include the e/ective mass m? N , it did keep positive and negative energy states in a plane wave decomposition, and handled pair theory correctly. As shown in [58,59], in perturbation theory this is equivalent to use of a nucleon e/ective mass in Walecka mean 0eld theory. The m? N obtained in the above way by Tjon and Wallace was ∼ 0:8mN at n = n0 , somewhat less than the 0:7mN that we advocate. Also we now have the scaling meson masses as ∼ 0:8mM at n=n0 . 4 Thus, their m? N dropped ∼ 3 times faster than the scaling meson masses. The linear scaling was, 3 however, consistent with present theory, which as we return to later, has m? N scaling 2 times faster than m? M. From the Tjon–Wallace results, we conclude that Dirac phenomenology is preserved well in all detail with the approximately linear scaling. In other words, their 0ne structure of the polarization variables survives not only BR scaling, but even scaling with e/ective nucleon mass scaling somewhat faster than the meson masses. There is no discernable e/ect of the “breaking” of this scaling by the pion e/ective mass being kept constant. We thus 0nd unfounded those claims that since conventional nuclear theory provides a successful description of the nuclear many-body problem, changes such as BR scaling should not be made in that it would upset the successful predictions. We have shown that much of nuclear structure—with a few notable exceptions which can be understood— are inert to the change of mass parameters.
5. In-medium pion decay constant The fact that most of nuclear structure physics do not “see” the BR scaling does not mean that all nuclear observables are inert to it. We have mentioned that the tensor force in nuclei has the scaling meson beating against the unscaling pion, which should leave a distinct imprint. In this 8
As noted [17], the vector coupling and associated constants are presumed to be nonscaling up to nuclear matter density.
G.E. Brown, M. Rho / Physics Reports 396 (2004) 1 – 39
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section we discuss several observables that probe the scaling f∗ =f , the bona-0de order parameter of chiral symmetry in the hadronic language. In Section 2.1, the notion of parametric density dependence in HLS theory was described. Now when such HLS Lagrangian is applied to nuclear matter, the IDD factor 1 can in turn be related to a Landau parameter which addresses many-body interactions as described in Section 2.2, an indication that the change of QCD vacuum in medium is intricately tied to many-body properties. This demonstrates that it is most likely to be futile to try to separate what one would attribute to QCD from what amounts to many-nucleon dynamics. Consider long wavelength Tuctuations on top of the “vacuum” de0ned by such parameters. As e/ective degrees of freedom, we may pick the pions and the nucleons and integrate out vector mesons and other heavy hadrons including scalars from the HLS Lagrangian. The resulting Lagrangian will take the same chiral symmetric form as in the free space except that the parameters of the Lagrangian intricately depend on density. With this Lagrangian, the power counting will be formally the same as in ChPT based on the Lagrangian whose parameters are de0ned in matter-free space. According to the discussion given in Section 2.3, we have, up to near nuclear matter density, ∗ 1(n) ≡ m∗V =mV ≈ F∗ (n)=F ≈ (qq(n) V =qq) V 1=2 ;
(25)
where F is the bare (parametric) pion decay constant. The (last) relation between the pion decay constant and the quark condensate comes from the Gell–Mann–Oakes–Renner mass formula for the pion in medium. The in-medium quark condensate can be estimated for low density in terms of the pion–nucleon sigma term >N using the low-energy theorem [60,61] ∗ =qq V =1− qq(n) V
>N n + ··· : m2 f2
(26)
Using the presently accepted value for the sigma term >N ≈ 45 MeV, we get ∗ =qq) V 1=2 ≈ (1 − 0:36n=n0 )1=2 ; (qq(n) V
(27)
which gives (qq(n V 0 )∗ =qq) V 1=2 ≈ 0:8 at nuclear matter density. On the other hand, the quantity 1 of (25) can be obtained by relating the vector-meson mass scaling to the Landau parameter F1 and then using the relation between the Landau parameter and the anomalous gyromagnetic ratio of heavy nuclei as well as the properties of nuclear matter [10,43]. One gets 1(n) ≈ (1 + 0:28n=n0 )−1 ;
(28)
where n0 is normal nuclear matter density. At nuclear matter density, this gives 1(n0 ) ≈ 0:78, about the same as (27). Thus, the two relations (27) and (28) agree up to nuclear matter density. In what follows, we shall adopt (28) since it directly reTects BR scaling. Furthermore, the model-independent relation (27) depends sensitively on the value of the N sigma term, the precise value of which is still highly controversial and uncertain. For a recent summary, see [62]. Given (28), one can make a few independent predictions. The 0rst quantity we can look at is Warburton’s jMEC [63] for axial-charge transitions in heavy nuclei. Using the chiral Lagrangian with
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G.E. Brown, M. Rho / Physics Reports 396 (2004) 1 – 39
the parametric dependence de0ned above, one can easily compute this in the leading chiral order, 9 that is, in tree order [64]. The prediction is that [14] jMEC (n = n0 =2) ≈ 1:63 ; jMEC (n = n0 ) ≈ 2:02 :
(29) jexp MEC
= 1:60 ± 0:05(2:01 ± 0:10) for the mass These should be compared with the observed values numbers A = 50(208). The next thing we consider is the recent experiment on deeply bound pionic atoms [65,66]. There is an on-going discussion as to whether this experiment is signalling “partial restoration” of chiral symmetry. There can be a variety of ways to approach this problem. In the framework we are adopting here with the chiral Lagrangian based on HLS/VM, what the experiment provides is an information on the only scale dependent parameter in the theory, namely, the ratio F∗ (n)=F at n . n0 . In the tree order, this ratio is given by (28) which at nuclear matter density is (F∗ (n0 )=F )2 = 0:61
(30)
with a theoretical uncertainty of ∼10% inherent in nuclear ?gl . This agrees with the value extracted —in the tree order—from the pionic atom data [66] (F∗ (n0 )=F )2 = 0:65 ± 0:05 :
(31)
One can understand this result obtained in the leading order ChPT with HLS/VM in a way more familiar to nuclear physicists, as follows based on the works by Friedman [67] and Weise and coworkers [68,69]. For instance, Friedman invokes density dependence in two places. One
? is to incorporate the ratio [69] f =f ≈ qq V ? =qq V ≈ 0:78 at n = n0 which enters into the Weinberg–Tomozawa isovector term in the pion–nucleon scattering amplitude. 10 This scaling can be identi0ed [17] as the “intrinsic” density dependence required by matching to QCD. The other medium modi0cation Friedman needs goes back to the relativistic impulse approximation used by Birbrair et al. [71] which gives small components of the Dirac wave functions for the nucleon enhanced by the factor F = mN =M (r) ;
(32)
where M (r) = mN + 12 [S(r) − V (r)] with S(r) and V (r) the vector mean 0elds. In fact, if we write M (r) = mN + S − 12 (S + V )
(33)
9 According to the chiral 0lter, the soft-pion term should dominate with next-order corrections suppressed for the axial-charge transitions. 10 This actually goes back to the work of Lutz et al. [70] for the value of f? obtained from the Gell–Mann–Oakes –Renner relation assumed to hold in matter with the pion mass unscaled by density. At low density, we expect this assumption to be valid. At present, the theoretical evidence for an unscaled pion mass comes (1) from two-color (SU (2)c ) QCD on lattice [45] although the error bars are a bit too big to con0rm the constancy and of course the two-color QCD may not reTect the real QCD and (2) from a skyrmion description [24]. Friedman invokes the in-medium pion decay constant expressed in terms of the pion–nucleon sigma term. As noted above, the numerical value turns out to be equivalent to 1(n0 ) used in this paper. The point we want to stress is that this quantity is the one that 0gures in BR scaling.
G.E. Brown, M. Rho / Physics Reports 396 (2004) 1 – 39
23
and note that up to nuclear matter density (S + V )=2 turns out to be only about 10% of the S in magnitude, 11 we see that M (r) m? (r), the nucleon e/ective (Landau) mass. ? In BR scaling [3], the nucleon e/ective mass scales more rapidly than m? or f because of pionic e/ects in the solitonic background going as gA? =gA . In the original formulation in terms of skyrmions [3], this factor entered when the Skyrme quartic term was taken into account for gA . Now the Skyrme quartic term is known to contain a lot more than what naively appears to result when heavy degrees of freedom, e.g., all heavy excitations of the vector-meson quantum numbers, are integrated out. It seems to represent physics ranging from part of pionic e/ects to extreme short-distance e/ects (e.g., proton decay [73]). (For discussions on some of these matters, see [34].) On the other hand, one can also calculate its e/ect from matching of chiral Lagrangian theory with sliding vacuum to Landau Fermi liquid theory as discussed in [10,42,43]. There this factor arose as a pionic contribution correlated with heavy modes in the vector-meson channel, −2 where F˜ () is the pionic contribution to the given by the formula gA? =gA = (1 − 13 F˜ 1 ()(m? 1 V =mV )) F1 Landau–Migdal parameter which has the value 13 F˜ 1 () = −0:153 at nuclear matter density. With ? ? 1(n0 ) = m? V (n0 )=mV ≈ 0:78, one obtains gA =gA ≈ 0:80 or gA ≈ 1:0. It must be mentioned here that how these descriptions are related is not understood yet and remains an open theoretical issue. Now in 0nite nuclei the gA? is found almost universally to be unity [74]. Thus, 1=2 ? m 1 ? mN =mN ≈ ≈ 0:7 at n = n0 : (34) gA m We suggest that this is equivalent to the nucleon mass scaling needed by Friedman. In our formulation, the phenomenological approach of Friedman corresponds to the leading-order treatment of our e/ective theory with the VM [7,14] for which the only scaling parameter is the pion decay constant with the pion mass unscaled. 12 Note that we arrived at (30) without making direct reference to the quark condensate. It was based on the “many-body” relation presumably valid up to and near nuclear matter density and not beyond the Fermi-liquid 0xed point. Relying on (25), one might naively conclude that the result signals a reduction of the quark condensate with density and hence might be taken as the signal that chiral symmetry is being restored. This inference is valid however only in the tree approximation used here and does not apply when one works at higher (loop) orders. The bare pion decay constant 0xed at the matching scale does not necessarily follow the quark condensate near chiral restoration as density is increased unless one takes into account the parametric dependence that follows the RG Tow. This is so even though the physical pion decay constant vanishes at the chiral transition in the chiral limit. With the e/ective Lagrangian that most of the people in the 0eld use, namely that in which the parametric dependence is absent, the behavior of the ratio as a function of density has little to do with chiral restoration, a point which seems to be often overlooked. 11
This observation is the basis of the suggestion made in [72] that Tuctuations in nuclear matter be computed around the “shifted vacuum” at which (S + V )=2 is exactly equal to zero. This “vacuum” at this point is close to the Fermi-liquid 0xed point. Note that this is analogous to the notion of Tuctuating around BR’s “sliding vacuum” [10,24,42,43]. 12 In HLS/VM with the explicit vector degrees of freedom, the scaling parameters are the gauge coupling g∗ , the vector-meson mass m∗V , the pion decay constant F∗ and the coeQcient a. However in medium a∗ ≈ 1, g∗ ≈ g, so m∗V goes as F∗ through HLS/VM relations. We are left with only one scaling factor as in the case where the vectors are integrated out.
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6. The VM and the photon coupling to the nucleon A highly signi0cant prediction of HLS/VM theory is that the photon couples to hadrons in medium quite di/erently from that in free-space. It has been shown by Harada and Sasaki [75] that the vector dominance in the photon coupling is strongly violated near the VM 0xed point and hence near the chiral transition point. While experiments at Je/erson Lab probe densities near that of nuclear matter and hence rather far from the VM 0xed point, the presence of nucleon in the medium is expected to drive the parameter a in HLS Lagrangian toward 1 from its value a = 2 at the vector-dominance regime. 13 This would have a strong rami0cation on properties of in-medium nucleon form factors. In this section, we brieTy discuss what can be expected. In HLS, the photon couples to the degrees of freedom involved in the theory as a A tr[J Q] ; ?L = −2eagF2 A tr[ Q] + 2ie 1 − (35) 2 where A is the photon 0eld, Q is the electric charge matrix Q = 13 diag(2 − 1 − 1) and J is the vector current made up of the chiral 0eld .. In HLS theory, baryons could be thought of arising as skyrmions in which case J can be identi0ed as the skyrmion current. Alternatively one could introduce “bare” baryon or quasiquark 0eld into the HLS scheme in which case, one can think of J as a “bare” baryon current. The usual vector dominance (VD) picture corresponds to taking a = 2 in which case there will be no direct coupling of the photon to the baryon just as there is none in the case of the pion (for which the second term of (35) would be of the form J = [9 ; ]). It turns out [77] that the VD at a = 2 is on an unstable trajectory of RG Tow of HLS theory with no connection to the trajectory that leads to the VM and that the fact that in nature the VDM seems to work in matter-free space and at NF = 3 is merely an “accident.” In fact, in the presence of matter (temperature or density), the Tow consistent with QCD is on the trajectory that leads to the VM 0xed point a = 1. Even in the absence of matter, a ≈ 1 seems close to nature [78] 14 although the VDM that works for the EM pionic form factor corresponds closer to a = 2. As is well known, the vector dominance picture is not a good one for the photon coupling to the nucleon, so a = 2 must hold poorly when nucleons are involved. Indeed, there is an indication that the photon coupling to a single nucleon is already near this 0xed point. Historically, a picture closely resembling this one was adopted by Iachello et al. [80] in 1973. There the authors assume that the -ray couples to nucleon more or less equally (at low momentum transfer) through the vector meson and directly to a compact core. In a recent reanalysis, Iachello [81] 0nds that the two-component description of [80] with a core size of ∼ 0:34 fm is in good agreement with the recent JLab data on the nucleon EM structure. The presence of the small core 15 in the proton of 0.2–0:3 fm is indicated also in the proton structure function in (deep) inelastic scattering [83]. It is intriguing that this two-component structure with a small core size is precisely the 50/50 picture that arises at the “magic angle” in the chiral bag model [34,76] used by Soyeur et al. [84,85] in analyzing nuclear form factors and that it corresponds to HLS/VM with the a = 1 0xed point. 13 The deviation from vector dominance in the photon coupling to the baryons has been known since some time. In fact, the chiral bag model [34,76] provided a natural mechanism for such a deviation. 14 Another remarkable evidence for a ≈ 1 in free space is found in the prediction of HLS/VM theory for the “chiral doubling” of heavy–light mesons observed in the recent BaBar/CLEO II discoveries [79]. 15 We recall that the small core size was the motivation of introducing the notion of “little bag” in 1979 [82].
G.E. Brown, M. Rho / Physics Reports 396 (2004) 1 – 39
25
The above consideration leads us to the prediction that in medium, the electromagnetic form factor will take the form for a = 1, e 1 e + H (q2 ) ; ∗ 2 2 2 1 − q =mV 2
(36)
where H is a slowly varying function of q2 (with q being the four-momentum transfer) with H (0)=1 and m∗V is the parametric mass that enters at 0nite density. The photon point q2 = 0 gives the correct charge. The -ray will couple to the dileptons in this half-way manner. Note that the dileptons discussed below will experience the same propagator suppression, namely, the correction factor F≈
1 + 1=(1 + Q2 =mV∗2 ) 1 + 1=(1 + Q2 =m2V )
(37)
with Q ≡ |˜q| but Q is generally small, the dileptons being nearly back to back, so this is probably unimportant. Thus, though vector dominance is violated (expected in the nucleon sector from other considerations than that of the VM), it is an adequate approximation in the dilepton calculation as we will discuss below.
7. “Sobar” con gurations and CERES dileptons There have been much discussions on the possible evidence for changes in hadron properties in the CERES dilepton data [86]. While the processes discussed above involve transition matrix elements with speci0c kinematics, the dilepton experiments measure spectral functions or more generally correlation functions averaged over density and temperature. For this purpose, we need to look at the spectral distribution in the hot and/or dense environment. In terms of the framework based on HLS/VM theory we are adopting in this paper, this means that we need to incorporate consistently into vector–vector correlation functions both quantum Tuctuations with the parametric masses and coupling constants and thermal loop and/or dense loop e/ects generated in the RG Tow. Now as pointed out in [87,88], this means, for the dilepton processes, considering both BR scaling 0guring in density-dependent parameters that results from the CSD and the mixing to the “sobar” con0guration N ∗ N −1 computed as “Tuctuations” on top of the soliton con0guration. This requires that the double-decimation be consistently implemented. We shall refer to this procedure as “BR/RW fusion.” In the Rapp–Wambach approach (abbreviated as R/W) [86], the fusion has not been implemented: There, the second decimation is replaced by con0guration mixing in lowest-order perturbation theory while the 0rst decimation CSD is left out. In several low-energy phenomena, the R/W -sobar provides most of the low-lying “” strength as we shall outline, B/R coming in as the intrinisc e/ect to somewhat increase the e/ect. This is because the -sobar has ∼ 20% of the -strength, at a low energy of 580 MeV. BR scaling can only move the remaining 80% at the parametric -mass to this energy at densities ¿n0 (nuclear matter density), and such high densities have not been investigated experimentally. The -sobar pushes the elementary up in energy, the states repelling each other; thus, in the region of the elementary the -sobar and the “defuse.” We shall see in Sections 7.1 and 8 that there is good empirical evidence for this, establishing that both R/W and B/R are present as required.
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G.E. Brown, M. Rho / Physics Reports 396 (2004) 1 – 39
7.1. The role of the “-sobar” in the 3 He (; 0 )ppn reaction Before we treat the CERES dilepton production which will require the fusion of the “-sobar” (or R/W) and BR scaling (i.e., B/R), we consider the process where the -sobar plays the primary role. Two recent papers by Lolos et al. [89,91] have discovered the fact that the [N ∗ (1520)N −1 ] excitation decays ∼ 20% of the time into a -meson, in agreement with the 15–30% listed in the − Particle Data Book and the ∼ 20% found by Langg[artner et al. [90]. We call the [N ∗ (1520)N −1 ]1 ; I =1 the “-sobar” because when measured in 0nite nuclei, rather than when produced on a single proton by, e.g., + p interactions, it takes on a collective character with increased -meson content due to the admixture of the elementary with density [87]. A clear evidence has been found that the production in the deuteron is dominated by the N ∗ (1520), an element of the -sobar. Of course, the nucleon density of the deuteron is so low that one could hardly expect any appreciable medium dependence from this nucleus. Greater production should be seen in 3 He, although the present experimental accuracy does not seem to be suQcient to show this. Here we give theoretical estimates for how much greater it should be with R/W alone and with B/R fused with R/W. These estimates can be easily extended to heavier nuclei. From the 3 He wave functions of Papandreou et al. [92], we easily see that the average density in 3 He is half that in nuclear matter. Although these authors could explain the experimental results with BR scaling alone, with our analysis we 0nd that they had to decrease the mass twice too much by their mean 0eld. As we shall develop, R/W which they also considered gives most of the experimental e/ects. From [87]—that we shall refer to as BLRRW—we see that the mixture of the elementary into the -sobar, which goes as the square of the matrix element divided by the energy di/erence increases linearly with the density n. This is because the -sobar is a collective state, a linear combination of the excitations of all nucleons in the nucleus up into the N ∗ (1520). The amount of admixed into the -sobar is the same as the amount of -sobar admixed into the . This is displayed, as function of density, in Fig. 1 of Kim et al. [88] in which it is shown that the sobar con0gurations can conveniently be incorporated into a massive Yang–Mills theory. For n = n0 =2, the Z factor comes out to be Z = 0:15 for R/W; fusing with B/R increases it to 0.23. 16 This gives the increased content of the in-medium -sobar, which then increases from 20% found by Langg[artner et al. (15–25% in the Particle Data Booklet) to 40% with the fused R/W and B/R. The recent unpublished calculation on the Regina 3 He(; 0 )ppn reaction by Rapp, reproduced here in Fig. 7 show that compared with experiment [93], the R/W results are spread widely about the unperturbed zero-density -sobar energy of 580 MeV, somewhat more upwards than downwards, by the large imaginary part which we estimate to be & 200, 150 MeV zero-density width of the plus some in-medium width. The -sobar width increases with energy spreading the strength more upwards. On the whole the R/W 0t is good, except that too much strength is predicted in the region up to the elementary at 770MeV. This will be improved by the fusing with B/R, the latter lowering the parametric mass in the Lagrangian by 77 MeV. The de0nite need for the fusing of 16 However, with the fusing of R/W and B/R, the vector dominance coupling of the should be decreased from m2 =gv to m2∗ =gv as gv —related to the hidden-gauge coupling—does not scale up to nuclear matter density and the square of this coupling enters into the cross-section. The latter turns out to be unchanged from the R/W result. Thus, the fusing of B/R with R/W does not change the low-energy result.
G.E. Brown, M. Rho / Physics Reports 396 (2004) 1 – 39
27
Fig. 7. R/W applied to the Regina 3 He(; 0 )ppn reaction. B/R would increase ∼ 50% the lower points where it fuses with R/W and would bring the elementary -meson mass down ∼ 10%, 770 MeV, at the upper side. The R/W calculations were kindly furnished to us by Ralf Rapp.
R/W and B/R is seen, therefore, most clearly in the region of the elementary where the push upwards by R/W must be compensated for by the downward shift from B/R. Here R/W and B/R “defuse” for the elementary . We return to this point in the next section on RHIC physics. Asked by the experimentalists, Rapp extended his calculations to higher densities than present in 3 He and 12 C in order to increase the medium e/ect. As noted, an ∼ 50% increase in R/W in the -sobar region is e/ected by the fusion with B/R, which we believe to explain the greater collectivity apparently seen. While the fusion of B/R with R/W does give a good description of the observed phenomena, the accuracy in measurements of the Langg[artner et al. and Regina–Tokyo groups is not great enough yet to show the necessary increase in 0 production cross-section per nucleon in going from proton to 3 He targets from medium e/ect. In Huber et al. [92], the cross-section for photo-production of the of 10:4 ± 2:5 b on the three 3 He nucleons is found for the 620–700 MeV range of photon energy (m∗ ∼ 500 MeV) whereas Langg[artner et al. 0nd ∼ 3:5 b=nucleon for 0 production in satisfactory agreement. In the 3 He (; 0 )ppn the low-mass strength is mostly explained by R/W, the B/R coming in to slightly enhance the low-mass strength, but chieTy to lower the high-mass strength, which in R/W alone fails to explain experiment. Thus, the higher-energy region in which R/W and B/R defuse shows de0nitively that both e/ects are present (see the next section). 7.2. The CERES: fusion of B/R and R/W We now go on to the fusion of the two e/ects described above in the dilepton production at CERN by the CERES collaboration in which R/W and B/R play about equal roles, the present error bar and binning in the experiment being too large to pin down their separate roles. Ralf Rapp (private communication) has carried out calculations fusing B/R and R/W, using vector dominance. The results are shown in Fig. 8. Although it has been shown in HLS/VM [75] that
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G.E. Brown, M. Rho / Physics Reports 396 (2004) 1 – 39
Fig. 8. Comparison of various calculations with the CERES dileptons. The 0gure was kindly furnished to us by Ralf Rapp.
vector dominance is violated at high temperature, as noted above, this should not a/ect the numerical results. From the Rapp results, one can say with some con0dence that the 0t is improved with the fusion of B/R and R/W. However, due to the error bars, one cannot say 0rmly that the fusion is required. As it stands, both R/W and B/R 0t the data just as well within the error bars. It is clearly diQcult to di/erentiate their separate roles by experiment in the low-energy regime. However, we shall see from the next section that this separation is straightforward in the high-energy region where they defuse. 8. RHIC resonances It should be clear in our discussion of the dileptons given above, also -mesons produced by -rays on 3 He and other nuclei, that it is intricate to separate the role of BR scaling from that of Rapp–Wambach. It is therefore signi0cant and exciting that a direct measurement of the -meson mass 17 m∗ could be made in a pristine atmosphere where temperature e/ects are small and where the density can be well reconstructed. We show in Fig. 9 the STAR/RHIC results [94]. The (preliminary) 0ts gave m = 0:698 ± 0:013 GeV for Au–Au ; = 0:729 ± 0:006 GeV for pp : (38) √ The collisions were measured at s = 200 GeV. There is a long history of the detailed spectral shape. The di/erence between its appearance in elementary reactions, e+ e− → + − or ! → ! + − 17
The “mass” involved here is the pole mass. However, the di/erence between the parametric and pole is of higher order in the power counting and can be taken to be negligible for the discussion.
G.E. Brown, M. Rho / Physics Reports 396 (2004) 1 – 39
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x102 18000 16000 14000 12000 10000 8000 6000 4000 2000 0
STAR Preliminary pp
Sum
1400
0 KS
1200
ω
1000
*0
STAR Preliminary Au+Au 40%-80%
Sum 0
KS ω *0
800
K
ρ0
600
ρ0
f0
400
f0
K
200 0 0.6
0.8
Invariant Mass
1 (GeV/c2)
1.2
0.6
0.8
Invariant Mass
1
1.2
(GeV/c2)
Fig. 9. The invariant mass distributions in pp and mid-central AuAu of the + − system, with a transverse momentum cut 0:2 ¡ pt ¡ 0:9 GeV, from Fachini [94]. Contributions from speci0c resonances are indicated by di/erent lines.
and hadroproduction reaction is well known. The latest Review of Particle Physics [95] averages the mass for these two sets of experiments separately, with a clear systematic di/erence of the order of 10 MeV between them: mleptoproduction = 775:9 ± 0:5 MeV ; = 766:9 ± 0:5 MeV : mhadroproduction
(39)
It was noticed back in the 1960s by Hagedorn and others that the particle composition in pp can be well reproduced by statistical models, see, e.g., Ref. [96]. The 0tted temperature is about the same as the chemical freezeout temperature of ∼ 165 MeV found at RHIC. In some sense the -meson must be born in a heat bath, with Boltzmann factors which cut o/ the high-energy end of the rather broad . Some discussion of this can be found in [97,98]. We expect that many papers will be written on this subject, which is not within the scope of this paper. We can only assume that perhaps half or less [98] of the drop in the -mass found in the Au–Au experiments comes from the heat-bath-related e/ects, and that these are about the same as in the pp experiment. Next, we discuss shifts in the -mass from forward scattering such as + → a1 → + . There are many of these which enter as principal values as discussed in [99,100] as well as in [97]. It seems diQcult to get more than a few MeV out of these, the sign probably attractive. We note that a substantial part of the upward push in these resonance calculations comes from the -sobar (N ∗ (1520)N −1 ) which gives the Rapp–Wambach e/ects. Thus, the RW and BR “defuse” for the elementary . In other words, the BR scaling must substantially overcome the Rapp–Wambach con0guration mixing in order to produce a substantial downward shift. In fact, the BR scaling shift will be, from Section 2.3 (see [17]), m∗ m (1 + 0:28n=n0 )−1 ;
(40)
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G.E. Brown, M. Rho / Physics Reports 396 (2004) 1 – 39 π
π ρ
π
π
Fig. 10. The product of probability of formation and decay into pions that can get out without being scattered must be optimal.
where n is the total baryon density in nonstrange particles. 18 This comes about because for low densities m∗ scales as f∗ while the quark mass and the pion mass do not scale. As a 0nal approximation in determining the density at which the -meson decays we might consider thermal (kinetic) freezeout, since the two ’s must come unscattered to the detector. The kinetic freezeout at RHIC can be obtained from a hydro-based 0t to the data 19 Tk ≈ 100 MeV; ≈ 81 MeV; N ≈ 380 MeV; K ≈ 167 MeV ; which translate into the pion density n ≈ 0:06 fm
−3
(41)
and
nN +NV ≈ 0:0075 fm−3
(42)
1 of nuclear matter density. or only 20 To obtain a better approximation we must optimize the product of the resonance formation and its decay, as shown in Fig. 10. Ignoring the lifetime of the in comparison with the other times— which underestimates the density at which the is formed since all the ’s will follow the 0reball expansion—the parameters for resonance formation and decay are found by Shuryak and Brown [97] to be
Tk ≈ 120 MeV; ≈ 62 MeV; N ≈ 270 MeV; K ≈ 115 MeV
(43)
with densities about 1.4 times higher than at kinetic freezeout. Taking into account that the density of excited baryons is double that of N + NV , and the above conditions, we 0nd the total density in nonstrange baryons to be 0:21n0 . We note that in BR scaling, basically the total density in nonstrange quarks and antiquarks should be used, whereas in RW con0guration mixing the assumption of these authors is that the resonances with quantum numbers built upon excited baryons will be only ∼ 50% as eQcient in con0guration mixing as those built upon the nucleon. Thus, at a total baryon density with 23 made up of excited 18
In [97], the mass shift was associated, in the language of linear sigma model, with the “amplitude-0eld” Tuctuation of the chiral 0eld—or radius 0eld R—not with the “phase-0eld” or pionic Tuctuation. It is perhaps worthwhile to point out that this way of looking at the mass shift is totally equivalent to the nonlinear sigma model approach of Brown and Rho [3] where the “dilaton” 0eld of trace anomaly plays the role of the amplitude 0eld in [97]. This point has been further clari0ed in [24]. In fact, it is shown that the approach of [3] is closer to the modern treatment based on HLS/VM theory of Harada and Yamawaki. In the language of HLS/VM, it is the intrinsic parametric dependence on density resulting from matching to QCD that plays an essential role in the mass scaling. The pionic Tuctuation in the theory is subject, of course, to low-energy theorems of chiral symmetry. 19 P. Kolb, private communication.
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baryons, they would add the 13 in N + NV . In this way, BR becomes somewhat larger than (and of opposite sign to) RW under RHIC condition. At the resulting density n ≈ 0:21n0 , and with the -mass dropping ∼ 22% at n0 , we 0nd a decrease of ∼ 43 MeV in m∗ . However, the -meson themselves—as well as the !’s, to a lesser extent—are also a source of scalar density, i.e., they are composed of constituent quarks. 20 Considering the ’s, we note that equilibrium through ↔ 2 continues basically down to thermal freezeout, and with ≈ 62 MeV, the pion fugacity will increase the number of ’s by e =T and the number of ’s by the square of this factor. With the greater multiplicity, but smaller Boltzmann factor than the ’s, we 0nd n =n ∼ 0:15, with n ∼ 0:08 fm−3 , or an n ∼ 0:013 fm−3 which considering that the presents 23 of the scalar source as a nucleon, gives the additional ∼ 10 MeV to the downward shift of the -meson. Thus, our total shift from BR scaling is ∼ 53 MeV, roughly 80% of the total observed downward shift. 21 This seems to account for most of the shift left over when the temperature-driven shift is subtracted away. The 150 MeV width of the is unchanged. Since the decay is p-wave, the width should decrease with the cube of the momenta of the pions into which the decays. This would cause an ∼ 30% decrease in width. The fact that the width does not change means that compensating increases must be furnished by the resonances, both those above and below the in mass contributing with equal sign to the width. From the upper part of Fig. 1 of Rapp and Gale [100], one can see that their resonances contribute ∼ 50 MeV to the total width for mass M = 700 MeV at T = 150 MeV whereas in [99], the increase is somewhat larger. In any case, the STAR results give a nice check that the increase in width of ∼ 50 MeV (collision broadening) is cancelled by the decrease in width from lower penetrability. The RHIC work has the great advantage over the dileptons that the energy resolution separates o/ the upper region in which BR and RW defuse, showing that BR must 0rst cancel RW and then add some net downward shift to the -mass, the general size predicted by BR scaling 0tting in nicely with the experimental results. 9. Eective forces in nuclei The Vlow-k (r) has supplanted the Kuo–Brown G-matrix [11] as e/ective interaction to be used in nuclear structure calculations. Integrating-out of the high-frequency parts of the two-body interaction supplants all of the paraphernalia of o/-shell particle energies, etc. [101]. Here we discuss how the CSD may change some of the results of the FLD. The most extensive, but still only partial, investigation of the problem we discuss here was carried out by Hosaka and Toki [102] for the 2s, 1d-shell matrix elements (with 16 O as closed shell). In this region of nuclei, up to 40 Ca, empirical two-body matrix elements which 0t well the experimental spectra have been determined. The most de0nite and important result of Hosaka and Toki is that 20
Since pion is not a source of scalar density, although composed of constituent quarks, cancellations must insure that its mass is not changed; these in turn insure that the pions are removed as source of scalar density. 21 We should note that the two scalings (27) and (28) which coincide for n = 0 and n = n0 give results that di/er by ∼ 13 MeV in the shift at n ≈ 0:21n0 relevant to the process in question. The shift given here is somewhat larger than that quoted by Shuryak and Brown [97]. This di/erence cannot be taken seriously as we really do not know precisely how the scaling interpolates between n = 0 and n0 .
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“the central G-matrix elements are already well reproduced by using the free-space parameters and that they are extremely sensitive to the masses of and ! mesons. This indeed provides a strong con? straint from the phenomenological side on the way the meson masses, particularly m? and m! should ? scale in medium. In fact, it turned out that they have to be correlated such that m =m ∼ m? ! =m! as long as the coupling constants are kept unchanged.” Note that the latter conclusion is consistent with what we have found in [17] and with the type-II analysis of nuclear matter by Song [42]. As for the former, it can be explained by the argument presented in the Introduction, namely the e/ect of the strong -meson tensor coupling which more than cancels the strong pionic tensor coupling at short distances and keeps the nucleon–nucleon potential small in magnitude (except for the strong short-range repulsion which is counterbalanced by the short-range attraction and integrated out to give Vlow-k (r)). Thus o/-shell e/ects are small. “Encouraged by this fact” (i.e., that with BR scaling the central G-matrix elements were well reproduced), Hosaka and Toki “have calculated the G-matrix elements using the various meson–nucleon masses scaled nearly the same way while keeping the pion strength unchanged. This time our interest is concerned with the LS and tensor matrix elements, since they are generally in poor agreement with nature if the free-space masses are employed.” By using the scaled masses, however, they “could not see signi0cant improvements in the comparison of the calculated and empirical matrix elements.” They 0nd that generally the LS matrix elements are enhanced by the increase of the , ! and contributions and the tensor matrix elements are suppressed by the increase in the repulsive components due to exchange. The little improvement from the dropping masses in the spin–orbit channel is disappointing, given the excellent agreement found by Tjon and Wallace with or without dropping masses in Figs. 5 and 6. However, the rapid oscillations characteristic of the Dirac phenomenology, as compared with the nonrelativistic equations, come about from the interference in contributions from di/erent densities in the nucleus. Such an e/ect would be missing in taking the scaled masses to be constant, independent of local density, as Hosaka and Toki did. It was actually the decrease in the in-medium tensor interaction found in the 1989 and 1990 B–R papers which provided an empirical guide for the dropping m? . The strong spin–orbit interaction, which is the basis for the shell model, was obtained in the Walecka-type relativistic mean 0eld calculations, a factor ∼ 2 at n = n0 greater than found in nonrelativistic Brueckner calculations. This spin–orbit interaction is very important in the single-particle energies, giving the splitting between the spin–orbit partners in the shell model. The Walecka theory gets suQcient spin–orbit splitting by using nucleon e/ective mass m? N ∼ 0:6mN , somewhat smaller ? than what we 0nd, m? ≈ 0:7m . We actually get m this low by assuming gA? ≈ 1 in medium. N N N Since one does not expect that the Gamow–Teller coupling constant (globally) goes below 1, this is the lowest we can expect. The spin–orbit interactions due to scalar and vector exchanges that Hosaka and Toki took are of the form
mS 2 Z1 (mS r) ; mN 3 2fV mV 2 gS2 LS VV = − mV + Z1 (mV r) ; 4 2 gV mN
VSLS
g 2 mS =− S 4 2
(44)
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where Z1 (x) = (1 + 1=x)
e−x : x
(45)
Note that the volume integrals of VS;LSV are independent of mS and mV , respectively, the masses going into rescaling the r 2 dr factor in the integral. 22 In mean 0eld, energies, etc. depend mainly on the 2 volume integrals. However, an additional m− N is present in the spin–orbit term (also in the tensor ? interaction). Taking mN → mN would strongly enhance both of these interactions, say, by a factor of ∼ 2 at n = n0 . Of course, the average density in the s; d-shell lies below n0 . As noted, the need for nucleon e/ective masses was clear in Walecka theory and these e/ective ? ? masses 0xed up the spin–orbit interaction. If m? N scaled like mS or mV , the scale invariance noted in Section 4 would still hold, being violated only by the constancy of the pion mass. But because of the “loop correction” m? gA? f? N ≈ (46) mN gA f ? in nuclei, the m? and since gA? tends to unity precociously N scales more rapidly than f just up to nuclear matter density. So the gA? =gA factor breaks the scale invariance, but judging from the Tjon–Wallace results, not in a serious way. We therefore conclude that the tensor force is the only component in nuclear force to be substantially changed by BR scaling, bearing out the 1990 work [2]. Along the same lines and even earlier [105] evidence had been found for an in-medium with a factor of 1.5 times that in the free G-matrix [106]. This came from a study of 447 sd-shell binding energy data. Assuming the strength of the -exchange potential be proportional to strength times (range)2 , this would correspond to m∗ ∼ 0:82m if the strength is kept constant and the range changed. In 2s, 1d-shell nuclei the average density is well below n0 , so one would not expect the m∗ to drop quite so much. In any case, Brown et al. [105] found this change to improve agreement with data. It appears diQcult to 0nd “smoking guns” in the structure of nuclei for 20% changes in hadron masses in going from n = 0 to n0 implied by BR scaling. In the case of !- and -exchange we saw from the Hosaka–Toki work that the e/ects cancel each other. However, the meson has no low-mass isoscalar partner, and one might look elsewhere than in the tensor interaction for e/ects, especially in the symmetry energy, which is so important in astrophysical applications. In mean 0eld theory the e/ect of the vector-coupled on the symmetry energy at nuclear matter density would be an increase by (m =m∗ )2 ≈ (0:8)−2 = 1:56. However, modi0cations in the -mass also a/ect the second-order tensor interaction, which, as we remarked in the Introduction, gives the main di/erence between 3 S0 and 1 S0 states. It was shown by Kuo and Brown [107] that the closure works well to approximate the second-order tensor interaction, i.e.,
VT 22
Q V2 VT ≈ T E Ee/
This is also true for the central potentials.
(47)
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with Ee/ ≈ 250 MeV. 23 Now VT2 1 2 (3 − !1 · !2 )(6 + 21 · 2 − 2S12 )VT(+) ≈− (r) ; Ee/ 250 MeV
(48)
where the 0nal factor is the square of the radial part of the tensor term. In getting (48), we have used the identity ·A ·B=A·B+i ·[A×B] and omitted the term linear in which will vanish when averaged over angle. We see that in the triplet state the second-order tensor gives the contribution −
24 V2 (r) ; 250 MeV T(+)
(49)
which explains why the triplet state is bound, and the singlet not. However, this is also the contribution to the symmetry energy ?Vsymm =
6!1 · !2 V2 (r) ; 250 MeV T(+)
(50)
which comes to ∼ 14 of the central contribution. Relativistic Brueckner–Hartree–Fock calculations with the strong -coupling of the Bonn potential [108] show the -contribution to ?Vsymm just cancels that from the vector coupling of the , leaving the pion exchange (mostly in second order) on the source of symmetry energy. Thus, we seem to be foiled, again, in nuclear structure physics a strong “smoking gun” for the BR scaling (or parametric dependence on density) of the mass. The conclusion is that in all nuclear structure observables so far probed, the parametric density dependence symptomatic of BR scaling is masked in an intricate way. This strongly suggests a “hidden symmetry” which seems to induce almost exact cancellation of possible smoking-gun signals, a phenomenon that is very much reminiscent of the Cheshire Cat phenomenon expounded in [34].
10. Conclusion: return of the Cheshire Cat The main conclusion we have arrived at in this paper is that up to date, no “smoking gun” signals have been observed in nuclear structure physics for BR scaling indicative of the spontaneous breaking or restoration of chiral symmetry, a basic element of QCD. This outcome, although somewhat disappointing and perhaps unappealing, is however very much in line with what we have been arguing since some years, namely, that low-energy strong interactions are strongly governed by the (approximate) Cheshire Cat Principle (CCP). Here we would like to enumerate a few cases where the CCP can be applicable. On the most fundamental level is the structure of the nucleon. As noted as early as in 1984 [110], the bag radius in the bag model of the nucleon, naively interpreted as the con0nement size, is a gauge artifact. Here the bag boundary provides the space-time point at which QCD and hadronic variables are matched. The CCP statement is that the physics should not depend on where the matching is made. How this comes about through the combination of quantum anomalies and topology 23
Note that Ee/ ¿ 175 MeV, the c.m energy corresponding p = 2:1 fm which is the cuto/ that 0gures for Vlow-k . In EFT, this means that the second-order tensor e/ect should reside predominantly in the local counter term.
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of skyrmions is detailed in [24]. What the Cheshire Cat phenomenon implies is that there is a continuous map between QCD degrees of freedom and hadronic degrees of freedom for low-energy hadronic properties. One can think of the HLS/VM aM la Harada and Yamawaki highlighted in this paper as a CCP in momentum space with an additional remarkable feature that is new, namely the presence and importance of the VM 0xed point. The VM may very well be present also in the chiral bag/skyrmion picture as conjectured in [24]. The important point regarding the VM 0xed point is that while physical processes away from the phase transition critical point—a few of which were considered above—may be given by “fusing” or “defusing” or by some combination of the two of the sobar and elementary-particle modes, thereby obstructing the clear evidencing of the “smoking gun,” the VM governs the scaling behavior of the physical parameters that enter the processes, such as, e.g., sending the vector-meson mass to zero in the chiral limit. How the chiral symmetry restoration and nuclear interactions manifest themselves in physical processes is therefore irrelevant. A case that is relevant at the next level of fundamental nature is the connection between the BR scaling 1 reTecting the property of QCD vacuum and the Landau parameter F1 as discussed in [9,42,43]. This illustrates a possible mapping between the complex vacuum structure of QCD characterized by the quark condensate in medium and many-body nuclear interactions embodied in the Landau parameters. If this identi0cation is correct, then it will suggest an inherent ambiguity in delineating QCD e/ects from many-body hadronic e/ects conventionally treated in the standard nuclear physics approach (SNPA) discussed in Section 3. Although not worked out in detail, we expect the issue of Gamow–Teller strengths in nuclei to be in a way quite similar to the relation between the BR scaling 1 and the Landau parameter F1 . It has been debated [109] since many years as to whether the “quenching” of Gamow–Teller strengths in nuclei as observed in giant Gamow–Teller resonances is due to “conventional” multi-particle– multi-hole e/ects (e.g., core polarization [111]) or to “exotic” e/ects (e.g., E-hole excitations [112]). The resolution to this debate [113] is that both are relevant in a way analogous to what happens in the Dirac phenomenology discussed in Section 4. As noted in Section 9, the tensor force predominantly excites (particle–hole) states at an energy ∼ 300 MeV which is comparable to the E-hole excitation energy. Because of subtle cancellations between various terms of the same scale involving multi-particle–multi-hole con0gurations and E-hole con0gurations, the e/ective Gamow–Teller coupling constant gA∗ for the transition to the lowest Gamow–Teller state can be calculated equally well by saturating—in the standard EFT language—the “counter term” by the E-hole con0gurations or by the multi-particle–multi-hole states or by both. Decimated down to the vicinity of the Fermi surface, both should reside in the “counter” term. If one were to look at the excitation functions going over a range of energies, one of course would have to be careful with the multitude of scales that 0gure for the speci0c excitations involved so as not to encounter the breakdown of the particular EFT one is using.
11. Uncited references [103,104].
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Physics Reports 396 (2004) 41 – 113 www.elsevier.com/locate/physrep
Physical and mathematical aspects of Gamow states O. Civitaresea;∗ , M. Gadellab a
Department of Physics, University of La Plata, 49 y 115, C.C. 67, 1900 La Plata, Argentina b Department of Theoretical Physics, University of Valladolid, 47011 Valladolid, Spain Received 1 March 2004 editor: G.E. Brown
Abstract Physical and mathematical aspects related to the description of resonant states are presented in a comprehensive way. The concepts concerning the representation of Gamow resonances are revisited in connection with a rigorous mathematical treatment, based on the use of MHller operators and rigged Hilbert spaces. The formalism is cast in a form amenable for applications to nuclear structure calculations in the continuum. c 2004 Elsevier B.V. All rights reserved. PACS: 03.65.Db; 25.40.Ny; 25.70.Ef Keywords: S-matrix theory and resonant states; Gamow states; Rigged Hilbert spaces; Average values on resonant states; Berggren and Bohm prescriptions
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Mathematical concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. The Hilbert space of scattering states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. The MHller wave operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. The spectral theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. A model for scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. The S-matrix formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6. Rigged Hilbert spaces (RHS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7. Examples of RHS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Physical concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Non-relativistic Gamow vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∗
Corresponding author. Tel.: +54-221-4-246062; fax: +54-221-4-252006. E-mail address: civitare@<sica.unlp.edu.ar (O. Civitarese).
c 2004 Elsevier B.V. All rights reserved. 0370-1573/$ - see front matter doi:10.1016/j.physrep.2004.03.001
42 44 46 47 49 49 51 52 55 60 60 62
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O. Civitarese, M. Gadella / Physics Reports 396 (2004) 41 – 113 3.2. 3.3. 3.4. 3.5.
Properties of Gamow vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Degenerate resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resonance scattering and Gamow vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Friedrichs model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1. The Friedrichs model in RHS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6. The Gamow vectors for the Friedrichs model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7. Double pole resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8. The choice of the space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9. Gamow states as continuous linear functionals over analytical test functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Vanishing average values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Complex average values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Real average values: Bohm interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Real average values: Berggren interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Relativistic Gamow vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. An exactly solvable model for unstable relativistic quantum <elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Resonances and Gamow vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67 67 69 74 79 80 82 85 86 90 91 91 92 95 96 96 101 103 104 104 106 110
1. Introduction In spite of the long-time elapsed since the discovery of decay phenomena in nuclei and their description in terms of resonances, the use of the concept in nuclear structure calculations has been hampered by an apparent contradiction with conventional quantum mechanics, the so-called probability problem. It refers to the fact that a state with complex energy cannot be the eigenstate of a self-adjoint operator, like the Hamiltonian, therefore resonances are not vectors belonging to the conventional Hilbert space. This report is devoted to the description of resonances, i.e. Gamow states, in an amenable mathematical formalism, i.e. Rigged Hilbert Spaces. Since we aim at further applications in the domain of nuclear structure and nuclear many-body problems, we shall address the issue in a physical oriented way, restricting the discussion of mathematical concepts to the needed, unavoidable, background. From a rigorous historical prospective, the sequence of events and papers leading to our modern view of Gamow vectors in nuclear structure physics includes the following series of publications: [17,103,111,74–76]. The use of Gamow states in conventional scattering and nuclear structure problems was advocated by Berggren, at Lund, in the 1960s [17] and lately [18], by Romo [103] and by Gyarmati and Vertse [111]. The notion was recovered years later, by Liotta, at Stockholm, in the 1980s [76], in connection with the microscopic description of nuclear giant resonances, alpha decay and cluster formation in nuclei. After these pioneering works, the use of Gamow states in nuclear structure calculations become widespread, in spite of the numerical and mathematical diEculties found in the implementation of calculations of realistic nuclear interactions in single-particle
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resonant basis. Recently, both at GANIL-Oak Ridge [74] and at Stockholm [75], the use of Gamow states in shell model calculations was reported. A parallel mathematical development took place, this time along the line proposed by Bohm and collaborators [20]. Curiously enough both attempts went practically unnoticed to each other for quite a long time until recently, when some of the mathematical and physical diEculties found in numerical applications of Gamow states were discussed on common grounds. The need to account for nuclear properties in the continuum, and the notion that the nuclear continuum can be replaced by few non-overlapping Gamow states in the so-called Berggren basis, i.e. single-particle basis which include bound single-particle states and few single-particle resonances and that can be used to perform approximate Tamm–DancoF (TDA) or random phase approximation (RPA) diagonalization of residual nuclear two-body interactions [76], drove the attention of nuclear structure physicist to the more mathematically oriented work of Bohm and collaborators. However, to the best of our knowledge, a review of these approaches is still missed in the literature. We have taken the existence of this gap as the motivation for the present work. Technically speaking, one may concentrate the pros and drawbacks of Berggren and Bohm approaches in the following: (i) The formalism developed by Berggren is oriented, primarily, to the use of a mixed representation where scattering states and bound states are treated on a foot of equality. The normally accepted notion that the continuous should always be taken into account as a sort of complementary subspace of the space spanned by the bound states and resonances was revisited by Berggren. In his approach a basis should contain bound states and few resonances, namely those which have a small imaginary part of the energy. Thus, it is a certain degree of ambiguity in the choice of the so-called narrow resonances [103]. No further requirements are imposed, as a departure from, for instance, the in
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Concerning the realization of the mathematical representation of Gamow states, we shall focus on the study of the Friedrichs model. This is a nice example of a solvable model where the use of generalized vectors in rigged Hilbert spaces becomes very simple and where the identi
2. Mathematical concepts The notion of time evolution, either in the Heisenberg or in the SchrKodinger representation, is a cornerstone in ordinary quantum mechanics. During the past century, since the discovery and formulation of quantum mechanics, the notion of time evolution has been exhaustively explored. However, this has not been the case with Gamow ideas on resonance phenomena, where the time evolution is determined by the presence of a complex parameter. Although the occurrence of this complex parameter does not present a serious formal diEculty, it is not possible to accommodate it in the framework of ordinary quantum mechanics. This may explain why this notion has been somehow neglected for a long period of time. Generally speaking, we may call resonance phenomena those for which the conventional de
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From this point of view, bound states (representing stable particles, with Lt = ∞) and resonances (representing unstable particles, with Lt=
For other problems, like interaction between matter and radiation, interaction between <elds, or even periodic interactions, the free Hamiltonian may appear in diFerent forms [1,100,5,7,67,9]. For a wide range of physical systems, the interaction is represented by a short ranged potential V (r). We shall make the following hypothesis on this potential: (i) V (r) is spherically symmetric. (ii) V (r) → 0, if r → ∞. (iii) V (r) decreases suEciently fast outside a domain. More stringent conditions on the potential can be found, for instance, in the domain of nuclear physics [32,48], where V (r) is also
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2.1. The Hilbert space of scattering states Once we have introduced the dynamics which produces the scattering phenomena, we should de
(1)
The continuous subspace of H with respect to H , which is Hc , has the following properties: (i) There are enough vectors in Hc in which H acts. The result of applying H to a vector in Hc is also a vector in Hc . We then say that H reduces 1 Hc . (ii) The operator H on Hc is self-adjoint and its spectrum is purely continuous. The space Hd is spanned by the bound states of H . Consequently, the scattering states must be included in Hc . However, for some Hamiltonians, not all states of Hc , in a dense subspace of Hc , should be considered as scattering states. It can be realized when the continuous spectrum of H has a fractal section. In such a case, Hc is decomposed into two mutually orthogonal parts, namely Hc = Hac ⊕ Hsc ; where the sub-indices ac and sc stand for absolutely continuous and singular continuous, respectively. The Hamiltonian H reduces both Hac and Hsc and the fractal section of the spectrum of H is the spectrum of the restriction of H to Hsc . This spectrum is called the singular continuous spectrum of H . The spectrum of H in Hac is the absolutely continuous spectrum of H . These names are taken from the measure theory [98]. Therefore, we have the following decomposition: H = Hd ⊕ Hac ⊕ Hsc :
(2)
Scattering states are not-bound regular vectors and therefore they belong to Hac . In addition, they should have
1
In mathematical terms, the restriction of the domain Dc of H to Hc is dense in Hc and H ∈ Hc , ∀ ∈ Dc . A subspace D of a Hilbert space H is dense if any vector in H can be arbitrarily approximated in norm by a vector in D. Thus, for an arbitrary ∈ H and for any j ¿ 0, it exists ∈ D such that 2
− ¡ j: In physical terms this inequality means that the state can be replaced by the state
within an accuracy j.
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47
2.2. The MHller wave operators We assume that any scattering state is asymptotically free in the past. For any scattering state , it exists a free state such that 3 lim {e−itH − e−itH0 } = 0 :
(3)
t →−∞
As the limit in a Hilbert space is taken with respect to its norm, this is equivalent to say that lim e−itH − e−itH0 = 0 :
(4)
t →−∞
Since the evolution operator is unitary, we have: lim − eitH e−itH0 = 0 :
(5)
t →−∞
It is natural to assume that any state can be prepared as a free state, i.e. is an arbitrary vector in H. Then, we can de
:
(6)
The existence of OUT depends on the potential V (r), see [100]. Analogously, we also assume that any scattering state is asymptotically free in the future. This means that for any scattering state ∈ Hac , there exists a free state ’ ∈ H such that lim {e−itH − e−itH0 ’} = 0 :
t →∞
(7)
If this happens for each ’ ∈ H, i.e., for each free state, then, there exists an operator IN de
(8)
Again, the existence of IN depends on V (r), see [100]. Along this paper we shall assume that no incoming scattering state can be trapped in the interaction region and become a bound state. Also, that no bound state can spontaneously decay and become a scattering state. This property is called asymptotic completeness. If asymptotic completeness applies, then the operators OUT and IN are unitary operators from H to Hac [1,100]. Therefore, their inverses exists from Hac to H. As these operators are unitary their inverses coincide with their adjoints and † 1 − IN = IN ;
if
† 1 − OUT = OUT :
The operators OUT and IN are the MHller wave operators. They have the following property: (0) = ’(0) ;
where can be either OUT or IN , then, () = ’() ; 3
From hereon we take ˝ = 1, unless indicated explicitly.
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where is an arbitrary time. The proof of this statement is very simple. Write eitH e−itH0 () = eitH e−i(t+)H0 (0) = e−iH ei(t+)H e−i(t+)H0 (0) ; taking limits as t goes to either −∞ or ∞, we have () = e−iH ’(0) = ’() : This formula can also be written as e−iH0 (0) = e−iH ’(0) = e−iH (0) : If the free state
(0) can be arbitrary chosen, we have that
e−iH0 = e−iH ;
(9)
or in in
(10)
In scattering theory we prepare a free state in the remote past. It evolves freely and at the time t this state would have been = (t) if no interaction is present. As the particle in the state (t) enters in the interaction region, the state becomes IN (t) = (t). When the particle leaves the interaction region, its state is given by ’(t) where OUT (t) = ’(t). The relation between the free incoming state (t) and the outgoing state ’(t) is given, at all times, by 1 ’(t) = − OUT IN (t) :
If we assume asymptotic completeness, for any free incoming state state ’. 4 As the Mller operators are unitary, 5 the operator 1 S := − OUT IN
(11) , there is a unique outgoing (12)
is also unitary. Observe that it maps H onto itself. The operator S is the S-operator. Eq. (11) can also be written as Se−itH0 (0) = e−itH0 ’(0) = e−itH0 S (0) : Since the choice of
(0) is arbitrary, we have that
Se−itH0 = e−itH0 S ;
(13)
or, in in
(14)
Eq. (14) shows that the S-operator commutes with the free Hamiltonian H0 . This property has important consequences which will be commented upon in the next subsection. 4
For the sake of convenience, we write ≡ (0) and ’ ≡ ’(0). Strictly speaking, from the mathematical side, the MHller operators are unitary from the absolutely continuous subspace Hac (H0 ) onto Hac (H ), for H0 and H , respectively. 5
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2.3. The spectral theorem Next, we are going to present, without proofs, important spectral properties of general self-adjoint operators. For practical purposes, we shall introduce here only those results of direct application to our formalism of resonance scattering. Let A be a self-adjoint operator on a Hilbert space H, a vector ∈ H is said to be a cyclic vector of A if the sequence {; A; A2 ; : : : ; An ; : : :} forms a basis, in general not orthonormal, of H. Self-adjoint operators having a cyclic vector, can be written as a multiplication operator, exactly like the position operator, on a certain Hilbert space. 6 If A has absolutely continuous spectrum only, i.e., if the decomposition of H with respect to A given in (2) is H = Hac with vanishing Hd and Hsc , then, d is the Lebesgue measure on (A) [1,98]. Summarizing, we have a unitary mapping U : U : H → L2 [(A); d] ; such that if
(15)
(x) ∈ L2 [(A); d] with x (x) ∈ L2 [(A); d], then
UAU −1 (x) = x (x) :
(16)
If S is an operator which commutes with A, then there exists a function S(x) such that USU −1 (x) = S(x) (x) for any (x) ∈ L2 [(A); d] with S(x) (x) ∈ L2 [(A); d] [1]. Should A not have a cyclic vector on H, then H can be decomposed as a direct sum of Hilbert spaces [84,98] H = H1 ⊕ H2 ⊕ · · · ⊕ Hn ; such that A is an operator and it has a cyclic vector on each of the Hi . Therefore, the spectral theorem applies on each of the Hi . In this case, we can always
In fact, the spectral theorem [98] states that there exists a measure d on the spectrum of A, (A), and a unitary operator U from H into L2 ((A); d), which is the space of square integrable functions from (A) into C, the set of complex numbers, with respect to the measure d, such that UAU −1 is the multiplication operator [1,98].
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beginning of Section 2. Then, (i) The Hamiltonians H0 and H = H0 + V are self-adjoint. (ii) As the potential is spherically symmetric, H commutes with the three components of the orbital angular momentum. In this case the Hilbert space H, which in our case is L2 [R3 ; d 3 r], the space of all integrable complex functions on the three-dimensional real space R3 , can be decomposed as ∞
H = ⊕ Hl ;
(17)
l=0
where Hl is the Hilbert space of all the states with value l of the angular momentum. The operators H0 and H are self-adjoint on each of the Hl . Let us take a <xed value of the orbital angular momentum l; thus Hl has a cyclic vector and therefore the spectral theorem applies on it. In particular, there exists a unitary mapping U from Hl to L2 (R+ ; dE) such that 7 UH0 U −1 l (E) = E l (E);
l
∈ Hl ;
(18)
where l is the component of the vector in the subspace Hl . The S-operator is a unitary operator on each of the Hl . Since the S-operator commutes with H0 , there exists a function Sl (E) such that USl U −1 l (E) = Sl (E) l (E) ;
(19)
where Sl is the restriction of S to Hl and l (E) ∈ L2 [R+ ; dE]. As U is unitary and S is also unitary on Hl , we conclude that multiplication by Sl (E) on L2 [R+ ; dE] is also a unitary application. This implies that [1] Sl (E) = ei l (E) ;
(20)
where l (E) is real for E real. An equation like (20) holds for all values of the angular momentum. The resulting functions l (E) are the phase shifts. Remark. From now on we shall work with a <xed value of the angular momentum and, for simplicity, we omit the label l unless otherwise stated. Note that the case l = 0 should be treated with care, since for it the centrifugal term vanishes in the radial form of both, the free and the total Hamiltonian. (iii) On H0 , both H0 and H have spectrum (H0 ) = (H ) = R+ . (iv) The MHller wave operators exists and asymptotic completeness holds for the pair {H0 ; H } on H0 . In particular, this means from (18) that U −1 H U −1 (E) = E (E) ;
(21)
where is either OUT or IN . Thus, there are two unitary operators that ful
V− = U IN :
R+ ≡ [0; ∞). Therefore, L2 (R+ ; dE) is the space of complex square integrable functions on R+ .
(22)
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51
Therefore, V± HV±−1 (E) = E (E) :
(23)
(v) The operator S(E) := Sl (E) can be analytically continued as described in the next subsection. 2.5. The S-matrix formalism We are now in the position to discuss the properties of the S-operator on Hl . Although we shall always work in the energy representation, for the√sake of clarity we shall discuss its properties on the p-plane, as a function of the variable p = + 2mE [90,20]. We call this function S(p) and it has the following properties: 1. As a consequence of causality [90], S(p) admits an analytic continuation S(z) on the complex plane. The possible singularities of S(z) are poles that may be of three kinds [20,90]: (i) Single poles in the positive imaginary axis of S(z) that correspond to the bound states of H . (ii) Single poles in the negative imaginary axis that correspond to virtual states. (iii) Pairs of poles, in principle of any order, in the lower half-plane. Each of the poles of a pair has the same negative imaginary part and the same real part with opposite sign. Thus, if pR is one of these two poles the other is −pR∗ where the star denotes complex conjugation. These poles are called resonance poles and in general there is an in
The fact that the real part of the complex conjugate poles be positive is a consequence of causality. If we de
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(ii) Let us call S(E − i0) the values of S(w) on the lower rim of the cut. The boundary values of S(w) on the upper rim of the cut are denoted by S(E + i0) and coincide with S(E). These values are related: S(E + i0) = S ∗ (E − i0) :
(24)
The above conditions (i) and (ii) are, in fact, the simplest which may be ascribed to the S matrix, in order to discuss the mathematical properties and the proper mathematical frame for Gamow vectors. The question about the existence of potentials which ful
(25)
is a rigged Hilbert space (RHS) if: (i) The intermediate space H is an in
9
We shall always assume that our Hilbert spaces are separable.
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the identity mapping i : → H ; such that i(’) = ’ for all ’ ∈ , is continuous. The topology in is not given by a norm, but in the most simplest cases of physical interest, by a countable in
(111)
(112)
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where " is a complex number and ’(!) ∈ L2 (R+ ). The scalar product of two vectors in H is given by ∞ " # ∗ ’∗ (!)3(!) d! : (113) =" #+ ’(!) 3(!) 0 Then, the action of the Hamiltonian H0 on !0 " H0 = ; !’(!)
is (114)
where !0 ¿ 0 is the eigenvalue of H0 and ’(!) is a function on R+ with the property !’(!) ∈ L2 (R+ ). Observe that the restriction of H0 to L2 (R+ ) is the multiplication operator: 0 0 H0 =! : (115) ’(!) ’(!) This operator has an absolutely continuous spectrum equal to R+ . Thus, H0 has an eigenvalue, !0 , imbedded in its continuous spectrum, R+ . The vector 1 |1 = (116) 0 is the state vector for the bound state. Observe that H0 |1 = !0 |1. After the de
V =
∞
0
(117) f(!) ’(!) d! ; ∗ "f (!)
(118)
where, $ is a positive real parameter, the coupling parameter, and f(!) is a function in L2 (R+ ) called the form factor. Let us denote by P the orthogonal projection to the subspace spanned by |1 and let Q be its orthogonal complement. Then, Q projects in (112) into its second component. We obviously have that P + Q = I;
PQ = QP = O ;
(119)
where I and O are the identity and the zero operator on H, respectively. Proposition F.1. We have that QVQ = O :
(120)
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Proof. Let us apply QVQ to QVQ
" ’(!)
= QV
in (112): 0 ’(!)
=Q
0
∞
f(!)’(!) d! = 0
0 0
:
(121)
This is often called the Friedrichs condition [47]. The Friedrichs condition is nothing else than a straightforward consequence of the model. Proposition F.2. The projections P and Q commute with H0 . Proof. It is suEcient to prove that PH0 = H0 P, since Q = I − P. " !0 " !0 " PH0 = ; =P !’(!) 0 ’(!) " " !0 " H0 P : = = H0 0 0 ’(!)
(122)
Under some conditions, that we are going to describe soon, the Friedrichs model has a resonance. The eFect of $V is to transform the bound state |1 of H0 in an unstable state. The real number !0 is transformed into a complex number !$ that goes to !0 as $ → 0. In order to describe resonances, we consider the reduced resolvent of H in |1 given by FH (z) = 1|
1 |1 ; z−H
(123)
where H is the total Hamiltonian given by (117). The complex valued function FH (z) has no singularities on the complex plane other than a branch cut coinciding with the spectrum 27 of H : (H ) ≡ R+ [66]. Under certain conditions on the form factor f(!), FH (z) admits analytic continuations from above to below and from below to above through the cut. These analytic continuations may have singularities which are associated with the resonances. We shall present here these ideas in detail. To start with, we shall introduce the following theorems: Theorem F.3. The reduced resolvent (123) is given by [47] −1 ∞ 1 |f(!)|2 1| d! : |1 = −z + !0 + $2 z−H z−! 0
(124)
Proof. The second resolvent identity [113] states that R(z; H ) = R(z; H0 ) − $R(z; H0 )VR(z; H ) ; 27
(125)
After the de
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where R(z; H ) =
1 ; z−H
R(z; H0 ) =
1 : z − H0
This gives P
1 1 1 1 P=P P : P − $P V z−H z − H0 z − H0 z − H
(126)
Inserting (119) in the second term in the right of (126), we have: P
1 1 (P + Q)V (P + Q) P z − H0 z−H =P
1 1 1 1 P+P P PVP QVP z − H0 z−H z − H0 z−H
+P
1 1 1 1 P+P P : PVQ QVQ z − H0 z−H z − H0 z−H
(127)
As P and Q commute with H0 and PQ = O, we have that P
1 Q=O ; z − H0
and therefore the second and fourth terms in the right-hand side of (127) vanish. We also have that Q
1 1 1 1 P=Q P : P − $Q V z−H z − H0 z − H0 z − H
(128)
The
1 1 1 (P + Q)V (P + Q) P = −$Q P z−H z − H0 z−H = −$Q
1 1 1 1 P − $Q P QVQ PVP z − H0 z−H z − H0 z−H
− $Q
1 1 1 1 P − $Q P : PVQ QVP z − H0 z−H z − H0 z−H
(129)
Again, the second and the third term in the right-hand side of (129) vanish. Due to the Friedrichs condition (120), also the
1 1 1 P : QVP P = −$ Q z−H z − H0 z−H
(130)
Inserting (130) into (127) and then, (127) into (126), we have P
1 1 1 1 P=P P P − $P PVP z−H z − H0 z − H0 z−H + $2 P
1 1 1 P : VQ QVP z − H0 z − H0 z−H
(131)
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If we multiply (131) to the left by (z − H0 )P, we can write 1 P=P (z − H0 ) P z − H0 and therefore,
1 1 2 P ; QVP P P = (z − H0 )P + $PVP − $ PVQ z − H0 z−H or 1 P P = [G(z)]−1 P ; z−H where
1 2 G(z) = (z − H0 ) P + $PVP − $ PVQ QVP : z − H0 Since, 1 1 1 1 QVP QV PVQ = PVQ z − H0 z − H0 0 0 0 0 1 1 Q = PVQ = PVQ z − H0 z − H0 f∗ (!) f∗ (!) ∞ |f(!)|2 0 d! z−! = PVQ f∗ (!) = P 0 0 z−! ∞ |f(!)|2 d! z−! = 0 ; 0 and PVP|1 = 0, we have that 2 G(z)|1 = z − !0 − $
0
∞
|f(!)|2 d! |1 : z−!
Consequently, 1 1 |1 = 1|P P|1 = 1|[G(z)]−1 |1 1| z−H z−H
−1 ∞ |f(!)|2 2 d! ; = z − !0 − $ z−! 0 which proves Theorem F.3. Now, let us assume that the function h(!) = |f(!)|2
(132)
(133)
(134)
(135)
(136)
(137)
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79
is entire analytic on the complex variable !. Then, the following result holds [51,47,66]: Theorem F.4. The function, ∞ |f(!)|2 d! ; 3(z) = z − !0 − $2 z−! 0
(138)
is a complex analytic function with no singularities on the complex plane other than a branch cut coinciding with the positive semi-axis R+ provided that 3(0) ¿ 0. It admits analytic continuations through the cut 3+ (z), from above to below 3− (z) from below to above. The continuation 3+ (z) has a zero at z0 with Im{z0 } ¡ 0, which is an analytic function on the coupling parameter $ on a neighborhood of zero. Analogously, 3− (z) has a zero at z0∗ , which is also an analytic function of $. The proof of this result is very technical and we refer the interested reader to the original sources [51,47,66]. The form of 3± (z) is ∞ |f(!)|2 2 3± (z) = z − !0 − $ d! ; (139) z − ! ± i0 0 where the signs plus and minus on the denominator of (139) indicate that the analytic function represented by the integral is an analytic continuation from above to below, +, or from below to above,−. Thus, z0 and z0∗ are, respectively, the zeroes of the equations: 3+ (z) = 0
and
3− (z) = 0 :
(140)
Theorem F.4 is also valid for other types of functions h(!). See [5]. 3.5.1. The Friedrichs model in RHS The notation for the Friedrichs model presented so far is not, in our opinion, the most practical for explicit calculations. We shall use a new notation, in the spirit of the Dirac notation, for which rigged Hilbert spaces (RHS) are particularly useful. According to the Gelfand–Maurin spectral theorem, there exists a RHS ⊂ H ⊂ × such that H0 ⊂ , H0 is continuous on and for any ! ∈ R+ , the absolutely continuous spectrum of H0 , there exists a |! ∈ × with H0 |! = !|!. As H0 has the eigenvector |1, the spectral decomposition of H0 is then ∞ !|!!| d! : (141) H0 = !0 |11| + 0
Then according to (118), V must have the following form: ∞ [f∗ (!)|!1| + f(!)|1!|] d! : V= 0
Also, the vector = "|1 +
(142)
in (112) can be written as
0
∞
’(!) |! d! :
(143)
Note that the set {|1; |!}, ! ∈ R+ , of generalized eigenvectors of H0 , forms a complete set in the sense that each ∈ can be written as a superposition of these vectors. Thus, each vector in has
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a component on the bound state |1 of H0 and an in
(144)
1|! = !|1 = 0 ;
(145)
!|! = ! |! = (! − ! ) :
(146)
To show (144), we observe that this is just the scalar product of vector (116) by itself. To show (146), let us consider the following scalar product in the old notation: ∞ 0 0 ’∗ (!)3(!) d! : (147) = ’(!) 3(!) 0 In the new notation, this scalar product should be written as ∞ ∞ ∞ ∞ ∗ ’ (!) !|d! 3(! ) |! d! = ’∗ (!)3(! )!|! d! d! : 0
0
0
0
(148)
Should Eqs. (147) and (148) coincide, then Eq. (146) must follow. Observe that the Dirac delta is referred to the integration from 0 to ∞. Finally (145) follows from the fact that the states |! span the lower component in (116) and functions in this component are orthogonal to |1. Then, if we apply (142) to (143), we have ∞ ∞ ∞ ∗ V = f (!)|! d!1|"|1 + f(!) ’(! )|1!|! d! d! 0
="
0
∞
f∗ (!)|! d! + |1
0
0
∞
0
f(!)’(!) d! :
(149)
This result coincides with (118). This shows how to operate in the new notation, which is more familiar to physicists. 3.6. The Gamow vectors for the Friedrichs model Here, we intend to get the explicit form of the Gamow vectors for the Friedrichs model. We make the following assumptions: (i) The function 3(z) has no singularities in the complex plane other than a branch cut coinciding with the positive semiaxis, i.e., the continuous spectrum of H . (ii) The function 3(z) can be analytically continued through the cut. These are the functions 3+ (z) and 3− (z) of the previous section. These extensions have poles located at the points z0 and z0∗ as stated before. (iii) Although resonance poles may, in principle, have arbitrary multiplicity, since in resonant scattering this is not forbidden by causality conditions [90], we shall assume
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81
Thus, we intend to obtain the decaying Gamow vector |f0 ∈ × + and the growing Gamow vector . In order to accomplish our goal, we do the following [13,8]: Let x be an arbitrary positive |f˜0 ∈ × − number (x ¿ 0) and write the eigenvalue equation (H − x) 8(x) = 0 :
(150)
Since x belongs to the continuous spectrum of H , 8(x) cannot be a normalized eigenvector of H and therefore, it should belong to either of the duals × ± . As |1 and |! form a complete system, we must have: ∞ (x; !)|! d! : (151) 8(x) = "(x)|1 + 0
If we apply H to (151), we obtain the following system of equations: ∞ (!0 − x)"(!) + $ (x; !)f∗ (!) d! = 0 ; 0
(! − x) (x; !) + $f(!)"(!) = 0 :
(152) (153)
To solve this system, we write "(!) in terms of (x; !) using (153) and carry the result to (152). We obtain an integral equation, which gives one solution of the form ∞ f(!) 1 ∗ |1 + $ |! d! : (154) 8+ (x) = |x + $f (x) 3+ (x) x − ! + i0 0 This is a functional in × + . When applied to a vector in + , it gives an analytic function on the lower half-plane. We say that 8+ (x) admits analytic continuation to the lower half-plane, in a weak sense. This continuation has a simple pole at z0 so that we can write on a neighborhood of z0 : 8+ (z) =
C + o(z) : z − z0
(155)
From (150) and (155), we get 0 = (H − z)8+ (z) =
1 (H − z)C + (H − z)o(z) ; z − z0
(156)
which gives (H − z0 )C = 0 ⇒ HC = z0 C :
(157)
Therefore, the residue C of 8(z) at the pole z0 coincides, save for an irrelevant constant, with the decaying Gamow vector |f0 . To calculate its explicit form, note that (155) on a neighborhood of z0 has the form ∞ f(!) constant |1 + $ |! d! + RT ; (158) 8+ (z) ≈ (z − z0 ) z − ! + i0 0 where RT stand for “regular terms”. Now, let us use the Taylor theorem to have 1 1 z − z0 = − + o(z) : z − ! + i0 z0 − ! + i0 (z0 − ! + i0)2
(159)
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By replacing (159) in (158), we get ∞ f(!) constant |1 + $ |! d! + RT : 8+ (z) ≈ (z − z0 ) z0 − ! + i0 0 Therefore, up to an irrelevant constant, we conclude that ∞ $f(!) C = |f0 = |1 + |! d! : z0 − ! + i0 0
(160)
(161)
The system given by Eqs. (152) and (153) has another solution that can be analytically continued in the upper half-plane. This solution gives, using the same technique, the growing Gamow vector |f˜0 : ∞ $f∗ (!) |! d! : (162) |f˜0 = |1 + z0∗ − ! − i0 0 Note that the Gamow vectors depend on the coupling constant $. 3.7. Double pole resonances The zeroes of 3± (z) at z0 and z0∗ can be multiple, and this means that 3+ (z) = (z − z0 )n G(z) with G(z) = 0 on a neighborhood of z0 . The question that arises now is whether these zeroes exist. We want to describe a typical situation, for which the zeroes at z0 and at z0∗ are double and obtain the Gamow vectors. Let us choose in (142) a form factor f(!) such that [5]: √ ! 2 (163) |f(!)| = P(!) with P(!) = (! − ")(! − "∗ ) :
(164)
Identities (163) and (164) determine the form factor f(!) up to a phase. We have √ ∗ √ √ 1 −z −" −" 2 − − : 3(z) = z − !0 − )$ ∗ P(z) "−" z − " z − "∗
(165)
If we make the change of variables: z = p2 ;
" = b2
(166)
and write ’(p) = 3(p2 ) = 3(z), we obtain ’(p) = p2 − !0 +
i)$2 : (b − b∗ )(p + b)(p − b∗ )
(167)
Then, if 3+ (z) has a double zero at z0 , ’(p) has a double zero at p0 with z0 = pR2 [5]. The condition that ’(p) has a double pole at p0 reads ’(p0 ) = 0;
’ (p0 ) = 0;
’ (p0 ) = 0 :
(168)
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The two complex equations (168) give a system of four real equations for six parameters, which are: !0 , $, Re p0 , Im p0 , Re b and Im b. We have two free parameters that, we choose to be !0 and $. We can write the other four parameters in terms of these two and obtain: 2 1=3 )$ 1=2 b = !0 + 2i (169) 16!0 and
p0 = ! 0 −
)$2 16!0
2=3
−
)$2 16!0
1=3 :
(170)
However, the parameters !0 and $ are not completely independent if we impose the condition Im p0 ¡ 0, since this implies that 2 2=3 )$ !0 − ¿0 : (171) 16!0 The condition Im p0 ¡ 0 comes from 3(0) ¿ 0 and we impose the latter in order to restrict the singularities of 3(z) to those on the branch cut [47,66]. The resonance poles are singularities of the analytic continuations of 3(z), or for ’(p) on the upper half-plane [66]. If we carry (169) and (170) to (167), we obtain ’(p) = −
(p − p0 )2 (p + p0∗ )2 ; (p + t)(p − t ∗ )
(172)
where t is a complex number diFerent from p0 or p0∗ . We observe that ’(p) has double zeroes at pR and pR∗ and therefore, so has 3(z) at z0 and z0∗ . Due to the fact that the analytic continuation of (154) exists on a neighborhood of z0 , we can now write C1 C2 8+ (z) ≈ + + RT (173) 2 (z − z0 ) z − z0 and 8+ (z) ≈ =
constant (z − z0 )2 constant (z − z0 )2
|1 + $
−$(z − z0 )
0
0
∞
f(!) |! d! z − ! + i0
∞
f(!) |! d! z0 − ! + i0 0 f(!) |! d! : (z0 − ! + i0)2
|1 + $
∞
Comparing (174) with (173), we obtain ∞ f(!) C1 = |1 + $ |! d! ; z0 − ! + i0 0 ∞ f(!) |! d! : C2 = −$ (z0 − ! + i0)2 0
(174)
(175) (176)
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We must show that C1 and C2 are indeed the Gamow vectors we are looking for. More precisely |f0 = C1 ;
|f1 = C2 :
(177)
Due to the properties of |f0 and |f1 presented in Sections 3.2 and 3.3, to get (177) we have to show that HC1 = zR C1 ;
HC2 = zR C2 + C1 :
(178)
From (175), there is no doubt that C1 = |f0 . To obtain the second relation in (178), we apply H to (176) and get ∞ ∞ f(!) HC2 = −$ d! ! |! ! | d! |! (z − ! + i0)2 R 0 0 ∞ ∞ f(!) 2 ∗ d! f (!)|1! |$ d! |! −$ (zR − ! + i0)2 0 0
∞ ∞ !f(!) |f(!)|2 2 |1 : (179) d! |! − $ d! = −$ (zR − ! + i0)2 (zR − ! + i0)2 0 0 The coeEcient of |1 in (179) is equal to one. To show it, let us go back to (139) and derive it with respect to z. We get ∞ |f(!)|2 3+ (z) = −1 − $2 d! : (180) (z − ! + i0)2 0 Since ’ (p0 ) = 0 implies 3 (z0 ) = 0 [5], we have that ∞ |f(!)|2 2 d! ; 3+ (z0 ) = 0 = −1 − $ (z0 − ! + i0)2 0
(181)
which supports our claim. After (181) and (179) can be written as ∞ (−! + zR )f(!) HC2 = |1 + $ d! |! (z0 − ! + i0)2 0 ∞
f(!) + z0 −$ d! |! (z0 − ! + i0)2 0 ∞
∞ f(!) f(!) |! + z0 −$ d! d! |! = |1 + $ z0 − ! + i0 (z0 − ! + i0)2 0 0 = C 1 + z 0 C2 ;
(182)
which shows that C2 = |f1 . On the subspace spanned by |f0 and |f1 , the total Hamiltonian H exhibits a block diagonal form: z0 1 : (183) H= 0 z0
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Analogously, one obtains for |f˜0 and |f˜1 the following expressions: ∞ f(!) |! ; d! ∗ |f˜0 = |1 + $ z − ! − i0 0 0 ∞ f(!) ˜ |f1 = −$ d! ∗ |! : (z0 − ! − i0)2 0
85
(184) (185)
Remark. Another model producing a double pole resonance has been discussed by MondragTon and coworkers [86,87,10]. In their model, a double pole resonance is produced in one-dimensional scattering with a double delta barrier. The potential has the form V=
) ) (x − a) + (x − b) ; " #
where a, b, " and # are real parameters, and it gives an in
(186)
2 + 2 + where the spaces S ∩ H± |R (see Section 2), we need that f∗ (!) ∈ S ∩ H± |R . In this case, both H0 and H have the following properties: (i) The spaces ± are reduced by both H0 and H . This means that H0 ± ⊂ ± and H ± ⊂ ± . (ii) Both Hamiltonians H0 and H are continuous on ± . (iii) Using the duality formula
A ’|F = ’|AF;
’ ∈ ;
F ∈ × ;
where A is either H0 or H , we can extend both operators to the duals × ± . Then, it is possible to × show that these extensions are continuous operators on ± , when we endow × ± with the weak 2 + topology (see [105]). The condition f∗ (!) ∈ S ∩ H± |R is however too restrictive (see [5]). For practical purposes, the function f(!) should be a real function of the real variable ! admitting analytic continuation to the whole complex plane with possibly a branch cut along the positive semiaxis. In this case, the property H ± ⊂ ± is false in general. There is, however, a way out as × H can be shown to be a continuous operator from × ± into ± , for many choices of f(!). Then, we assume in general that × H × ± ⊂ ± :
(187)
The possibility of de
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the analytical continuation of the S-matrix in the energy representation, S(E), through the spectrum of H , coincide with the poles of the analytic continuations of the reduced resolvent through the cut [15]. These poles coincide with the zeroes of the functions 3± (z). These poles appear in complex conjugate pairs. In the simplest case, we have only two and are simple: z0 = ER − i=2 and z0∗ = ER + i=2 with ER ¿ 0, ¿ 0. Since z0 and z0∗ are the poles of the analytic continuation of S(E), Gamow vectors can be de
n = 1; 2; 3 : : : :
The scalar product on +∞ ˆ ˆ ˆ dE ˆ ∗ (E)(E) ; | = −∞
(188)
(189)
provides the following norm: ˆ 2 = | ˆ ˆ :
(190)
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The set of norms (188) along the norm (190) endow with a metric topology, with the following properties: (i) Since is a subspace of L2 (R) and it possesses the norm of L2 (R), it is clear that the canonical injection i : → L2 (R);
ˆ ˆ i[(z)] = (z);
ˆ ∈ ∀(z)
is continuous. (ii) Since has the Hilbert space norm of L2 (R), it then results that any Cauchy sequence in is also a Cauchy sequence in L2 (R). The converse is, however, not true as we have on an in
(191)
Thus, (191) is a RHS (although the functionals on are linear and not anti-linear). We should remark that, according to the Gelfand Maurin theorem, any hermitian operator A on admitting one and only one self-adjoint extension on L2 (R) (hermitian operators with this property are called essentially self-adjoint operators) has a complete system of eigenvectors in whose respective eigenvalues are in the Hilbert space spectrum of A. Related to the RHS in (191), we have another RHS which is obtained with the use of the inverse ˆ Fourier transform on (191). For any (E) ∈ , we have +∞ 1 ˆ ˆ (t) = F−1 {(E)} = eiEt (E) dE : (192) 2) −∞ Since the inverse Fourier transform of a Schwartz function is also a Schwartz function, for each ˆ (E), the (t) in (192) is a Schwartz function. We shall call ˜ the vector space of functions of the form (192). Thus, ˜ = F−1 : ˜ We can endow ˜ with The inverse Fourier transform F−1 is an one-to-one mapping from onto . − 1 ˜ ˆ ˆ the topology transported from into by F . This means that if 1 ; 2 ; : : : ; ˆ n ; : : : is a Cauchy ˜ The sequence in , then, 1 = F−1 ˆ 1 ; 2 = F−1 ˆ 2 ; 3 = F−1 ˆ 3 ; : : : is a Cauchy sequence in . ˜ then, space is also dense in L2 (R) and, if ˜ is the space of all linear continuous functionals on , ˜ ⊂ L2 (R) ⊂ ˜ is a RHS. We can extend F−1 as a mapping from onto ˜ by means of the duality formula:
ˆ F ˆ −1 F ˆ = | ˆ : |F = F−1 |F
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At this point, the connection between this formalism and the formalism presented in Section 3.1 is established. In fact, schematically, RHS ↔ ⊂ L2 (R) ⊂ ;
RHS ↔ ˜ ⊂ L2 (R) ⊂ ˜ ; RHS ↔ ± ⊂ H ⊂ × ± : Remark. The relation between S, S (space of linear continuous functionals on S also called space of tempered distributions), , , ˜ and ˜ is given by: ⊂ S ⊂ L2 (R) ⊂ S ⊂ and ˜ ⊂ S ⊂ L2 (R) ⊂ S ⊂ ˜ :
Note that if S(x) ∈ S , the Dirac formula gives ∞ S(y) (x − y) dy : S(x) = −∞
Analogously, we can obtain the value of ˆ c (z) ∈ at any complex value z (with Im z = 0), if we note that the ultra-distributions in can be represented as complex analytic functions with a branch cut on a real interval. Therefore, the desired formula is [107] +∞ 1 ˆ ˆ c (z) = 1 (E) dE ; (193) 2)i −∞ E − z where ˆ (E) = ˆ c (E + i0) − ˆ c (E − i0) : (194) ˆ ˆ and c (E + i0) and c (E − i0) are, respectively, the boundary values from above to below and from below to above of the complex analytic function ˆ c (z) (to illustrate the point see Eqs. (39) and (193)). Now, let H be the total hamiltonian and assume that it has a simple continuous spectrum from E0 to E1 (in general H is semibound so that E1 = ∞). Due to the above-mentioned Gelfand–Maurin theorem, there exists functionals |E ∈ such that H |E = E|E for (almost in the sense of the Lebesgue measure) all E ∈ (E0 ; E1 ). Now, let zR = ER − i=2 and zR∗ = ER + i=2 its complex conjugate just as in previous sections. Let us take those c (z) ∈ such that the diFerence (194) vanishes outside the interval (E0 ; E1 ). In this case, formula (193) can be written as E1 1 ˆ c (z ∗ ) = 1 ˆ (E) dE ; (195) R 2)i E0 E − zR∗ and also as [ ˆ c (zR )]∗ =
1 2)i
E1
E0
1 [ ˆ (E)]∗ dE : zR − E
(196)
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It is customary to write the diFerence ˆ (E) (see (194)) in the Dirac notation as ˆ (E) = E| ˆ . Then from (195), we have ˆ (z ∗ ) = 1 R 2)i
E1
E0
1 E| dE E − zR∗
(197)
and using the convention |E∗ = E| , we have from (196): [ ˆ (zR )]∗ =
1 2)i
E1
1 |E dE : zR − E
E0
(198)
We now de
1 f˜0 | = zR∗ | = 2)i
E1
1 E| dE E − zR∗
(199)
1 |E dE : zR − E
(200)
[ ˆ (zR )]∗ = |f0 :
(201)
E0
and |f0 = |zR =
1 2)i
E1
E0
In consequence: ˆ c (z ∗ ) = f˜ | ; R 0
Compare the form of these functionals to (67). Vectors (199) and (200) are, respectively, the left and right Gamow vectors in this presentation. For n = 0; 1; 2; : : : ; also note that 1 H |f0 = 2)i n
= If
1 2)i
E1
1 H n |E dE zR − E
E0
E1
1 E n |E dE : zR − E
E0
(202)
(z) is an arbitrary function in , we have |H n |f0 =
1 2)i
E1
E0
En |E dE = zRn |f0 ; zR − E
so that if we omit the arbitrary function H n |f0 = zRn |f0 ;
n = 0; 1; 2; : : : :
(203)
(z), we have (204)
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Analogously, for n = 0; 1; 2; : : : ; we have f˜0 |H n = zRn f˜0 | :
(205)
The time evolution is given by |e−iHt |f0 = e−izR [ ˆ (zR∗ )]∗ = e−itER e−t=2 |f0 :
(206)
Of course, if we omit the arbitrary e−iHt |f0 = e−itER e−t=2 |f0 :
(z) ∈ , we have: (207)
This exponential evolution is valid for all values of time and therefore it does not lead to semigroup time evolution. Consequently, this description based on ultra-distributions is not suitable for a description of quantum irreversibility, in the sense of Prigogine [13,96,97], as it is the description based on Hardy functions. 4. Observables Generally speaking, by quantum mechanics we mean the theory and its interpretation. Central to this point is the assignment of mean values to observables acting on states. So far, we have not addressed this problem in relation to Gamow states and we shall devote this section to present the state of the art. As it will become evident from our discussion, the notion that the use of Gamow states violates the quantum mechanical probabilistic interpretation is wrong. As we said in the Introduction, and as it has been shown in the previous section, probability considerations are avoided in RHS. The question whether it is legitimate to de
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−∞ to ∞. In this case, we can de
(208)
Yet this result is not acceptable from the physical point of view. Due to the uncertainty principle we cannot measure simultaneously the real part of zR , which is the resonant energy, and its imaginary part, which is proportional to the inverse of the half life. Thus, zR cannot be the average of any measurement process and cannot be accepted as the energy average. Thus, the above generalization, which in principle seems to be the most natural de
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Actually, the same problem arises when dealing with the complex scaling formalism [101,108]. Gamow vectors are, in complex scaling, eigenvectors with complex eigenvalues of an analytically dilated non-self-adjoint Hamiltonian H (*), where * is a complex parameter such that H (* = 0) = H is the total, interacting, hamiltonian H = H0 + V . These complex eigenvalues, zR , are independent of * [114], and coincide with poles of the S-matrix in the energy representation, as it was shown in [4]. Therefore, they are truly resonance eigenvalues. However, the corresponding eigenvectors |f* are dependent on *, as dictated by the eigenvalue equation H (*)|f* = zR | f* :
(209)
The eigenvector |f* is a normalized vector on Hilbert space and, therefore, its norm can be unity. Then, a natural de
(210)
Since |f* is normalized the above expectation value is well de
(211)
S ∩ H+ ⊂ H+ ⊂ (S ∩ H+ )× :
(212)
and Let us de
−1 × 1 ˜ = (*− ) U − IN |f0
D
1 −1 × = (*+ ) U − OUT |f0 :
(213)
and (214) G
and From (213) and (214) is not possible to obtain the explicit form of these objects, let us take arbitrary vectors ± ∈ ± and consider the functions − (E)
−1 1 = (*− ) U − IN
−
and
+ (E)
1 −1 = (*+ )U − OUT
+
;
and their complex conjugates: #
− (E)
=[
− (E)]
∗
∈ S ∩ H+2
and
# + (E)
=[
+ (E)]
∗
2 ∈ S ∩ H− :
D
. In order to get
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93
2 We recall that ± (E) ∈ S∩H± , and they are, therefore, de
2 × ∈ (S ∩ H− )
and
D
∈ (S ∩ H+2 )× :
−1 × 1 −1 × −1 Since the operators (*− ) U − IN and (*+ ) U OUT are one to one and onto transformations G D 2 × between dual spaces, the functionals and represent Gamow vectors in (S ∩ H− ) and 2 × (S ∩ H+ ) , respectively. We see that: (i) These Gamow vectors are represented by square integrable functions in L2 (R). Moreover, by using the de
2 ∈ H−
and
D
∈ H+2 :
(218)
2 × (ii) The operator that represents the Hamiltonian on the RHS S ∩ H± ⊂ H± ⊂ (S ∩ H± ) is given by 2
1 −1 −1 (*+ ) U − *+ ; OUT H OUT U
on S ∩ H+2 and −1 1 −1 (*− )U − *− IN H IN U 1 −1 2 on S ∩ H− . Observe that − OUT H OUT = IN H IN = H0 . Since U diagonalizes H0 , we have that −1 2 + |R (E (E) = E (E)). Thus, *± E *± are UH0 U −1 = E is the multiplication operator on S ∩ H± 2 the multiplication operators on S ∩ H± , respectively. These operators are the restrictions of the multiplication operator on L2 (R), which we also denote by E. Now, let us observe that neither E E nor E D = (2)i)−1 ; (219) E G = −(2)i)−1 ∗ E − zR E − zR
are square integrable functions on L2 (R). This means that, although the Gamow vectors may be represented by square integrable functions, these functions are not in the domain of the operator Eˆ (i.e. the Hamiltonian). (iii) Formulas (219) acquire meaning in the RHS given by (211) and (212). If we use the duality formula E± (E)|F± = ± (E)|E F± ;
2 ∀± (E) ∈ S ∩ H± ;
2 × ∀F± ∈ (S ∩ H± )
(220)
2 × we extend the action of the operator E to the duals (S ∩H± ) and, then, (219) becomes meaningful.
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Now, we are in the position to give a de
="
1 ; E − zR∗
D
="
1 ; E − zR
(221)
where " is a complex constant to be <xed by the normalization condition ∞ dE G 2 D 2 2 = =" = "2 ) = 1 : (222) 2 2 −∞ (E − ER ) + (=2) √ Therefore, " = 1= ). Once we have chosen this normalization criterium, we can propose expressions for the mean value of the energy on the Gamow vectors:
G
|E|
G
and
D
|E|
D
:
(223)
Let us evaluate these values. Since E D (E) = E D (E), we have that ∞ 1 E E dE 1 ∞ 2 D D |E| = dE = : ∗ ) −∞ E − zR E − zR ) −∞ E −ER 2 + 1 =2
(224)
The change of variables x=
E − ER ; =2
(225)
transforms the last integral in (224) into ∞ x dx ER ∞ d x + : ) −∞ x2 + 1 2) −∞ x2 + 1
(226)
The
D
|E|
D
= ER
(227)
G
|E|
G
= ER :
(228)
and
Remark. (a) Observe that the integral in (224) does not converge although its Cauchy mean value does exists. Therefore, identities (227) and (228) are indeed Cauchy mean values. (b) We see that this de
D
|
G
= 0.
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4.4. Real average values: Berggren interpretation Berggren’s approach to the mean value of the Hamiltonian on a Gamow state [18] can be formulated in a manner which is very similar to Bohm’s. Following [33,34], we shall not use Hardy functions to construct the Gelfand triplets. Instead, we consider here another triplet ˜ ⊂ H ⊂ ˜× for which the space ˜× consists of tempered ultra-distributions [33,34,107]. See Section 3.9. One of the advantages of this presentation is that the Gamow vectors are normalized in the sense that the product f˜0 |f0 exists and is given by [33,34]: 28
E1 − ER E0 − ER 1 ˜ arctan − arctan : (230) f0 |f0 = 2 4) This normalization allows also to de
(232)
We can also de
· (E − ER )2 + 2 arctan
E1 − ER
!
1 − arctan
E0 − ER
!" :
In the limit E1 → +∞; E0 → −∞ the above equation yields =) P(E) = (E − ER )2 + 2
(233)
(234)
which is the Breit–Wigner form proposed by [20,18]. The coincidence between (232) for the mean value of the Gamow states and Bohm’s one, presented in the previous sub-section, comes from a re-interpretation of Berggren’s de
This normalization is not unique. In Ref. [13], Antoniou and Prigogine have proposed f˜0 |f0 = 1.
(236)
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Now, let A be an arbitrary observable. We can de
(237)
l;l
If we replace A by H we obtain, straightforwardly, the value ER for this average. Instead, Berggren de
(238)
which means that Berggren’s approximation coincides with Bohm’s to the
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By boldface letters, we denote three dimensional vectors. Four-dimensional vectors in Minkowski space are denoted by roman style letters. The products of two vectors in Minkowski space are characterized by a dot. In Minkowski space we use the metric (+− − −). For example, k·x=k0 x0 −kx. The Lorentz invariant measure in (239) is d k˜ =
d3 k ; (2))3 2!(k)
!(k) = (k2 + M 2 )1=2 :
(240)
The creation and annihilation operators in (239) satisfy the usual commutation relations: [a(k); a† (k )] = (2))3 2!(k) (k − k ) :
(241)
The Hamiltonian of the <eld ’(x) is given by † ˜ (k)a(k) : HM = d k!(k)a
(242)
We shall consider, in addition, a simple bilocal scalar <eld, (x ; q), with continuous mass spectrum. The notion of bilocal <eld was introduced long ago by Yukawa [115] and Markov [83] in their discussion of the extended elementary particles. It is used here in a diFerent context. The <eld depends also on an additional real variable q representing an internal degree of freedom. We shall start by considering the classical <eld and then quantize it. For simplicity we shall assume that the classical <eld (x ; q) is an even function of q. This means that (x ; −q) = (x ; q) :
(243)
For the mass operator M, we choose the simplest possible form: M2 = 4m2 −
92 : 9q2
(244)
The spectrum of the mass operator is [2m; ∞). The <eld Klein–Gordon equation:
(x ; q) satis<es the following generalized
( − M2 ) (x ; q) = 0 ; where form:
(245)
is the usual d’Alambert operator. The solution of Eq. (245) can be written in the following (x ; q) =
4
d k
∞
−∞
d@
cos q@ −ik x 2 e (k − 4m2 − @2 )B(k ; @) ; (2))4
(246)
where we took into account (243). The amplitude B(k ; @) is also an even function of @. Integrating the r.h.s. of (246) over k0 gives ∞ d 3 k cos @q (B∗ (k; @)eik ·x + B(k; @)e−ik ·x ) ; d@ (247) (x ; q) = (2))4 2E(k; @) −∞ where k = (E; k) and E(k; @) = [4m2 + @2 + k2 ]1=2 :
(248)
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We can change the variables in (247) so that E is the new independent variable instead of @: @ = (E 2 − k2 − 4m2 )1=2 ; then,
(x ; q) =
0
∞
d@ dE = ; E @
dE d 3 k cos @(k )q ∗ (B (k; E)eik ·x + B(k; E)e−ik ·x ) : (3))4 @(k )
(249)
(250)
Once we have the classical <eld written as in (250), we can make use of the standard quantization rules [19,106] to obtain the quantum <eld ∞ dE d 3 k cos @(k )q † [B (k; E)eik ·x + B(k; E)e−ik ·x ] : (x ; q) = (251) 4 @(k ) (2)) 0 where the creation B† (k; E) and annihilation operators B(k; E) satisfy the following commutation relations: [B(k; E); B† (k ; E )] = (2))4 @(k ) 4 (k − k ) : The Hamiltonian for the bilocal <eld (x ; q) is given by d 3 k dE EB† (k; E)B(k; E) : Hm = (2))4 @(k; E) We now introduce the quadratic interaction Hamiltonian as ∞ Hint = −$ d 3 x dq (x; q)f(q)’(x) ; −∞
(252)
(253)
(254)
where we assume that the even function f(q) is a Lorentz scalar and has the Fourier transform: f(q) = dy "(y) cos yq : (255) The function f(q) plays the same role as the form factor in the Friedrichs model. To avoid divergencies, we choose it so that it has a good asymptotic behavior. With this choice of the interaction, the total Hamiltonian becomes d 3 k dE EB† (k; E)B(k; E) P0 = (2))4 @(k; E) d 3 k dE $"(@(k; E)) d3 k † !(k)a (k)a(k) + + (2))3 2!(k) (2))3 2! @(k; E) ×(a(k) + a† (−k))(B† (k; E) + B(−k; E)) :
(256)
Note that the function "(·) in the third integral in (256) coincides with the function "(·) that appears in (255). The three momentum for the interaction <eld is given by d3 k k d 3 k dE † kB a† (k)a(k) : (k; E)B(k; E) + (257) P= (2))4 @(k; E) (2))3 2!(k)
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The analogy between the Hamiltonian (256) and the interacting Hamiltonian in the usual Friedrichs model is obvious. As a matter of fact, we are considering here an inAnite collection of Friedrichs models, each corresponding to a value of the momentum k. Our next step is to diagonalize the four momentum (256) and (257). This means that we are looking for creation, b† (E; k), and annihilation, b(E; k), operators such that the four momentum components P can be written as d 3 k dE k b† (E; k)b(E; k) ; (258) P = (2))4 @(E; k) where @ is given in (249). To achieve it, we write the eigenvalue equation [7]: [P ; b† (E; k)] = k b† (E; k) ;
(259)
where k = (E; k) and the P are given by (256) and (257). To solve (259), i.e., to obtain the creation operators b† (E; k), we make the following ansatz: † b (E; k) = dE (T (E; E ; k)B† (E ; k) + R(E; E ; k)B(E ; −k)) + t(E; k)a† (k) + r(E; k)a(−k) ;
(260) b† (E; k)
B† (E; k);
is a linear combination of which obviously means that we are assuming that † † B(E; k); a (k) and a(k). In order to obtain b (E; k), we insert (260) into (259) to obtain four equations in the undetermined amplitudes T (E; E ; k), R(E; E ; k), t(E; k) and r(E; k) [7]. The formal solution, for positive energies, of the eigenvalue equation (259) is [7] $ b† (E; k) = C
B† (E; k) + 2)$"(@(E; k))G(E; k)
$"(@(E ; k)) (E ; k) × dE @(E ; k)
B† (E; k) B(E; −k) − × E − E E +E
(E + !(k))a† (k) + (E − !(k))a(−k) : − 2!(k)
(261)
Let us analyze this result. The functions !(k), "(·) and @(E; k) have been de
=(E; k) =
∞
E0
dE 2E
<(E ; k) ; E2 − E2
(263)
and E0 = (4m2 + k2 )1=2 . Therefore, G(E; k) depends on <(E; k) = 2)
$2 "2 (@(E; k)) ; @(E; k)
(264)
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through =(E; k), since <(E; k) is given by Eq. (264). Note that <(E; k) depends on "(·) and hence on the form factor f(q). The properties of G(E; k) and =(E; k) are discussed in [7]. Formulas (233) and (234) are also valid for complex values of E and the Green function G(E; k) is analytic in E 2 with a branch cut in [E02 ; ∞). The Green function admits analytic continuations from above to below G(E + i0; k), i.e., from the upper to the lower half plane, and from below to above G(E − i0; k) through the cut. If M ¿ 2m, G(E + i0; k) has a pole at E 2 = k2 + c2 and G(E − i0; k) has a pole at the conjugate point E 2 = k2 + c∗2 , where: c2 = 2 − i ;
(265)
and and are real positive numbers which depend on the form factor f(q) [7]. The equation for the complex pole c2 is: 2 <(E ; k) dE 2 =0 : (266) !2 (k) − E 2 − E − E2 Compare this equation with (140). Eq. (266) can be written as 2 <(E ; k) dE 2 − i)<(E; k) = 0 ; (267) M 2 − (E 2 − k2 ) − PV E − E2 where PV stands for the Cauchy principal value. For small values of the coupling constant $, we can omit the integral term and the result is √ 2 2 2 2 2 2 2 2 ["( M − 4m )] √ c = M − i)<(E; k)|E 2 −k2 =M 2 = M − 2i) $ : (268) M 2 − 4m2 From (268), we note that the pole appears if M ¿ 2m only. Let us go back to the formal solution (261). Along this solution of (259) for positive energies, there exists another formal solution for negative energies which is given by [7] $"(@(E; k)) b(E; k) = C B(E; k) + 2)$"(@(E; k))G(E; k) dE @(E; k)
(E + !(k))a(k) + (E − !(k))a+ (−k) B(E; k) B+ (E; −k) − : (269) × − E − E E + E 2!(k) This is the annihilation operator. The signs of the boundary conditions G(E ± i0; k) that provide both analytic continuations of the Green function, coincide with the signs in the denominator of the Lippmann–Schwinger equations [90]. These denominators have the sign plus for the incoming and the sign minus for the outgoing equation respectively. Therefore, the solution of the eigenvalue equation (230) with incoming boundary conditions is obtained by replacing G(E; k) by G(E + i0; k) in (261) and (269). The incoming operators are $"(@(E ; k)) † † bin (E; k) = B (E; k) + 2)$"(@(E; k))G(E + i0; k) dE @(E ; k)
† B(E ; −k) (E + !(k))a† (k) + (E − !(k))a(−k) B (E; k) − − (270) × E − E − i0 E + E 2!(k)
O. Civitarese, M. Gadella / Physics Reports 396 (2004) 41 – 113
and
bin (E; k) = B(E; k) + 2)$"(@(E; k))G(E − i0; k) ×
B† (E ; −k) B(E ; k) − E − E + i0 E + E
−
dE
$"(@(E ; k)) @(E ; k)
(E + !(k))a(k) + (E − !(k))a† (−k) 2!(k)
101
: (271)
Here, E ¿ E0 (k) = (4m2 + k2 )1=2 and we have chosen the constant C = 1 in (268). The outgoing operators, b†out (E; k) and bout (E; k), are obtained with the change +i0 → −i0 in (270) and (271), respectively. These operators have the following commutation relations [7]: [bin (E; k); b†in (E ; k )] = [bout (E; k); b†out (E ; k)] ; = (2))4 @(E; k) (E − E ) 3 (k − k ) ; G(E + i0; k) ; G(E − i0; k) G(E − i0; k) : [bin (E; k); b†out (E ; k )] = (2))4 @(E; k) (E − E ) 3 (k − k ) G(E + i0; k) [bout (E; k); b†in (E ; k )] = (2))4 @(E; k) (E − E ) 3 (k − k )
(272)
All other commutators vanish. These operators are the solutions to the diagonalization problem in the sense of Eq. (258) holds, i.e., d 3 k dE P = k b†in (E; k)b in (E; k) : (273) (2))4 @(E; k) out out Now, we are ready to obtain the Gamow vectors for the unstable <eld. 5.2. Resonances and Gamow vectors Let us start with the following remark [7]: Let us consider the vacuum state before the interaction, is switched on. It is characterized by the following equations: B(E; k)|0 = 0
and
a(k)|0 = 0 :
(274)
This is not the vacuum for the interacting <eld, since [7] b(E; k)|0 = 0 ;
(275)
i.e. there arise a new vacuum state which will be a superposition of states with an arbitrary number of particles of B and a-types. The new vacuum C can be obtained from the old one by means of a transformation of the type |C = eV |0 ;
(276)
where V is a quadratic functional of creation operators B† (E; k) and a† (k). It results that bin (E; k)|C = 0 :
(277)
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Due to the form (269) of b†in (E; k), the state +in (E; k) = b†in (E; k)|C
(278)
has a pole as a function of E when we make the analytic continuation from above to below. This pole has been found to be at the point [7] zR = (k + 2 − i)1=2 ;
(279)
and the width and therefore the lifetime depend on k. The state +in (E; k) can be written in a neighborhood of zR as 1 +in (E; k) = ’G (k) + regular part : (280) E − zR in The residue ’Gin (k) has the following properties [7]: (i) It is an eigenvector of the total Hamiltonian P0 with eigenvalue zR , P0 ’Gin (k) = zR ’Gin (k) :
(281)
Therefore ’Gin (k) is the Gamow vector associated to the resonance with resonance pole at zR . (ii) As a consequence of (i), ’Gin (k) decays exponentially: e−itP0 ’Gin (k) = e−itzR ’Gin (k) :
(282) ’Gin (k)
cannot be a normalizable vector in a Hilbert (Fock) space. Properties (i) and (ii) show that As happens in the non-relativistic case, we need to rig the Fock space in order to
(283)
where C is the set of complex numbers and ⊕ and ⊗ means direct sum and tensor product, respectively. This Fock algebra admits a topology which is obtained from the topology on [54,7] and that is stronger (has more open sets) than the topology of the Hilbert space on the Fock space H⊗ = C ⊕ H ⊕ (H ⊗ H) ⊕ (H ⊗ H ⊗ H) : : : : (iii) Then, the dual F× of F is obtained. We have the RHS F ⊂ H⊗ ⊂ F ×
(284)
In our particular case, we construct F as follows:
(285)
O. Civitarese, M. Gadella / Physics Reports 396 (2004) 41 – 113
103
(iii) S(R3 ) represents the space of all complex functions on the three dimensional real space R3 having the same properties as S in (ii). Usually, we call S and S(R3 ) the one-dimensional and the three-dimensional Schwartz space, respectively. (iv) The topology on is then obtained from the topologies on the Schwartz spaces. This is a technical matter [93], but this topology makes both and F metric spaces, i.e., the topology can be obtained from a metric. A typical function of (H2− ∩S)⊗S(R3 ) is of the form ’(E; k). For a <xed value of the momentum k, this is a function of the energy and belongs to H2− ∩ S. For a <xed value of the energy E, ’(E; k) ∈ S(R3 ). Thus, if ’(E; k) is in the space (H2− ∩ S) ⊗ S(R3 ), then, ’(E; k) represents an one particle state. This state is equally well represented by the bra vector d 3 k dE ’| = (286) C|bout (E; k)’(E; k) : (2))4 @(E; k) The mapping given by d 3 k dE ’(E; k) → C|bout (E; k)’(E; k) ; (2))4 @(E; k) is one to one and, hence, the space is the space of vectors of the form (286). Once we have , we have the triplet (284). The action of the Gamow vector ’Gin (k) on the arbitrary vector of given by (286) is c G ’|’in (k) = ’(E; k) ; (287) ,− (−E) E=zR
where [7] ,± (E) are, respectively, the solutions of the equations given by ,± (E; k),± (−E; k) = G± (E; k) :
(288)
Note that the Gamow vector is a functional that vanishes on the vacuum and on the space of two or more particles. This model is exactly solvable because the interaction between <elds is quadratic. Due to the presence of a mass spectrum in the second <eld, the interaction contains non-trivial features like those studied here. Relativistic Gamow vectors can be obtained approximately when the interaction between <elds is not quadratic [6]. 6. Conclusions In this work we have presented the essentials of the mathematical and physical interpretation of resonances. We have shown that the mathematical diEculties arising from the probabilistic interpretation of resonances, in the context of ordinary quantum mechanics, are easily removed by considering the resonances as vectors belonging to rigged Hilbert spaces. The chain of mathematical steps needed to reach this result has been discussed by: (i) de
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de
Acknowledgements We want to express our gratitude to our colleagues, who have contributed to the development of the <eld and from whom we have had the privilege of learning the techniques presented in this work. Since these is a very long list, indeed, we warmly and gratefully thanks all of them in the persons of Profs. J.P. Antoine, I.E. Antoniou, A. Bohm, C.G. Bollini, M. Castagnino, H.D. Doebner, R. Laura, R.J. Liotta, A. MondragTon, Yu. Melnikov, T. Petroski, the late I. Prigogine, G. Pronko, Z. Suchanecki, and Drs R. de la Madrid and S. Wickramasekara. This work has been supported by The Junta de Castilla y Leon, Project VA 085/02, the FEDER-Spanish Ministry of Science and Technology Projects DGI BMF 2002-0200 and DGI BMF2002-3773 and by the CONICET of Argentina.
Appendix A At the end of Section 2.6, we have introduced the Gelfand–Maurin spectral theorem. Let us go back to formula (32) and omit the arbitrary vector ’ ∈ and take f(A) ≡ I , the identity operator. Then, (32) leads to = |$$| d$ : (A.1) (A)
The function ($) = $| = |$∗ is the wave function for the pure state in the A-representation. If B is another observable, the Gelfand–Maurin theorem for B implies the existence of a new RHS
O. Civitarese, M. Gadella / Physics Reports 396 (2004) 41 – 113
⊂ H ⊂ × such that B is continuous on and × and for all f(B) = f(b)|bb| db ;
∈ , we have
(B)
where b ∈ (B) and B|b = b|b, |b ∈ × and = |bb| db ;
(B)
(A)
(A.2)
(A.3)
(B)
where (b) = b| = |b∗ is the wave function of we use (A.3) in (A.1) to conclude that = |bb|$$| db d$ ;
105
in the B-representation. If further,
∈ , (A.4)
If A = B, obviously b|$ = ($ − b). In the general case, A = B and the kernel b|$ obviously depend on A and B. Multiplying (A.4) to the left by the bra $|, we obtain $| = $|bb| db : (A.5) (B)
which gives a direct relation between the wave functions in the B representation and in the A representation. This formula is invertible. If we multiply (A.1) to the left by the bra b|, we obtain that b| = b|$$| d$ : (A.6) (A)
A typical example is A = Q and B = P the one-dimensional position and momentum operators, respectively. We know that these two operators have purely continuous spectrum covering the real axis. Then, Q|x=x|x, ∀x ∈ R and P|p=p|p, ∀p ∈ R. The kets |x and |p are antilinear continuous functionals on the Schwartz space S [26]. We also know that the Fourier transform changes the wave function for a pure state from the position representation into the momentum representation. Then, we have ∞ x| = x|pp| dp ; (A.7) −∞
which implies that 1 x|p = √ e−ipx : 2)
(A.8)
The Gelfand–Maurin theorem can be extended to a set of commuting observables. Here, we shall focus our attention to the case in which H; L2 and Lz are a set of commuting observables, where ˜L ≡ (Lx ; Ly ; Lz ) represents the orbital angular momentum. Then, there exists a RHS ⊂ H ⊂ ×
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such that the eigenvalue equations H |E; l; lz = E|E; l; lz ; L2 |E; l; lz = l(l + 1)|E; l; lz ; Lz |E; l; lz = lz |E; l; lz
(A.9)
have solution in the antidual of the three dimensional Schwartz space S(R3 ). Another possibility for a set of commuting observables is the triplet of components of the three-dimensional position, {Qx ; Qy ; Qz }, or momentum operator, {Px ; Py ; Pz }. In these two latter cases, the eigenvalue equations Qx |x = x|x;
Qy |x = y|x;
Px |p = px |p;
Py |p = py |p;
Qz |x = z|x ; Pz |p = pz |p
(A.10) (A.11)
with x = (x; y; z) and p = (px ; py ; pz ), also have solutions in the antidual of S(R3 ). Note that if (x) ∈ S(R3 ), then in the H; L2 ; Lz representation, the wave function for the state (x) is given by E; l; lz |xx| d 3 x (A.12) E; l; lz | = R3
with x| = (x). Eq. (A.12) is invertible: l ∞ x|E; l; lz E; l; lz | d 3 x x| = lz =−l l=0
R3
(A.13)
Appendix B Along this paper, we have de
O. Civitarese, M. Gadella / Physics Reports 396 (2004) 41 – 113
107
Let (x) be a pure quantum state in S(R3 ). Using the Gelfand–Maurin theorem for a system of commuting observables, we have that l ∞ |E; l; lz E; l; lz | dE : (B.2) = lz =−l l=0
R3
If we apply H to (B.2), we have that l ∞ H |E; l; lz E; l; lz | dE : H = lz =−l l=0
Then, x|H =
R3
l ∞ lz =−l l=0
R3
x|H |E; l; lz E; l; lz | dE :
(B.3)
(B.4)
As x|H is the wave function in coordinate representation of H , the SchrKodinger equation yields ˝2 x|H = − F + V (x) x| ; (B.5) 2m where F is the three dimensional Laplacian. If we use (B.13) in (B.5), we have that l ∞ ˝2 x|H = − F + V (x) x|E; l; lz E; l; lz | dE ; 2m R3
(B.6)
lz =−l l=0
where the brackets in (B.6) shows that only the kernel x|E; l; lz depends on the variable x. Comparing (B.6)–(B.4) and taking into account that E; l; lz | is arbitrary, we conclude that ˝2 F + V (x) x|E; l; lz : x|H |E; l; lz = − (B.7) 2m Since V (x) = V (r) is spherically symmetric, we can use spherical coordinates x = (r; *; ). Then, the SchrKodinger equation (B.7) in spherical coordinates reads ˝2 l(l + 1) ˝2 1 9 2 9 r; *; |H |E; l; lz = − r + + V (r) r; *; |E; l; lz 2m r 2 9r 9r 2mr 2 = Er; *; |E; l; lz :
(B.8)
If we separate the radial and angular dependence, we obtain 1 r; *; |E; l; lz = r|El *; |l; lz = Gl (r; E)Yl; lz (*; ) ; (B.9) r where Yl; lz (*; ) are the spherical harmonics. Connecting (B.8) and (B.9), we obtain for the radial part: ˝2 d 2 ˝2 l(l + 1) − + + V (r) Gl (r; E) = EGl (r; E) : (B.10) 2m dr 2 2mr 2
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Therefore, Gl (r; E) is the solution to the radial part of the SchrKodinger equation with orbital angular momentum equal to l. If we choose l = 0 and write G(r; E) = G0 (r; E), we get −
˝2 d 2 G(r; E) + V (r)G(r; E) = EG(r; E) : 2m dr 2
(B.11)
The solutions of (B.11) are the Dirac kets, with <xed E, in the energy representation. The general solution of (B.11) is given by " eikr + #1 e−ikr ; 0¡r¡a ; 1 G(r; E) = "2 eiQr + #2 eiQr ; (B.12) a¡r¡b ; F1 eikr + F2 e−ikr ; b ¡ r ¡ ∞ ; where "1; 2 ; #1; 2 and F1; 2 are arbitrary constants that depend on E and which we shall <x by using boundary conditions and # # 2m 2m E; Q = (E − V0 ) : (B.13) k= 2 ˝ ˝2 Taking into account the form of the potential, the boundary conditions are [79,80]: G(0; E) = 0 ;
(B.14)
i.e., the wave function vanishes at the origin, and G(a − 0; E) = G(a + 0; E) ; G (a − 0; E) = G (a + 0; E) ; G(b − 0; E) = G(b + 0; E) ; G (b − 0; E) = G (b + 0; E) ;
(B.15)
which implies the continuity of the function and its
˝2 d 2 G(r; z0 ) + V (r)G(r; z0 ) = z0 G(r; z0 ) : 2m dr 2
(B.16)
O. Civitarese, M. Gadella / Physics Reports 396 (2004) 41 – 113
The solution G(r; z0 ) of (B.16) has the form (B.12) with # # 2m 2m k = k0 = z0 ; Q = Q 0 = (z0 − V0 ) : 2 ˝ ˝2 We impose to the solutions of (B.16) the boundary conditions: G(0; z0 ) = 0 ;
109
(B.17)
(B.18)
i.e., the wave function vanishes at the origin; G(a − 0; z0 ) = G(a + 0; z0 ) ; G (a − 0; z0 ) = G (a + 0; z0 ) ; G(b − 0; z0 ) = G(b + 0; z0 ) ; G (b − 0; z0 ) = G (b + 0; z0 ) ;
(B.19)
which implies the continuity of the function G(r; z0 ) and its
for k → ∞ :
(B.20)
See also [71]. The meaning of this boundary condition is the following: take the latter formula of (B.19) and note that F1 and F2 must depend on k and, therefore, on z. Then, the S-matrix is written in the form [89,64] F1 (k) S(k) = − : (B.21) F2 (k) Resonances are placed at the poles of S(k) for Im k ¡ 0, Real k = 0 and this corresponds to the zeroes of F2 (k) with identical properties. Thus, if a resonance is placed at z0 = (˝2 k02 )=(2m), then, F2 (k0 ) = 0 and reciprocally. This justi<es characterization (B.20) for resonance states. Thus, a solution of the eigenvalue equation HG(r; z) = zG(r; z) represents a Gamow vector for the square well potential (B.1) and l = 0 if and only if it ful
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Multiplying the formal solution in Eq. (25) by Vˆ we have that 1 Tˆ 2B (E + )|k = Vˆ |k + Vˆ Tˆ 2B (E + )|k : + E − Hˆ 0
(28)
Since this equation holds for an arbitrary plane wave |k and because these plane waves form a complete set of states we have the following operator equation for the two-body T -matrix 1 Tˆ 2B (z) = Vˆ + Vˆ (29) Tˆ 2B (z) : z − Hˆ 0 This equation is called the Lippmann–Schwinger equation and from its solution we are able to determine the scattering properties of the potential Vˆ . To see this we 8rst note that from the de8nition of the T -matrix in Eq. (27), together with Eq. (26), it follows immediately that m f(k ; k) = − k |Tˆ 2B (2 +k )|k : (30) 4˝2 Therefore, we indeed see that the two-body T -matrix completely determines the scattering amplitude. The Lippmann–Schwinger equation for the two-body T -matrix can be solved in perturbation theory in the potential. This results in the so-called Born series given by Tˆ 2B (z) = Vˆ + Vˆ Gˆ 0 (z)Vˆ + Vˆ Gˆ 0 (z)Vˆ Gˆ 0 (z)Vˆ + · · · ;
(31)
where Gˆ 0 (z) =
1 ; z − Hˆ 0
(32)
is the noninteracting propagator of the atoms. By using, instead of the true interatomic interaction potential, a pseudopotential of the form 4a˝2 (x − x ) ; V (x − x ) = (33) m the 8rst term in the Born series immediately yields the correct result for the scattering amplitude at low energies and momenta, given in Eq. (10). Such a pseudopotential should therefore not be used to calculate higher-order terms in the Born series, but should be used only in 8rst-order perturbation theory. The poles of the T -matrix in the complex-energy plane correspond to bound states of the potential. To see this we note that the formal solution of the Lippmann–Schwinger equation is given by 1 ˆ Tˆ 2B (z) = Vˆ + Vˆ V : (34) z − Hˆ After insertion of the complete set of eigenstates | & of Hˆ = Hˆ 0 + Vˆ we have | & & | Tˆ 2B (z) = Vˆ + Vˆ Vˆ ; z −
& &
(35)
where the summation over & is discrete for the bound-state energies & ¡ 0, and represents an integration for positive energies that correspond to scattering solutions of the SchrPodinger equation, so explicitly we have that | | dk ˆ | k(+) k(+) | ˆ Vˆ Vˆ + V V : (36) Tˆ 2B (z) = Vˆ + z − (2)3 z − 2 k
R.A. Duine, H.T.C. Stoof / Physics Reports 396 (2004) 115 – 195
127
From this equation we clearly see that the two-body T -matrix has poles in the complex-energy plane, corresponding to the bound states of the potential. In addition, the T -matrix contains a branch cut on the positive real axis due to the continuum of scattering states. As an example, we note that for s-wave scattering the T -matrix T 2B (2 k+ ) ≡ k |Tˆ 2B (2 k+ )|k is independent of the angle between k and k. From the relation between the T -matrix and the scattering amplitude, and the expression for the latter in terms of the phase shift, we have for low positive energies 1 4˝2 m (mE=˝2 ) cot(( mE=˝2 )) − i mE=˝2
1 4a˝2 ; m 1 + ia (mE=˝2 ) − (are4 mE=2˝2 )
T 2B (E + ) = −
(37)
where we made use of the expansion in Eq. (17). From this result we deduce by analytic continuation that
1 4a˝2 2B T (z) : (38) m 1 − a −(mz=˝2 ) − (are4 mz=2˝2 ) Clearly, for large and positive scattering length the T -matrix has a pole at negative energy Em = −˝2 =ma2 , in complete agreement with our previous discussions. Summarizing, we have found that the scattering length of an attractive potential well can have any value and depends strongly on the energy of the weakliest bound state in the potential. In principle therefore, if we have experimental access to the energy di4erence of this bound state and the continuum threshold we are able to experimentally alter the scattering length and thereby the e4ective interactions of the atoms. In the single-channel case this is basically impossible to achieve. In a multichannel system, however, the energy di4erence is experimentally accessible, which makes the low-energy e4ective interactions between the atoms tunable. In the next section we discuss this situation. 2.3. Example of a Feshbach resonance We consider now the situation of atom–atom scattering where the atoms have two internal states [96]. These states correspond, roughly speaking, to the eigenstates of the spin operator S of the valence electron of the alkali atoms. The e4ective interaction potential between the atoms depends on the state of the valence electrons of the colliding atoms. If these form a singlet the electrons are, in principle, allowed to be on top of each other. For a triplet this is forbidden. Hence, the singlet potential is generally much deeper than the triplet potential. Of course, in reality the atom also has a nucleus with spin I which interacts with the spin of the electron via the hyper8ne interaction ahf Vhf = 2 I · S (39) ˝ with ahf the hyper8ne constant. The hyper8ne interaction couples the singlet and triplet states. Moreover, in the presence of a magnetic 8eld the di4erent internal states of the atoms have
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R.A. Duine, H.T.C. Stoof / Physics Reports 396 (2004) 115 – 195 V (r)+ε+>
Bound state
∆µΒ
VS (r)+∆µΒ VT (r) V (r)+ε−>
R
Fig. 5. Feshbach resonance in a two-channel system with square-well interaction potentials. The triplet potential VT (r) is indicated by the thick dashed line. The singlet potential that contains the bound state responsible for the Feshbach resonance is indicated by the thin dashed line. Due to the Zeeman interaction with the magnetic 8eld, the energy di4erence between the singlet and triplet is equal to T'B. The interactions in the open and closed hyper8ne channels are indicated by V↑↑ (r) and V↓↓ (r), respectively.
a di4erent Zeeman shift. In an experiment with magnetically trapped gases, the energy di4erence between these states is therefore experimentally accessible. Putting these results together, we can write down the SchPodinger equation that models the above physics ˝ 2 ∇2 Vhf − m + VT (r) − E T (r) =0 : (40) ˝ 2 ∇2 S (r) + T'B + VS (r) − E Vhf − m Here, VT (r) and VS (r) are the interaction potentials of atoms with internal state |T and |S , respectively, and T'B is their di4erence in Zeeman energy due to the interaction with the magnetic 8eld B, with T' the di4erence in magnetic moment. In agreement with the above remarks, |T is referred to as the triplet channel, whereas |S is referred to as the singlet channel. The potentials VT (r) and VS (r) are the triplet and singlet interaction potentials, respectively. As a speci8c example, we use for both interaction potentials again square well potentials, −VT; S if r ¡ R ; VT; S (r) = (41) 0 if r ¿ R ; where VT; S ¿ 0. For convenience we have taken the range the same for both potentials. Furthermore, we assume that the potentials are such that VT ¡ VS and that VS is just deep enough such that it contains exactly one bound state. Finally, we assume that 0 ¡ Vhf VT ; VS ; T'B. The potentials are shown in Fig. 5. To discuss the scattering properties of the atoms, we have to diagonalize the hamiltonian for r ¿ R, in order to determine the incoming channels, which are superpositions of the triplet and singlet states |T and |S . Since the kinetic energy operator is diagonal in the internal space of the
R.A. Duine, H.T.C. Stoof / Physics Reports 396 (2004) 115 – 195
atoms, we have to 8nd the eigenvalues of the hamiltonian 0 Vhf ¿ H = : Vhf T'B
129
(42)
These are given by
T'B 1 ± (T'B)2 + (2Vhf )2 : 2 2 The hamiltonian H ¿ is diagonalized by the matrix cos sin Q() = ; −sin cos ¿ =
±
according to Q( ¿ )H ¿ Q−1 ( ¿ ) =
¿
−
0
0
¿
+
(43)
(44)
;
(45)
which determines tan ¿ = −2Vhf =T'B. We de8ne now the hyper8ne states |↑↑ and |↓↓ according to | ↑↑ |T = Q( ¿ ) ; (46) | ↓↓ |S which asymptotically represent the scattering channels. In this basis the SchrPodinger equation for all r reads ˝ 2 ∇2 V↑↓ (r) − m + V↑↑ (r) − E ↑↑ (r) =0 ; (47) ˝ 2 ∇2 ↓↓ (r) ¿ ¿ − V↑↓ (r) + + − − + V↓↓ (r) − E m ¿ and we have de8ned the potentials according to where the energy E is measured with respect to − 0 V↑↑ (r) V↑↓ (r) VT (r) (48) = Q( ¿ ) Q−1 ( ¿ ) : V↑↓ (r) V↓↓ (r) 0 VS (r) Since all these potentials vanish for r ¿ R we can study scattering of atoms in the states |↑↑ ¿ ¿ T'B and − 0. and |↓↓ . Because the hyper8ne interaction Vhf is small we have that + Moreover, for the experiments with magnetically trapped gases we always have that T'BkB T where kB is Boltzmann’s constant and is T the temperature. This means that in a realistic atomic gas, in which the states |↑↑ and |↓↓ are available, there are in equilibrium almost no atoms that scatter via the latter state. Because of this, the e4ects of the interactions of the atoms will be determined by the scattering amplitude in the state |↑↑ . If two atoms scatter in this channel with energy E kB T T'B they cannot come out in the other channel because of energy conservation. Therefore, the indices ↑↑ refers to an open channel, whereas ↓↓ is associated with a closed channel. The situation is further clari8ed in Fig. 5.
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To calculate the s-wave scattering length in the open channel we have to solve the SchrPodinger equation. In the region r ¿ R the solution is of the from ¿ ikr u↑↑ (r) Ce + De−ikr ; (49) = ¿ (r) u↓↓ Fe− r ¿ ¿ − − )=˝2 − k 2 and, because we have used the same notation as in Eq. (12), the where = m( + s-wave phase shift is again determined by Eq. (13). In the region r ¡ R the solutions are of the form ¡ ¡ ¡ u↑↑ (r) A(eik↑↑ r − e−ik↑↑ r ) ; (50) = ¡ ¡ ¡ (r) u↓↓ B(eik↓↓ r − e−ik↓↓ r ) where ¡ k↑↑ = ¡ = k↓↓
¿ ¡ m( − − − )=˝2 + k 2 ; ¿ ¡ m( − − + )=˝2 + k 2 ;
(51)
and ¡
± =
T'B − VT − VS 1 (VS − VT − T'B)2 + (2Vhf )2 : ∓ 2 2
are the eigenvalues of the matrix V −V T hf H¡ = : Vhf T'B − VS
(52)
(53)
In order to determine the phase shift we have to join the solution for r ¡ R and r ¿ R smoothly. This is done most easily by transforming to the singlet-triplet basis {|T ; |S } since this basis is independent of r. Demanding the solution to be continuously di4erentiable leads to the equations ¡ ¿ u↑↑ (R) u↑↑ (R) Q−1 ( ¡ ) = Q−1 ( ¿ ) ; ¡ ¿ (R) (R) u↓↓ u↓↓ 9 −1 ¡ Q ( ) 9r
¿ ¡ (r) u↑↑ u↑↑ (r) 9 −1 ¿ Q ( ) = ; ¡ ¿ (r) r=R 9r (r) r=R u↓↓ u↓↓
(54)
where tan ¡ = 2Vhf =(VS − VT − T'B). These four equations determine the coeKcients A; B; C; D and F up to a normalization factor, and therefore also the phase shift and the scattering length. Although it is possible to 8nd an analytical expression for the scattering length as a function of the magnetic 8eld, the resulting expression is rather formidable and is omitted here. The result for the scattering length is shown in Fig. 6, for VS = 10˝2 =mR2 , VT = ˝2 =mR2 and Vhf = 0:1˝2 =mR2 , as a function
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131
2
1
a/R
0
-1
-2
-3 4.4
4.45
4.5
4.55
4.6
4.65
4.7
4.75
4.8
4.85
4.9
∆µB/( h2/mR2)
Fig. 6. Scattering length for two coupled square-well potentials as a function of T'B. The depth of the triplet and singlet channel potentials is VT = ˝2 =mR2 and VS = 10˝2 =mR2 , respectively. The hyper8ne coupling is Vhf = 0:1˝2 =mR2 . The dotted line shows the background scattering length abg .
of T'B. The resonant behavior is due to the bound state of the singlet potential VS (r). Indeed, solving the equation for the binding energy in Eq. (21) with V0 = −VS we 8nd that |Em | 4:62˝2 =mR2 , which is approximately the position of the resonance in Fig. 6. The di4erence is due to the fact that the hyper8ne interaction leads to a shift in the position of the resonance with respect to Em . The magnetic-8eld dependence of the scattering length near a Feshbach resonance is characterized experimentally by a width TB and position B0 according to TB : (55) a(B) = abg 1 − B − B0 This explicitly shows that the scattering length, and therefore the magnitude of the e4ective interatomic interaction, may be altered to any value by tuning the magnetic 8eld. The o4-resonant background scattering length is denoted by abg and is, in our example, approximately equal to the scattering length of the triplet potential VT (r). Using the expression for the scattering length of a square well in Eq. (18) for = 1, we 8nd that abg −0:56R. Furthermore, we have for our example that the position of the resonance is given by B0 4:64˝2 =mT'R2 and that the width is equal to TB −0:05˝2 =mT'R2 . Next, we calculate the energy of the molecular state for the coupled-channel case which is found by solving Eq. (47) for negative energy. In particular, we are interested in its dependence on the magnetic 8eld. In the absence of the hyper8ne coupling between the open and closed channel we simply have that m (B) = Em + T'B. Here, Em is the energy of the bound state responsible for the Feshbach resonance, that is determined by solving the single-channel SchPodinger equation for the singlet potential. This bound-state energy as a function of the magnetic 8eld is shown in Fig. 7 by the dashed line. A nonzero hyper8ne coupling drastically changes this result. For our example the bound-state energy is easily calculated. The result is shown by the solid line in Fig. 7 for the same parameters as before. Clearly, close to the resonance the dependence of the bound-state energy on the magnetic 8eld is no longer linear, as the inset of Fig 7 shows. Instead, it turns out to be
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0
εm/(h− 2 /mR2 )
-0.005 0 -1e-05
-0.01
-2e-05 -3e-05 -4e-05
-0.015
-5e-05 -6e-05 4.63685 -0.02 4.615
4.62
4.625
4.63
4.63702 4.635
4.64
∆µB/( h−2/mR2)
Fig. 7. Bound-state energy of the molecular state near a Feshbach resonance for two coupled square-well interaction potentials. The solid line and the inset show the result for Vhf = 0:1˝2 =mR2 . The dashed line corresponds to Vhf = 0. The other parameters are the same as in Fig. 6.
quadratic. Moreover, the magnetic 8eld B0 where the bound-state energy is equal to zero is shifted with respected to the case where Vhf = 0. It is at this shifted magnetic 8eld that the resonance is observed experimentally. Moreover, for magnetic 8elds larger than B0 there no longer exists a bound state and the molecule now decays into two free atoms due to the hyper8ne coupling, because its energy is above the two-atom continuum threshold. Close to resonance the energy of the molecular state turns out to be related to the scattering length by
m (B) = −
˝2 ; m[a(B)]2
(56)
as in the single-channel case. As we will see in the next sections, the reason for this is that close to resonance the e4ective two-body T -matrix again has a pole at the energy in Eq. (56). This important result will be proven analytically in Section 4. First, we derive a description of the Feshbach resonance in terms of coupled atomic and molecular quantum 8elds. 3. Many-body theory for Feshbach-resonant interactions In this section we derive the e4ective quantum 8eld theory that o4ers a description of Feshbachresonant interactions in terms of an atom–molecule hamiltonian. We start from a microscopic atomic hamiltonian that involves atoms with two internal states, i.e., we consider a situation with an open and a closed channel that are coupled by the exchange interaction. The 8rst step is to introduce a quantum 8eld that describes the bound state in the closed channel, which is responsible for the Feshbach resonance. This is achieved using functional techniques by a so-called Hubbard– Stratonovich transformation and is described in detail in Section 3.1. This section is somewhat
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133
technical and may be omitted in a 8rst reading of this paper. The most important result is a bare atom–molecule quantum 8eld theory that is presented in Section 3.2. In Section 3.3 we subsequently dress the coupling constants of this bare atom–molecule theory with ladder diagrams, to arrive at the desired e4ective quantum 8eld theory that includes all two-atom physics exactly. The Heisenberg equations of motion of this e4ective 8eld theory are presented in Section 3.4. 3.1. Bare atom–molecule theory Without loss of generality we can consider the simplest situation in which a Feshbach resonance arises, i.e., we consider a homogeneous gas of identical atoms in a box of volume V . These atoms have two internal states, denoted by | ↑ and | ↓ , that are described by the 8elds *↑ (x; +) and *↓ (x; +), respectively. The atoms in these two states interact via the potentials V↑↑ (x − x ) and V↓↓ (x − x ), respectively. The state |↓ has an energy T'B=2 with respect to the state |↑ due to the Zeeman interaction with the magnetic 8eld B. The coupling between the two states, which from the point of view of atomic physics is due to the di4erence in singlet and triplet interactions, is denoted by V↑↓ (x − x ). Putting everything together we write the grand-canonical partition function for the gas as a path integral given by [97–99] 1 ∗ ∗ ∗ ∗ Zgr = d[*↑ ]d[*↑ ]d[*↓ ]d[*↓ ] exp − S[*↑ ; *↑ ; *↓ ; *↓ ] : (57) ˝ Since we are dealing with bosons, the integration is over all 8elds that are periodic on the imaginarytime axis ranging from zero to ˝-, with ˝ Planck’s constant and -=1=kB T the inverse thermal energy. The Euclidian action is given by ˝- 9 9 S[*∗↑ ; *↑ ; *∗↓ ; *↓ ] = d+ dx *∗↑ (x; +)˝ *↑ (x; +) + *∗↓ (x; +)˝ *↓ (x; +) 9+ 9+ 0 ∗ ∗ +H [*↑ ; *↑ ; *↓ ; *↓ ] ; (58) with the grand-canonical hamiltonian functional given by H [*∗↑ ; *↑ ; *∗↓ ; *↓ ] 2 2 1 ˝∇ ∗ −'+ dx *↑ (x ; +)V↑↑ (x − x )*↑ (x ; +) *↑ (x; +) = − 2m 2 2 2 T'B ˝∇ ∗ + −' + dx*↓ (x; +) − 2m 2 1 + dx *∗↓ (x ; +)V↓↓ (x − x )*↓ (x ; +) *↓ (x; +) 2 1 dx dx *∗↑ (x; +)*∗↑ (x ; +)V↑↓ (x − x )*↓ (x ; +)*↓ (x; +) + c:c: ; + 2
dx*∗↑ (x; +)
(59)
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∆µB open channel
Fig. 8. Illustration of a Feshbach resonance. The upper potential curve corresponds to the closed-channel interaction potential V↓↓ (x − x ) that contains the bound state responsible for the Feshbach resonance, indicated by the dashed line. The lower potential curve corresponds to the open-channel interaction potential V↑↑ (x − x ).
where ' is the chemical potential of the atoms. Note that this hamiltonian functional is the grandcanonical version of the hamiltonian in Eq. (47). The indices ↑ and ↓ now refer again to single-particle states, and the two-particle hyper8ne states are denoted by |↑↑ and |↓↓ , respectively. The closedchannel potential is assumed again to contain the bound state responsible for the Feshbach resonance, as illustrated in Fig. 8. As a 8rst step towards the introduction of the molecular 8eld that describes the center-of-mass motion of this bound state, we introduce the complex pairing 8eld /(x; x ; +) and rewrite the interaction in the closed channel as a Gaussian functional integral over this 8eld, given by ˝- 1 ∗ ∗ d+ dx dx *↓ (x; +)*↓ (x ; +)V↓↓ (x − x )*↓ (x ; +)*↓ (x; +) exp − 2˝ 0 ˝- 1 ∗ d+ dx dx [/∗ (x; x ; +)*↓ (x ; +)*↓ (x; +) ˙ d[/ ]d[/] exp − 2˝ 0 −1 + *∗↓ (x ; +)*∗↓ (x; +)/(x; x ; +) − /∗ (x; x ; +)V↓↓ (x − x )/(x; x ; +)]
:
(60)
This step is known as a Hubbard–Stratonovich transformation [97,99] and decouples the interaction in the closed channel. In the BCS-theory of superconductivity this Hubbard–Stratonovich transformation introduces the order parameter for the Bose–Einstein condensation of Cooper pairs into the theory. This order parameter is proportional to the macroscopic wave function of the condensate of Cooper pairs. In our case, as we shall see, the above Hubbard–Stratonovich transformation introduces the order parameter for Bose–Einstein condensation of the molecular state responsible for the Feshbach resonance.
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135
The functional integral over the 8elds *∗↓ (x; +) and *↓ (x; +) has now become quadratic and we write this quadratic part as ˝˝ ˝ dx [*∗↓ (x; +); *↓ (x; +)] d+ dx d+ − 2 0 0
(x ; + ) * ↓ −1 ; (61) ×G↓↓ (x; +; x ; + ) · *∗↓ (x ; + ) where the so-called Nambu-space Green’s function for the closed channel obeys the Dyson equation −1 1 G↓↓ (x; +; x ; + ) = G0;−↓↓ (x; +; x ; + ) −
↓↓ (x; +; x
; + ) :
The noninteracting Nambu-space Green’s function is given by
−1 0 G0; ↓↓ (x; +; x ; + ) −1 ; G0; ↓↓ (x; +; x ; + ) = 1 (x ; + ; x; +) 0 G0;−↓↓ where
˝2 ∇2 T'B 9 + − ' G0; ↓↓ (x; +; x ; + ) = −˝(+ − + )(x − x ) ; ˝ − 9+ 2m 2
(62)
(63)
(64)
is the single-particle noninteracting Green’s function. The self-energy is purely o4-diagonal in Nambu space and reads
0 (x; x ; +) ˝ ↓↓ (x; +; x ; + ) = (+ − + ) · ; (65) 0 ∗ (x; x ; +) where (x; x ; +) ≡ /(x; x ; +) + V↑↓ (x − x )*↑ (x; +)*↑ (x ; +) :
(66)
Note that a variation of the action with respect to the pairing 8eld shows that /(x; x ; +) = V↓↓ (x − x )*↓ (x)*↓ (x ) ;
(67)
which relates the auxiliary pairing 8eld to the wave function of two atoms in the closed channel. Roughly speaking, to introduce the 8eld that describes a pair of atoms in the closed-channel bound state we have to consider only contributions from this bound state to the pairing 8eld. Close to resonance it is this contribution that dominates. Note that the average of the pairing 8eld in Eq. (67) indeed shows that the pairing 8eld is similar to the macroscopic wave function of the Cooper-pair condensate. However, in this case we are interested in the phase / = 0 and therefore need to consider also Juctuations. Since the integration over the 8elds *∗↓ (x; +) and *↓ (x; +) involves now a Gaussian integral, it is easily performed. This results in an e4ective action for the pairing 8eld and the atomic 8elds that
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describes the open channel, given by 2 2 ˝- 9 ˝∇ −' d+ dx *∗↑ (x; +)˝ *↑ (x; +) + *∗↑ (x; +) − S e4 [*∗↑ ; *↑ ; /∗ ; /] = 9+ 2m 0 1 ∗ + dx *↑ (x ; +)V↑↑ (x − x )*↑ (x ; +) *↑ (x; +) 2 1 ˝−1 − d+ dx dx [/∗ (x; x ; +)V↓↓ (x − x )/(x; x ; +)] 2 0 +
˝ −1 )] : Tr[ln(−G↓↓ 2
(68)
Because we are interested in the bare atom–molecule coupling we expand the e4ective action up to quadratic order in the 8elds /∗ (x; x ; +) and /(x; x ; +). Considering higher orders would lead to atom–molecule and molecule–molecule interaction terms that will be neglected here, since in our applications we always deal with such a small density of molecules that the mean-8eld e4ects caused by these interactions are negligible. Hence, we expand the e4ective action by making use of −1 1 Tr[ln(−G↓↓ )] = Tr[ln(−G0;−↓↓ )] −
∞ 1 Tr[(G0; ↓↓ m m=1
↓↓ )
m
] :
(69)
This results for the part of the e4ective action that is quadratic in /∗ (x; x ; +) and /(x; x ; +) in ˝ 1 ˝∗ dy dy d+ dx dx d+ S[/ ; /] = − 2 0 0 ×/∗ (x; x ; +)˝G/−1 (x; x ; +; y; y ; + )/(y; y ; + ) ;
(70)
where the Green’s function of the pairing 8eld obeys the equation G/ (x; x ; +; y; y ; + ) = ˝V↓↓ (x − x )(x − y)(x − y )(+ − + ) 1 ˝- dz dz [V↓↓ (x − x )G0; ↓↓ (x; +; z; + ) d+ − ˝ 0 ×G0; ↓↓ (x ; +; z ; + )G/ (z; z ; + ; y; y ; + )] :
(71)
From this equation we observe that the propagator of the pairing 8eld is related to the many-body T -matrix in the closed channel. More precisely, introducing the Fourier transform of the propagator to relative and center-of-mass momenta and Matsubara frequencies 0n = 2n=˝-, denoted by G/ (k; k ; K; i0n ), we have that MB G/ (k; k ; K; i0n ) = ˝T↓↓ (k; k ; K; i˝0n − T'B + 2') ;
(72)
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137
where the many-body T -matrix in the closed channel obeys the equation 1 MB T↓↓ (k; k ; K; z) = V↓↓ (k − k ) + V↓↓ (k − k ) V k
×
[1 + N ( K=2+k − ' + (T'B=2)) + N ( K=2−k − ' + (T'B=2))] z − K=2+k − K=2−k
MB ×T↓↓ (k ; k ; K; z) :
(73)
with N (x) = [e-x − 1]−1 the Bose distribution function. Here, V↓↓ (k) = dxV↓↓ (x)eik·x denotes the Fourier transform of the atomic interaction potential. This equation describes the scattering of a pair of atoms from relative momentum k to relative momentum k at energy z. Due to the fact that the scattering takes places in a medium the many-body T -matrix also depends on the center-of-mass momentum K, contrary to the two-body T -matrix introduced in the previous section, which describes scattering in vacuum. The kinetic energy of a single atom is equal to k = ˝2 k2 =2m. The factor that involves the Bose–Einstein distribution function arises because the probability of a process where a boson scatters into a state that is already occupied by N1 bosons is proportional to 1 + N1 . The reverse process is only proportional to N1 . This explains the factor 1 + N1 + N2 = (1 + N1 )(1 + N2 ) − N1 N2 ;
(74)
in the equation for the many-body T -matrix [100]. The many-body T -matrix is discussed in more detail in the next section when we calculate the renormalization of the interatomic interactions. For now we only need to realize that, for the conditions of interest to us, we are always in the situation where we are allowed to neglect the many-body e4ects in Eq. (73) because the Zeeman energy T'B=2 strongly suppresses the Bose occupation numbers for atoms in the closed channel. This is certainly true for the experimental applications of interest because in the current experiments with magnetically-trapped ultracold gases the Zeeman splitting of the magnetic trap is much larger than the thermal energy. This reduces the many-body T -matrix equation to the Lippmann–Schwinger equation in Eq. (29) for the two-body T -matrix in the closed 2B channel T↓↓ (k; k ; z − K =2), which, in its basis-independent operator formulation, reads ˆ ˆ Tˆ 2B ↓↓ (z) = V ↓↓ + V ↓↓
1 Tˆ 2B ↓↓ (z) ; z − Hˆ 0
(75)
with Hˆ 0 = pˆ 2 =m. As we have seen previously, this equation is formally solved by ˆ ˆ Tˆ 2B ↓↓ (z) = V ↓↓ + V ↓↓
1 Vˆ ↓↓ ; z − Hˆ ↓↓
(76)
with Hˆ ↓↓ = Hˆ 0 + Vˆ ↓↓ . From the previous section we know that the two-body T -matrix has poles at the bound states of the closed-channel potential. We assume that we are close to resonance and hence that one of these bound states dominates. Therefore, we approximate the two-body T -matrix by ˆ |2m 2m | Vˆ ↓↓ ; Tˆ 2B ↓↓ (z) V ↓↓ z − Em
(77)
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where the properly normalized and symmetrized bound-state wave function 2m (x) ≡ x|2m obeys the SchrPodinger equation 2 2 ˝∇ + V↓↓ (x) 2m (x) = Em 2m (x) : (78) − m It should be noted that this wave function does not correspond to the dressed, or true, molecular state which is an eigenstate of the coupled-channels hamiltonian and determined by Eq. (47). Rather, it corresponds to the bare molecular wave function. The coupling V↑↓ (x − x ) of this bare state with the continuum renormalizes it such that it contains also a component in the open channel. Moreover, as we have already seen in the previous section, this coupling also a4ects the energy of this bound state. Both e4ects are important near the resonance and are discussed in detail later on. We are now in the position to derive the quadratic action for the quantum 8eld that describes the bare molecule. To do this, we consider 8rst the case that the exchange interaction V↑↓ (x − x ) is absent. Within the above approximations, the two-point function for the pairing 8eld is given by k|Vˆ ↓↓ |2m 2m |Vˆ ↓↓ |k /(k; K; i0n )/∗ (k ; K; i0n ) = −2˝ : (79) i˝0n − K =2 − Em − T'B + 2' We introduce the 8eld *m (x; +), that describes the bound state in the closed channel, i.e, the bare molecule, by considering con8gurations of the pairing 8eld such that √ (80) /(x; x ; +) = 2V↓↓ (x − x )2m (x − x )*m ((x + x )=2; +) : Using this we have that *m (K; 0n )*∗m (K; 0n ) =
˝ ; −i˝0n + K =2 + Em + T'B − 2'
(81)
which shows that the quadratic action for the bare molecular 8eld is, in position representation, given by ˝- 9 ˝ 2 ∇2 ∗ ∗ S[*m ; *m ] = d+ dx*m (x; +) ˝ (82) − + Em + T'B − 2' *m (x; +) : 9+ 4m 0 In the absence of the coupling of the bare molecular 8eld to the atoms, the dispersion relation of the bare molecules is given by ˝!k (B) = k =2 + Em + T'B :
(83)
As expected, the binding energy of the bare molecule is equal to m (B) = Em + T'B. The momentum dependence of the dispersion is due to the kinetic energy of the molecule. To derive the coupling of this bare molecular 8eld to the 8elds *∗↑ (x; +) and *↑ (x; +) it is convenient to start from the e4ective action in Eq. (68) and to consider again only terms that are quadratic in the self-energy. Integrating out the pairing 8elds leads to an interaction term in the action for the 8eld describing the open channel, given by ˝ 1 ˝ dy dy [V↑↓ (x − x )*∗↑ (x; +)*∗↑ (x ; +) d+ dx dx d+ 2 0 0 (4) ×G↓↓ (x; x ; +; y; y ; + )V↑↓ (y − y )*↑ (y; + )*↑ (y ; + )] ;
(84)
R.A. Duine, H.T.C. Stoof / Physics Reports 396 (2004) 115 – 195
(4)
G
=
+
T
139
MB
Fig. 9. Diagrammatic representation of the two-particle Green’s function in the closed channel. The solid lines correspond to single-atom propagators.
where the two-atom four-point Green’s function is given diagrammatically in Fig. 9. For our purposes it is, for the same reasons as before, suKcient to neglect the many-body e4ects on this propagator and to consider again only the contribution that arises from the bound state in the closed channel. This gives for the Fourier transform of this Green’s function ∗ (k)2 (k ) 2m m (4) G↓↓ (k; k ; K; 0n ) ; (85) i˝0n − K =2 − T'B − Em + 2' where 2m (k) is the Fourier transform of the bound-state wave function. After substitution of this result into Eq. (84) the resulting interaction term is decoupled by introducing the 8eld *m (x; +) with the quadratic action given in Eq. (82). This procedure automatically shows that the bare atom √ –molecule coupling constant is equal to V↑↓ (k)2m (k)= 2. 3.2. Bare atom–molecule hamiltonian In the previous section we have derived, from a microscopic atomic hamiltonian, a bare atom–molecule theory for the description of a Feshbach resonance. It is determined by the action ˝- 9 9 ∗ ∗ ∗ ∗ S[*↑ ; *↑ ; *m ; *m ] = d+ dx *↑ (x; +)˝ *↑ (x; +) + *m (x; +)˝ *m (x; +) 9+ 9+ 0 (86) + H [*∗↑ ; *↑ ; *m ; *∗m ] ; where the bare or microscopic atom–molecule hamiltonian functional is given by H [*∗↑ ; *↑ ; *m ; *∗m ] 2 2 1 ˝∇ ∗ ∗ −'+ dx *↑ (x ; +)V↑↑ (x − x )*↑ (x ; +) *↑ (x; +) = dx*↑ (x; +) − 2m 2 2 2 ˝∇ ∗ + T'B + Em − 2' *m (x; +) + dx*m (x; +) − 4m (87) + dx dx [g↑↓ (x − x )*∗m ((x + x )=2; +)*↑ (x ; +)*↑ (x; +) + c:c:] ; √ and the bare atom–molecule coupling is given by g↑↓ (x) = V↑↓ (x)2m (x)= 2, where V↑↓ (x) is the coupling between the open and closed atomic collision channel of the Feshbach problem, that has its origin in the exchange interaction of the atoms. Note also that the atom–molecule coupling is proportional to the wave function 2m (x) for the bound molecular state in the closed channel responsible for the Feshbach resonance.
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Physically, the microscopic hamiltonian in Eq. (87) describes bosonic atoms in the open channel of the Feshbach problem in terms of the 8elds *∗↑ (x; +) and *↑ (x; +). These atoms interact via the interaction potential V↑↑ (x − x ). Apart from this background interaction, two atoms in the gas can also form a molecular bound state in the closed channel with energy Em that is detuned by an amount of T'B from the open channel. This bare molecular state is described by the 8elds *∗m (x; +) and *m (x; +). The most important input in the derivation of Eq. (87) is that the energy di4erence between the various bound states in the closed channel is much larger than the thermal energy, so that near resonance only one molecular level is of importance. This condition is very well satis8ed of almost all the atomic gases of interest. An exception is 6 Li, which has two Feshbach resonances relatively close to each other [101,65]. The derivation presented in the previous section is easily generalized to this situation, by introducing an additional molecular 8eld to account for the second resonance. To point out the di4erences of our approach with work of other authors a few remarks are in order. First of all, our starting point was the microscopic two-channel atomic hamiltonian in Eq. (59), from which we derived the microscopic atom–molecule hamiltonian in Eq. (87). As we started with the full interatomic interaction potentials, the atom–molecule coupling constant and atom–atom interaction have momentum dependence which cut o4 the momentum integrals encountered in perturbation theory. Because of this, no ultraviolet divergencies are encountered at any order of the perturbation theory, as we will see in the next section. This contrasts with the model used by Kokkelmans and Holland [81], and Mackie et al. [80], who use a phenomenological atom–molecule hamiltonian with delta-function interactions and therefore need a renormalization procedure to subtract the ultraviolet divergencies. In an application of the above microscopic atom–molecule hamiltonian to realistic atomic gases we have to do perturbation theory in the interaction V↑↑ (x − x ) and the coupling g↑↓ (x − x ). Since the interatomic interaction is strong, this perturbation theory requires an in8nite number of terms. Progress is made by realizing that the atomic and molecular densities of interest are so low that we only need to include two-atom processes. This is achieved by summing all ladder diagrams as explained in detail in the next section. 3.3. Ladder summations From the bare or microscopic atom–molecule theory derived in the previous section we now intend to derive an e4ective quantum 8eld theory that contains the two-atom physics exactly. This is most conveniently achieved by renormalization of the coupling constants. Moreover, the molecules acquire a self-energy. Both calculations are done within the framework of perturbation theory to bring out the physics involved most clearly. It is, however, also possible to achieve the same goal in a nonperturbative manner by a second Hubbard–Stratonovich transformation. Because we are dealing with a homogeneous system, it is convenient to perform the perturbation theory in momentum space. Therefore, we Fourier transform to momentum space, and expand the atomic and molecular 8elds according to *↑ (x; +) =
1 ak; n eik·x−i!n + ; (˝-V )1=2 k; n
(88)
R.A. Duine, H.T.C. Stoof / Physics Reports 396 (2004) 115 – 195
and *m (x; +) =
1 bk; n eik·x−i!n + ; 1=2 (˝-V )
141
(89)
k; n
respectively. The even Matsubara frequencies !n = 2n=˝- account for the periodicity of the 8elds on the imaginary-time axis. With this expansion, the grand-canonical partition function of the gas is written as a functional integral over the 8elds ak; n and bk; n and their complex conjugates. It is given by 1 Zgr = d[a∗ ]d[a]d[b∗ ]d[b] exp − S[a∗ ; a; b∗ ; b] ; (90) ˝ where the action S[a∗ ; a; b∗ ; b] is the sum of four terms. The 8rst two terms describe noninteracting atoms and noninteracting bare molecules, respectively, and are given by Sa [a∗ ; a] = (−i˝!n + k − ')a∗k; n ak; n ; (91) k; n
and Sm [b∗ ; b] =
(−i˝!n + k =2 + Em + T'B − 2')b∗k; n bk; n :
(92)
k; n
The atomic interactions are described by the action 1 1 Sint [a∗ ; a] = V↑↑ (k − k )a∗K=2+k; n=2+m a∗K=2−k; n=2−m 2 ˝-V K;k;k n; m; m
×aK=2+k ; n=2+m aK=2−k ; n=2−m ;
(93)
where V↑↑ (k) is the Fourier transform of the interatomic interaction potential. This Fourier transform vanishes for large momenta due to the nonzero range of the interatomic interaction potential. The last term in the action describes the process of two atoms forming a molecule and vice versa, and is given by 1 Scoup [a∗ ; a; b∗ ; b] = g↑↓ (k)[b∗K; n aK=2+k; n=2+m aK=2−k; n=2−m + c:c:] ; (94) (˝-V )1=2 K;k n; m
where g↑↓ (k) is the Fourier transform of the bare atom–molecule coupling constant. This coupling constant also vanishes for large momenta since the bare molecular wave function has a nonzero extent. We 8rst discuss the renormalization of the microscopic atomic interaction V↑↑ (k), due to nonresonant background collisions between the atoms. The 8rst term that contributes to this renormalization is of second order in the interaction. It is found by expanding the exponential in the path-integral expression for the grand-canonical partition function in Eq. (90). To second order in the interactions this leads to 1 1 2 ∗ 1 ∗ ∗ ∗ Zgr = d[a ]d[a] 1 − Sint [a ; a] + 2 Sint [a ; a] + · · · exp − Sa [a ; a] : (95) ˝ 2˝ ˝
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R.A. Duine, H.T.C. Stoof / Physics Reports 396 (2004) 115 – 195 K/2+k K/2+k
K/2−k
K/2+k
K/2−k
K/2 k T +
K/2+k′ (a)
K/2−k′ K/2+k′
+...=
K/2−k′
MB
=
+
T
MB
MB
T
K/2+k′
K/2−k′
(b)
Fig. 10. (a) Ladder diagrams that contribute to the renormalization of the interatomic interaction. (b) Diagrammatic representation of the Lippmann–Schwinger equation for the many-body T -matrix. The solid lines correspond to single-atom propagators. The wiggly lines correspond to the interatomic interaction V↑↑ .
After the decoupling of the eight-point function resulting from the square of the action Sint [a∗ ; a] with the use of Wick’s theorem, it gives rise to various terms in the perturbation theory which can be depicted by Feynman diagrams [99,98]. As mentioned already, we only take into account the ladder Feynman diagram. This diagram is given by the second term of the Born series depicted in Fig. 10(a), and corresponds to the expression 1 − V↑↑ (k − k )G0; a (K=2 + k ; i!n=2+m ) ˝-V k ;m
×G0; a (K=2 − k ; i!n=2−m )V↑↑ (k − k ) ;
(96)
−˝ ; −i˝!n + k − '
(97)
where G0; a (k; i!n ) =
is the noninteracting propagator of the atoms. After performing the summation over the Matsubara frequencies we 8nd that, to second order, the renormalization of the interatomic interactions is given by [1 + N ( K=2+k − ') + N ( K=2−k − ')] 1 V↑↑ (k − k ) V↑↑ (k − k ) → V↑↑ (k − k ) + V i˝!n − K=2+k − K=2−k + 2' k
×V↑↑ (k − k ) ;
(98)
which is 8nite due to the use of the true interatomic potential. In comparing this result with the 8rst two terms of the Born series for scattering in vacuum in Eq. (31), we see that the only di4erence between the two-body result and the above result is the factor involving the Bose distributions. This so-called statistical factor accounts for the fact that the scattering takes place in a medium and is understood as follows. The amplitude for a process where an atom scatters from a state with occupation number N1 to a state with occupation number N2 contains a factor N1 (1 + N2 ). The factor N1 simply accounts for the number of atoms that can undergo the collision, and may be understood from a classical viewpoint as well. However, the additional factor (1 + N2 ) is a result of the Bose statistics of the atoms and is therefore called the Bose-enhancement factor. For fermions this factor would correspond to the Pauli-blocking factor (1 − N2 ), reJecting the fact that a fermion
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is not allowed to scatter into a state that is already occupied by an identical fermion. In calculating the Feynman diagram we have to take into account the forward and backward scattering processes, which results in the statistical factor in Eq. (98). Continuing the expansion in Eq. (95) and taking into account only the ladder diagrams leads to a geometric series, which is summed by introducing the many-body T -matrix in the open channel. It is given by 1 MB T↑↑ (k; k ; K; z) = V↑↑ (k − k ) + V↑↑ (k − k ) V k
×
[1 + N ( K=2+k − ') + N ( K=2−k − ')] MB T↑↑ (k ; k ; K; z) : z − K=2+k − K=2−k
(99)
Its diagrammatic representation is given in Fig. 10(b). For the moment we neglect the many-body e4ects on the scattering atoms and put the Bose-distribution functions equal to zero. This assumption is valid at temperatures far below the critical temperature [102]. This reduces the many-body T -matrix 2B to the two-body T -matrix T↑↑ (k; k ; z − K =2). For the low temperatures of interest to us here, we are allowed to take the external momenta equal to zero. For small energies we 8nd, using the result in Eq. (38), that the e4ective interaction between the atoms reduces to 2B T↑↑ (0; 0; i˝!n − K =2 + 2')
4abg ˝2 1 = : m 1 − abg −m(i˝!n − K =2 + 2')=˝2 − abg rbg m(i˝!n − K =2 + 2')=2˝2
(100) Here abg and rbg are the scattering length and the e4ective range of the open-channel potential V↑↑ (x), respectively. Although these could in principle be calculated with the precise knowledge of this potential, it is much easier to take them from experiment. For example, the magnitude of the scattering length can be determined by thermalization-rate measurements [4]. The e4ective range is determined by comparing the result of calculations with experimental data. We will encounter an explicit example of this in Section 6. The next step is the renormalization of the microscopic atom–molecule coupling constant. Using the same perturbative techniques as before, we 8nd that the e4ective atom–molecule coupling is given in terms of the bare coupling by 1 MB gMB (k; K; z) = g↑↓ (k) + T (k; k ; K; z) V ↑↑ k
×
[1 + N ( K=2+k − ') + N ( K=2−k − ')] g↑↓ (k ) ; z − K=2+k − K=2−k
(101)
and is illustrated diagrammatically in Fig 11. Neglecting again many-body e4ects, the coupling constant becomes g2B (k; z − K =2) with 1 2B 1 g2B (k; z) = g↑↓ (k) + T↑↑ (k; k ; z) g↑↓ (k ) : (102) V z − 2 k k
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g
+
+
+...=
+
= gMB T
MB
Fig. 11. Renormalization of the atom–molecule coupling constant by interatomic interactions. The solid lines correspond to single-atom propagators. The wiggly lines corresponds to the interatomic interaction V↑↑ .
+
+
+...=
+
T
MB
=
Fig. 12. Molecular self-energy. The solid lines correspond to single-atom propagators. The wiggly lines corresponds to the interatomic interaction V↑↑ .
From the above equation we infer that the energy dependence of this coupling constant is the same as that of the two-body T -matrix. This result is easily understood by noting that for a contact 2B potential V↑↑ (k) = V0 and we simply have that g2B = g↑↓ T↑↑ =V0 . Hence we have for the e4ective atom–molecule coupling g2B (0; i˝!n − K =2 + 2')
=g
1 − abg
1
[ − m(i˝!n − K =2 + 2')=˝2 ] − [abg rbg m(i˝!n − K =2 + 2')=2˝2 ]
:
(103)
where g is the e4ective atom–molecule coupling constant at zero energy. The latter is also taken from experiment. We come back to this point in Section 4.1 where we discuss the two-atom properties of our e4ective many-body theory. Finally, we have to take into account also the ladder diagrams of the resonant part of the interaction. This is achieved by including the self-energy of the molecules. It is in 8rst instance given by the expression [1 + N ( K=2+k − ') + N ( K=2−k − ')] MB 2 6MB (K; z) = g↑↓ (k) g (k; K; z) ; (104) V z − K=2+k − K=2−k k
and shown diagrammatically in Fig. 12. We neglect again many-body e4ects which reduces the self-energy in Eq. (104) to 62B (z − K =2) with 62B (z) = 2m |Vˆ ↑↓ Gˆ ↑↑ (z)Vˆ ↑↓ |2m ;
(105)
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where the propagator Gˆ ↑↑ (z) is given by Gˆ ↑↑ (z) =
1 ; z − Hˆ ↑↑
(106)
with the hamiltonian pˆ2 + Vˆ ↑↑ ≡ Hˆ 0 + Vˆ ↑↑ : Hˆ ↑↑ = m
(107)
We insert in Eq. (105) a complete set of bound states | with energies E and scattering states | (+) k that obey the equation in Eq. (25). This reduces the self-energy to 1 1 dk 2B 2 2 6 (z) = | 2m |Vˆ ↑↓ | | + | 2m |Vˆ ↑↓ | (+) ; (108) k | 3 z − E (2) z − 2 k where we replaced the sum over the momenta k by an integral. Using Eq. (102) and the equation for the scattering states we have that 1 g2B (k; 2 k+ ) = √ 2m |Vˆ ↑↓ | 2
(+) k
:
(109)
Neglecting the energy dependence due to the contribution of the bound states since their binding energies are always large compared to the thermal energy, we have, using the result for the atom–molecule coupling constant in Eq. (103), the intermediate result 1 dk 2B 62B (z) = 2 |g (0; 2 k+ )|2 : (110) (2)3 z − 2 k The remaining momentum integral yields the 8nal and for our purposes very important result 2B (z) ≡ 62B (z) − 62B (0) ≡ 62B (z) + (T'B0 + Em ) ˝7m g2 m −mz = − 2 2 −2 abg − 2rbg 4 ˝ ˝2
√ √ i abg rbg i abg rbg √ − log + i abg log − abg − 2rbg abg − 2rbg
2 abg rbg mz mz × 2 3rbg − 2abg − 2 ˝ 2˝
mz abg rbg mz 2 −1 abg − 2rbg 1 + abg (abg − rbg ) 2 + ; × ˝ 2˝2
(111)
where we have denoted the energy-independent shift 62B (0) in such a manner that the position of the resonance in the magnetic 8eld is precisely at the experimentally observed magnetic-8eld value B0 . This shift is also shown in the results of the calculation of the bound-state energy of the coupled square wells in Fig. 7.
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3.4. E8ective atom–molecule theory Putting the results from the previous section together, we 8nd that the atom–molecule system is described by the e4ective action S e4 [a∗ ; a; b∗ ; b] = (−i˝!n + k − ')a∗k; n ak; n k; n
+
1 1 2B Tbg (i˝!n − K =2 + 2') 2 ˝-V K;k;k n; m; m
×a∗K=2+k; n=2+m a∗K=2−k; n=2−m aK=2+k ; n=2+m aK=2−k ; n=2−m 2B ∗ + [ − i˝!n + k =2 + (B) − 2' + ˝7m (i˝!n − k =2 + 2')]bk; n bk; n k; n
+
1 g2B (i˝!n − K =2 + 2') 1=2 (˝-V ) K;k n; m
×[b∗K; n aK=2+k; n=2+m aK=2−k; n=2−m + c:c:] ;
(112)
2B (z) ≡ where (B) ≡ T'(B − B0 ) is the so-called detuning. From now on we use the notation Tbg 2B 2B 2B T↑↑ (0; 0; z), and g (z) ≡ g (0; z). Since these coupling constants are the result of summing all ladder diagrams, these diagrams should not be taken into account again. In the next section we discuss how the coupling constants are determined from experiment. To consider also the real-time dynamics of the system we derive the Heisenberg equations of motion for the 8eld operators ˆ a (x; t) and ˆ m (x; t), that annihilate an atom and a molecule at position x and time t, respectively. Their hermitian conjugates are the creation operators. To determine the Heisenberg equations of motion for these 8eld operators, we 8rst have to perform an analytic continuation from the Matsubara frequencies to real frequencies. To ensure that the physical quantities and equations of motion are causal, this has to be done by a so-called Wick rotation. This amounts to the replacement of the Matsubara frequencies by a frequency with an in8nitesimally small and positive imaginary part
i!n → !+ :
(113)
This leads to a subtlety involving the analytic continuation of the square root of the energy in the various expressions. Due to the branch cut in the square root we have that √ −i˝!n → −(˝!+ ) = −i ˝! : (114) The last expression on the right-hand side of this equation is valid for ˝! on the entire real axis. To obtain the equation of motion in position and time representation, we have to Fourier transform back from momentum and frequency space. This amounts to the replacement ˝! − K =2 → i˝
˝ 2 ∇2 9 + : 9t 4m
(115)
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147
Note that this combination of time and spatial derivatives is required due to the Galilean invariance of the theory. For simplicity we assume that we are so close to resonance that we are allowed to neglect the energy dependence of the e4ective atomic interactions and the e4ective atom–molecule coupling. Moreover, for notational convenience we take only the leading-order energy dependence of the molecular self-energy into account. Higher orders are straightforwardly included but lead to somewhat complicated notations in the position and time representation. The leading-order energy dependence of the self-energy is, after the Wick rotation to real energies, given by m3=2 √ (+) ˝7m (E) −g2 i E : (116) 2˝3 The additional superscript indicates that we are dealing with the retarded self-energy, i.e., the (+) 2B self-energy evaluated at the physically relevant energies E + so that ˝7m (E) ≡ ˝7m (E + ). Note that for positive energy E this result is in agreement with the Wigner-threshold law. This law gives the rate for a state with well-de8ned positive energy to decay into a three-dimensional continuum. Within the above approximations, the Heisenberg equations of motion for the coupled atom–molecule model read 2 2 4abg ˝2 ˆ † 9 ˆ a (x; t) ˝∇ ˆ ˆ a (x; t) i˝ = − + a (x; t) a (x; t) 9t 2m m + 2g ˆ †a (x; t) ˆ m (x; t) ;
3=2 ˝ 2 ∇2 ˝ 2 ∇2 ˆ 9 ˆ m (x; t) 9 2 m ˆ2 = − + (B(t)) − g i˝ i i˝ + m (x; t) + g a (x; t) ; 9t 4m 2˝3 9t 4m
(117)
where we have also allowed for a time-dependent detuning. In the next section we show that these equations correctly reproduce the Feshbach-resonant scattering amplitude and the binding energy of the molecule. Moreover, we apply the e4ective theory derived in this section to study many-body e4ects on this binding energy, above the critical temperature for Bose–Einstein condensation. 4. Normal state In this section we discuss the properties of the gas in the normal state. In the 8rst section, we consider the two-atom properties of our many-body theory. Hereafter, we discuss the equilibrium properties that follow from our theory. In the last section, we investigate many-body e4ects on the energy of the molecular state, above the critical temperature for Bose–Einstein condensation. 4.1. Two-atom properties of the many-body theory In this section we show that our e4ective 8eld theory correctly contains the two-atom physics of a Feshbach resonance. First, we show that the correct Feshbach-resonant atomic scattering length is obtained after the elimination of the molecular 8eld. Second, we calculate the bound-state energy and show that it has the correct threshold behavior near the resonance. To get more insight in the nature of the molecular state near resonance, we also investigate the molecular density of states.
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4.1.1. Scattering properties To calculate the e4ective interatomic scattering length, we have to eliminate the molecular 8eld from the Heisenberg equations of motion in Eq. (117). Since the scattering length is related to the scattering amplitude at zero energy and zero momentum, we are allowed to put all the time and spatial derivatives in the equation of motion for the molecular 8eld operator equal to zero. This equation is now easily solved, which leads to ˆ m (x; t) = − g ˆ 2 (x; t) : (118) (B) a Substitution of this result into the equation for the atomic 8eld operator leads for the interaction terms to 4abg ˝2 ˆ † ˆ ˆ ˆ† ˆ a (x; t) a (x; t) a (x; t) + 2g a (x; t) m (x; t) m 4abg ˝2 2g2 ˆ † (x; t) ˆ a (x; t) ˆ a (x; t) : = − a (B) m
(119)
From this result we observe that we have to take the renormalized atom–molecule coupling constant at zero energy equal to g = ˝ 2abg TBT'=m, so that we have 4abg ˝2 2g2 4a(B)˝2 − = ; m (B) m where we recall that the scattering length near a Feshbach resonance is given by TB ≡ abg + ares (B) : a(B) = abg 1 − B − B0
(120)
(121)
Since both the width TB and the background scattering length abg are known experimentally, the knowledge of the di4erence in magnetic moment between the open and the closed channel T' completely determines the renormalized coupling constant g. Since the open and the closed channel usually correspond to the triplet and singlet potential, respectively, we always have that |T'| 2'B , with 'B the Bohr magneton. More precise values of the di4erence in magnetic moments are obtained from coupled-channels calculations using the interatomic interaction potentials [10,14,81,103]. From the above analysis we see that the correct Feshbach-resonant scattering length of the atoms is contained in our theory exactly. Next, we show that our e4ective theory also contains the correct bound-state energy. 4.1.2. Bound-state energy The energy of the molecular state is determined by the poles of the retarded molecular propagator Gm(+) (k; !). It is given by ˝ Gm(+) (k; !) = : (122) (+) + ˝! − k =2 − (B) − ˝7m (˝! − k =2) For positive detuning (B) there only exists a pole with a nonzero and negative imaginary part. This is in agreement with the fact that the molecule decays when its energy is above the two-atom continuum threshold. The imaginary part of the energy is related to the lifetime of the molecular state. For negative detuning the molecular propagator has a real and negative pole corresponding to
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149
the bound-state energy. More precisely, in this case the poles of the molecular propagator are given by ˝! = m (B) + k =2, where the bound-state energy is determined by solving for E in the equation (+) E − (B) − ˝7m (E) = 0 :
(123)
In general, this equation cannot be solved analytically but is easily solved numerically, and in Section 6 we discuss its numerical solution for the parameters of 85 Rb. Close to resonance, however, we are allowed to neglect the e4ective range of the interactions. This reduces the retarded self-energy of the molecules to √ g2 m3=2 i E (+) : (124) ˝7m (E) − 2˝3 1 − i|abg | mE=˝2 Moreover, the bound-state energy is small in this regime and we are allowed to neglect the linear terms in the energy with respect to the square-root terms. This reduces the equation for the bound-state energy in Eq. (123) to √ g2 m3=2 i E = (B) : (125) 2˝3 1 − i|abg | mE=˝2 This equation is easily solved analytically, and yields the result
m (B) = −
˝2 ; m[a(B)]2
(126)
which analytically proves the numerical result in Eq. (56). This numerical result was obtained for the speci8c case of two coupled attractive square wells. The above analytic proof, which does not depend on the details of the potential, shows that the result is general. The same result is found by noting that after the elimination of the molecular 8eld the e4ective on-shell T -matrix for the atoms in the open channel is given by 2 2B T 2B (E + ) = Tbg (E + ) + |g2B (E + )|2 Gm(+) ( mE=˝2 ; E) : (127) ˝ Close to resonance this expression reduces to
2 1 4a (B)˝ res : (128) T 2B (E) m 1 + iares (B) mE=˝2 The pole of this T -matrix, which gives the bound-state energy, is indeed equal to the result in Eq. (126) close to resonance. 4.1.3. Molecular density of states The molecular density of states is obtained by taking the imaginary part of the retarded molecular propagator [98], i.e., 1 Im[Gm(+) (k; !)] : 9m (k; !) = − (129) ˝ For simplicity, we discuss here only the situation that we are close to resonance, and therefore approximate the retarded molecular self-energy by the square-root term resulting from Wigner’s threshold law as given in Eq. (116). The extension to situations further of resonance are straightforward.
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ρm (k,ω)
Z δ (−hω-εm-εk/2)
negative detuning positive detuning
δ+εk/2
εm+εk/2
hω
−
Fig. 13. Molecular density of states. The solid line shows the density of states for negative detuning. Since there is a true bound state in this case there is a pole in the density of states. For positive detuning the density of states is approximately a Lorentzian as shown by the dashed line.
For the case of negative detuning, the molecular density of states is shown by the solid line in Fig. 13 and has two contributions. One arising from the pole at the bound-state energy and the second from the two-atom continuum. Within the above approximation, it is given by 9m (k; !) = Z(B)(˝! − k =2 − m (B))
(g2 m3=2 =2˝3 ) ˝! − k =2 1 + (˝! − k =2) ; [˝! − k =2 − (B)]2 + (g4 m3 =42 ˝6 )(˝! − k =2)
(130)
with Z(B) the so-called wave-function renormalization factor −1
(+) 97m (˝!) Z(B) = 1 − 9!
1+
˝!= m (B)
g2 m3=2 4˝3 | m (B)|
−1
:
(131)
This factor goes to zero as we approach the resonance and it becomes equal to one far o4 resonance. Physically, this is understood as follows. Far o4 resonance the bound state of the coupled-channels hamiltonian in Eq. (47), i.e., the dressed molecule, is almost equal to the bound state of the closed-channel potential and has zero amplitude in the open channel. This corresponds to the situation whereZ(B) 1. As the resonance is approached, the dressed molecule contains only with an amplitude Z(B) the closed-channel bound state, i.e., the bare molecule. Accordingly, the contribution of the open channel becomes larger and gives rise to the threshold behavior of the bound-state energy in Eq. (126). Of course, the square of the wave function of the dressed molecule is normalized to one. This is expressed by the sum rule for the molecular density of states, d(˝!)9m (k; !) = 1 : (132) In detail, the dressed molecular state with zero momentum is given by |2m ; dressed = Z(B)bˆ†0 |0 + Ck aˆ†k aˆ†−k |0 : k
(133)
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Here, the second-quantized operator bˆ†0 creates a molecule with zero momentum. It acts on the vacuum state |0 . The bare molecular state is therefore given by |2m = bˆ†0 |0 . The operator aˆ†k creates an atom with momentum ˝k and hence the coeKcient Ck denotes the amplitude of the dressed molecular state to be in the open channel of the Feshbach problem. To gain more insight in the nature of the dressed molecular state we calculate the coeKcients Ck in perturbation theory. Neglecting the o4-resonant background interactions and the energy dependence of the atom–molecule coupling constant, the hamiltonian appropriate for our purposes is, in terms of the above operators, given by Hˆ = Hˆ am + Hˆ coup ; with Hˆ am =
k aˆ†k aˆk +
k
and
(134) k k
2
+ (B) bˆ†k bˆk ;
g ˆ† Hˆ coup = √ [bK aˆK=2+k aˆK=2−k + h:c:] : V K;k
(135)
(136)
The zeroth-order state around which we perturb is the bare molecular state |2m with energy (B). In 8rst order in g the dressed molecular state is given by 1 Hˆ coup bˆ†0 |0 |2m ; dressed = Z(B) bˆ†0 |0 + ˆ (B) − H am g 1 = Z(B) bˆ†0 |0 + √ aˆ†k aˆ†−k |0 ; (137) (B) − 2
V k k where Z(B) = 1 − O(g2 ). This result shows that, to 8rst order in g, the coeKcients Ck are given by 1 g : (138) Ck = √ V (B) − 2 k We now calculate the wave-function renormalization factor Z(B) in a di4erent manner by demanding that the dressed molecular wave function is properly normalized, i.e., 2m ; dressed|2m ; dressed = 1 :
(139)
This leads to 1 = Z(B) +
(+) 1 ((B)) 2g2 97m = Z(B) − : 2 V 9! [(B) − 2 k ] k
(140)
The factor of two corresponds to the two contributions arising from the matrix element 0|aˆk aˆ−k aˆ†k aˆ†−k |0 . From this result we 8nd that the wave-function renormalization factor is given by
−1 (+) (+) 97m ((B)) ((B)) 97m 1− ; (141) Z(B) = 1 + 9! 9! in agreement with the result in Eq. (131) to second order in g.
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Note that the total number of atoms in the dressed molecular state should be equal to two. The number of atoms is given by † † bˆk bˆk dressed + aˆk aˆk dressed ; (142) N =2 k
k
where · · · dressed ≡ 2m ; dressed| · · · |2m ; dressed . For the number of atoms with momentum ˝k we have that aˆ†k aˆk dressed =
1 4g2 ; V [(B) − 2 k ]2
from which, with the use of Eq. (140), we 8nd that † aˆk aˆk dressed = 2 − 2Z(B) :
(143)
(144)
k
Using bˆ†k bˆk dressed = Z(B)k; 0 we have indeed that the total number of atoms N = 2, as required. If the magnetic 8eld varies not too rapidly, we are allowed to make an adiabatic approximation to the Heisenberg equation of motion for the bare molecular 8eld operator in Eq. (117). This amounts to introducing a molecular 8eld ˆ m (x; t) that annihilates a dressed molecule, i.e., a molecule with internal state given by Eq. (133). This is achieved as follows. In frequency and momentum space the action for the bare molecular 8eld is given by d! ∗ (+) S[*∗m ; *m ] = *m (k; !)[˝! − k =2 − (B) − ˝7m (˝! − k =2)]*m (k; !) : (145) (2) k
Next, we expand this action around the pole of the propagator m (B). To linear order, this yields the result d! *∗m (k; !) *m (k; !) S[*∗m ; *m ] [˝! − k =2 − m (B)] : (146) (2) Z(B) Z(B) k
From this equation we see that the 8eld that describes the dressed molecule is given by *m = *m = Z(B). This leads to the following action for the dressed molecular 8eld in position and time representation ˝ 2 ∇2 9 ∗ ∗ − m (B) *m (x; t) : (147) S[*m ; *m ] = dt dx*m (x; t) i˝ + 9t 4m
More importantly, the terms that describe the coupling between the atoms and the molecules are multiplied by a factor Z(B). In detail, the coupled Heisenberg equations of motion for the atomic and dressed molecular 8eld operators are given by [42] 2 2 4abg ˝2 ˆ † ˝∇ 9 ˆ a (x; t) ˆ a (x; t) ˆ a (x; t) = − + i˝ (x; t) a 9t 2m m + 2g Z(t) ˆ †a (x; t) ˆ m (x; t) ; 2 2 9 ˆ m (x; t) ˝∇ + m (t) ˆ m (x; t) + g Z(t) ˆ 2a (x; t) ; i˝ = − 9t 4m
(148)
R.A. Duine, H.T.C. Stoof / Physics Reports 396 (2004) 115 – 195
153
where Z(t) ≡ Z(B(t)), and m (t) ≡ m (B(t)). In the derivation of the above coupled equations we have assumed that we are allowed to make an adiabatic approximation for the renormalization factor Z(B) and that we can evaluate it at every time at the magnetic 8eld B(t). In principle there are retardation e4ects due to the fact that the dressed molecular state does not change instantaneously. Following the above manipulations for time-dependent magnetic 8eld we see that these e4ects can be neglected if 9 ln Z(t) | m (t)| : ˝ (149) 9t In principle, the Heisenberg equation of motion for the molecular 8eld operator also contains an imaginary part due to the fact that the dressed molecule can decay into a pair of atoms with opposite momenta. The rate for this process will be small, however, under the condition given in Eq. (149). We will come back to this process when we consider its e4ect on the coherent atom–molecule oscillations. For positive detuning the molecular density of states has only a contribution for positive energy. For large detuning it is in 8rst approximation given by 9m (k; !) =
˝;m (B)=2 ; [(˝! − k =2 − (B))2 + (˝;m (B)=2)2 ]
(150)
where the lifetime of the molecular state is de8ned by g2 m3=2 ;m (B) = (B) : (151) ˝4 As expected, the density of states is, in the case of positive detuning, approximately a Lorentzian centered around the detuning with a width related to the lifetime of the molecule. It is shown in Fig. 13 by the dashed line. 4.2. Equilibrium properties The equilibrium properties of the gas are determined by the equation of state, which relates the total density of the gas to the chemical potential. This equation can be calculated in two ways, either by calculating the thermodynamic potential and di4erentiating with respect to the chemical potential, or by directly calculating the expectation value of the operator for the total density. We discuss both methods, which should, of course, yield the same result. Nevertheless, to show the equivalence is a subtle matter. First, we calculate the thermodynamic potential [104]. Within our approximations, in 8rst instance it is given by the expression 1 1 0('; T ) = Tr[ln(G0;−a1 )] + Tr [ln(Gm−1 )] : (152) Here, we recall that G0; a (k; i!n ) is the noninteracting atomic propagator of the atoms in Eq. (97). The full molecular propagator is given by Gm (k; i!n ) =
−˝ ; 2B (i˝! − =2 + 2') −i˝!n + k =2 + (B) − 2' + ˝7m n k
(153)
154
R.A. Duine, H.T.C. Stoof / Physics Reports 396 (2004) 115 – 195
+
=
+
+
+
+...
Fig. 14. Diagrams contributing to the thermodynamic potential of the gas. The noninteracting atomic and molecular propagators are denoted by the solid and dashed thin lines, respectively. The full molecular propagator is given by the thick dashed line. The bare and renormalized atom–molecule coupling constants are denoted by the open and 8lled triangles, respectively.
with the molecular self-energy given in Eq. (111). The so-called ring diagrams that contribute to the thermodynamic potential in our approximation are given in Fig. 14. The full molecular propagator is denoted by the thick dashed line and the noninteracting molecular propagator is denoted by the thin dashed line. The noninteracting atomic propagators are indicated by the thin solid lines. The total atomic density is calculated by using the thermodynamic identity N = −90('; T )=9', which results in 1 9 1 1 n=− − ln[-(−i˝!n + k =2 + (B) − 2' ˝-V i!n − ( k − ')=˝ 9' -V n n k
2B + ˝7m (i˝!n
k
− k =2 + 2'))] :
(154)
After performing the summation over the Matsubara frequencies in this expression, the 8rst term corresponds to the density of an ideal gas of bosons. The second term in Eq. (154) is more complicated and should, in principle, be dealt with numerically. For negative detuning we can gain physical insight, however, by expanding the propagator around its pole at the molecular binding energy m (B). This leads to the approximation 9 2B ln[-(−i˝!n + k =2 + (B) − 2' + ˝7m (i˝!n − k =2 + 2'))] 9' =
2B ) (i˝!n − k =2 + 2')] −2[1 − (˝7m 2B (i˝! − =2 + 2') −i˝!n + k =2 + (B) − 2' + ˝7m n k
2 ; −i˝!n + k =2 + m (B) − 2'
(155)
2B where we used the expression for the residue of the pole in Eq. (131), and (˝7m ) (E) ≡ 2B 9˝7m (E)=9E. With this approximation the sum over the Matsubara frequencies in Eq. (154)
R.A. Duine, H.T.C. Stoof / Physics Reports 396 (2004) 115 – 195
(a)
(b)
(c)
(d)
155
Fig. 15. Examples of approximations for (a) the atomic propagator and (c) the molecular propagator. The corresponding ring diagrams that contribute to the thermodynamic potential are shown in (b) and (d), respectively.
is performed easily and leads to the result 2 1 1 + n=− ˝-V i!n − ( k − ')=˝ i!n − ( k =2 + m (B) − 2')=˝ n k
1 [N ( k − ') + 2N ( k =2 + m (B) − 2')] : = V
(156)
k
This important result shows that in equilibrium in the normal state and for negative detuning the gas in 8rst approximation behaves as an ideal-gas mixture of atoms and dressed molecules. The same result is found if we neglect in the Heisenberg equations of motion for the atomic and dressed molecular 8eld operators in Eq. (148) the interaction terms and calculate the total density in equilibrium. Instead of calculating the thermodynamic potential and di4erentiating with respect to the chemical potential we can also calculate the total density directly by using n = −Ga (x; +; x; ++ ) − 2Gm (x; +; x; ++ ) :
(157)
An important di4erence between directly calculating the density in this manner and calculating it indirectly from the thermodynamic potential is that we should use in Eq. (157) not the noninteracting atomic propagator. Instead, we should use an approximation to the atomic propagator that contains the same self-energy diagrams as the diagrams shown in Fig. 14. Conversely, in calculating the thermodynamic potential with the use of Eq. (152) we should not use the full atomic propagator. The reason for this is that if we calculate ring diagrams with this propagator we 8nd diagrams which are already contained in the ring diagram of the full molecular propagator. The following explicit example clari8es this further. If we use for the atomic propagator the approximation given diagrammatically in Fig. 15(a), the ring diagram that contributes to the thermodynamic potential is given in Fig. 15(b). On the other hand, if we use for the molecular propagator the approximation given in Fig. 15(c) the resulting ring diagram, given in Fig. 15(d), is exactly the same as Fig. 15(b). Clearly, to avoid double
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R.A. Duine, H.T.C. Stoof / Physics Reports 396 (2004) 115 – 195
Fig. 16. Self-energy of the atoms. The solid and dashed thick lines correspond to the full atomic and molecular propagators, respectively. The 8lled triangles correspond to the renormalized atom–molecule coupling constant.
counting problems in the calculation of the thermodynamic potential we should take only one of these diagrams into account. However, if we calculate the density directly from the atomic and molecular propagators we should use both the diagrams given in Fig. 15(a) and (c). We now argue that by directly calculating the density, again for negative detuning, we indeed recover the result in Eq. (156). We 8rst calculate the contribution arising from the molecular propagator. It is found to be equal to 1 nm ≡ −Gm (x; +; x; ++ ) = − Gm (k; i!n ) ˝-V n k 1 1 1 d(˝!)9m (k; !) = V ˝- n i!n − (˝! − 2')=˝ k dk d(˝!)9m (k; !)N (˝! − 2') : = (158) (2)3 Taking into account only the pole in the density of states leads to the result dk N ( k =2 + m (B) − 2') : nm = Z(B) (2)3
(159)
At 8rst sight this result seems a factor Z(B) to small to agree with the result in Eq. (156). However, we have, in fact, already seen in Eq. (144) that the contributions from the atoms to the density results in a term proportional to 2 − 2Z(B). Taking this into account, the result from the direct calculation agrees with the result in Eq. (156) obtained previously. A di4erent way for obtaining the factor 2−2Z(B) in the atomic density is to include the self-energy diagram shown in Fig. 16 in the atomic propagator. The corresponding mathematical expression in 8rst instance is given by 4g2 ˝7a (k; i!n ) = − Gm (k + q; i!n+m )Ga (k; i!n ) : (160) V q; n To understand the physics of this expression, we note that if we neglect the energy and momentum dependence of the molecular propagator we have that Gm (k; i!n ) −˝=(B). Within this approximation the self-energy is given by 8na ares (B)˝2 =m, which corresponds precisely to the Feshbach-resonant part of the self-consistent Hartree–Fock self-energy of the atoms, as expected from the diagram in Fig. 16.
R.A. Duine, H.T.C. Stoof / Physics Reports 396 (2004) 115 – 195
157
The full calculation of the expression for the self-energy in Eq. (160) is complicated due to the fact that we have to use the full atomic and molecular propagators, which makes the calculation self-consistent. To illustrate in perturbation theory that we are able to reproduce the result in Eq. (144) let us simply take the noninteracting atomic and molecular propagators. The self-energy is then given by 4g2 N ( q − ') − N ( k+q =2 + (B) − 2') ˝7a (k; i!n ) = : (161) V q i˝!n − ( k+q =2 − q + (B) − ') To compare with the two-atom calculation for negative detuning performed in the previous section, we must take only one other atom present with momentum −˝k, and no molecules. The self-energy of the atom with momentum ˝k is then given by ˝7a (k; i!n ) =
1 4g2 : V i˝!n − ((B) − k − ')
(162)
With this self-energy the retarded propagator of the atoms is given by ˝ Ga(+) (k; !) = : + 2 ˝! − k − (4g =V )[˝!+ + k − (B)]−1
(163)
It has two poles, one close to k , and one close to (B). The residue of the latter is given by Zk
1 4g2 ; V [2 k − (B)]2
(164)
in agreement with the result in Eq. (143). Moreover, we have that Zk = 2 − 2Z(B) :
(165)
k
Hence, the total density of the atoms is given by (2 − 2Z(B)) 1 N ( k =2 + (B) − 2') + N ( k − ') : na V V k
(166)
k
Together with the molecular density from Eq. (159) that becomes Z(B) N ( k + (B) − 2') ; nm V
(167)
k
the total density 2nm + na is again equal to the result in Eq. (156) to lowest order in the interactions. 4.3. Applications In this section we present results on the properties of the normal state of the gas. First, we calculate the density of atoms and molecules as a function of the detuning, at a 8xed temperature. Second, we calculate the density of atoms and molecules, and the temperature of the gas, as a function of detuning at 8xed entropy and total density. This calculation is of interest because it gives the outcome of a magnetic-sweep experiment through the Feshbach resonance in the adiabatic approximation. Finally, we calculate the critical temperature for Bose–Einstein condensation as a function of the detuning, at 8xed total density.
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4.3.1. Density of atoms and molecules As we have seen, the density of the gas is most easily calculated by means of Eq. (154). We report all our results as a function of the detuning in units of the energy g4 m3 =42 ˝6 . The temperature is given in units of the critical temperature for Bose–Einstein condensation of an ideal gas of atoms with a total density n, i.e., T0 =
3:31˝2 n2=3 : mkB
(168)
We compare the exact results, found from numerically performing the summation over Matsubara frequencies in Eq. (154), to the ideal-gas mixture result in Eq. (156). However, this last equation was derived for negative detuning where there is a real pole in the molecular Green’s function. We extend this result to positive detuning by describing the molecular gas at positive detuning as an ideal gas of molecules with a positive bound-state energy given by the energy at which the molecular density of states in Eq. (129) has its maximum. It turns out that, for small detuning, this maximum is at ˝2 =ma2 . Furthermore, this approximation implies that we ignore the physical e4ects of the lifetime of the molecule, that is nonzero for positive detuning, on the equilibrium properties of the gas. These lifetime e4ects are however included in the exact result in Eq. (154). In Fig. 17 the results of the calculation of the density as a function of the detuning are shown, for a temperature of T = 2T0 . Fig. 17(a) shows the fraction of atoms and Fig. 17(b) shows the number of atoms in molecules, i.e., twice the fraction of molecules, as a function of the detuning. The solid and the dotted lines show the exact result for a total atomic density of n = 1011 cm−3 and n = 1012 cm−3 , respectively. As expected, for negative detuning most of the atoms in the gas are bound to molecules. One should note, however, that we are still in the situation where we are allowed to neglect the e4ect of molecule–molecule interactions, as well as atom–molecule interactions, as assumed in the derivation of the microscopic atom–molecule theory in Section 3. The reason for this is the following. The interactions between a dressed molecule and an atom, and the interactions among dressed molecules are known to be proportional to the scattering length a(B) [105,106]. Taking into account the unitarity limit of the corresponding T -matrices, the mean-8eld e4ects of these interactions are estimated to play a role only in a regime very close to the resonance [106]. Moreover, in this regime the theory presented in the present paper is not applicable anymore, as the system enters the strong-coupling regime where little is known quantitatively. At positive detuning, most of the atoms are free. Moreover, we 8nd that the width in detuning of this crossover regime is approximately equal to the temperature. The dashed lines show the ideal-gas mixture result for a density of n = 1011 cm−3 , i.e., the result that does not incorporate the e4ects of the nonzero lifetime of the molecules at positive detuning. For negative detuning, we observe that this ideal-gas result becomes equal to the exact result. This implies that the pole approximation in Eq. (155) is indeed a reasonable approximation suKciently far from resonance. An important conclusion is therefore that, for suKciently negative detuning, we are allowed to treat the gas as an ideal-gas mixture of atoms and dressed molecules with binding energy m (B). For positive detuning, the ideal-gas result di4ers substantially from the exact result. In particular, for relatively large detuning, the ideal-gas calculation considerably underestimates the number of molecules. The exact result shows that there is, even at relatively large detuning, a signi8cant fraction of molecules in the gas. This is the result of the 8nite lifetime of the molecules in this case. Physically, this comes about because the molecular density of states for positive detuning has signi8cant spectral
R.A. Duine, H.T.C. Stoof / Physics Reports 396 (2004) 115 – 195
159
fraction of atoms
1 0.8 0.6 0.4 0.2 0
2 x fraction of molecules
(a) -0.001
(b)
-0.0005
0
0.0005
0.001
0
0.0005
0.001
1 0.8 0.6 0.4 0.2 0 -0.001
-0.0005
4 π2
−
h
6
δ/g4 m3
Fig. 17. (a)–(b) Fraction of atoms and fraction of atoms in molecules as a function of the detuning at a 8xed temperature of 2T0 , for two di4erent total densities. The solid line is the exact result that includes all two-atom physics, and particular the e4ects of the nonzero lifetime of the molecule at positive detuning, for a total atomic density of n = 1011 cm−3 . The dashed line shows the result for this density if we approximate the gas by an ideal-gas mixture of atoms and dressed molecules. The dotted line shows the exact result for a total density of n = 1012 cm−3 .
weight at low energies. In equilibrium, this leads to a signi8cant fraction of molecules. For even larger positive detuning, the ideal-gas result reduces again to the exact result. 4.3.2. Adiabatic sweep through the resonance We now calculate the number of atoms and molecules in the gas during an adiabatic sweep in the magnetic 8eld, such that the detuning changes from positive to negative. The condition for adiabaticity is that the entropy of the gas is constant. The entropy is given by 90 S =− : (169) 9T The total number of atoms is, of course, also constant throughout the sweep. As we have seen, for suKciently large absolute values of the detuning, the gas is well-described by an ideal-gas approximation. For simplicity, we will therefore treat the gas here as an ideal-gas mixture since we are mostly interested in the 8nal density of atoms and molecules and the 8nal temperature of the gas after the sweep, for which an ideal-gas treatment is suKcient [107]. In Fig. 18 the results of the calculation of the fraction of atoms and twice the fraction of molecules is presented. The total atomic density is taken equal to n=1013 cm−3 . The solid lines show the result for an initial temperature of T = 2T0 , and the dashed lines show the result for an initial temperature
R.A. Duine, H.T.C. Stoof / Physics Reports 396 (2004) 115 – 195
fraction of atoms
160
2 x fraction of molecules
(a)
(b)
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.003
-0.002
-0.001
-0.003
-0.002
-0.001
0
0.001
0.002
0
0.001
0.002
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
4π
2
−
h
6
δ/g m 4
3
Fig. 18. (a)–(b) Fraction of atoms and twice the fraction of molecules as a function of the detuning for an adiabatic sweep through the resonance. The total atomic density is equal to n = 1013 cm−3 . The solid lines show the result for an initial temperature of T = 2T0 . The dashed lines show the result for T = 4T0 .
of T = 4T0 . As we go from positive to negative detuning, most of the atoms in the gas are converted to molecules. The range of detuning where the conversion takes place is proportional to the initial temperature of the gas, as expected. In Fig. 19 the temperature is plotted as a function of the detuning for the two initial temperatures T = 2T0 and 4T0 . The total density is again equal to n = 1013 cm−3 . Clearly, the gas is heated as the detuning is changed from positive to negative. This is easily understood, since molecules form as the detuning is changed from positive to negative values, and their binding energy is released as kinetic energy into the gas. 4.3.3. Critical temperature Finally, we calculate the critical temperature for Bose–Einstein condensation of the atom–molecule mixture, at a 8xed total atomic density. The results are presented in Fig. 20, for a total density of n = 1013 cm−3 . The solid line shows the exact calculation and the dashed line shows the ideal-gas mixture result. For positive detuning and far from resonance, we are essentially dealing with an atomic gas. Hence we have in this regime that TBEC = T0 . For suKciently negative detuning we are dealing with a gas of molecules with twice the atomic mass, and hence we have that TBEC = 2−5=3 T0 . The feature in the critical temperature at zero detuning turns out to be a signature of a true thermodynamic phase transition, between a phase with a single Bose–Einstein condensate of molecules and a phase containing two Bose–Einstein condensates, one of atoms and one of molecules, as was
R.A. Duine, H.T.C. Stoof / Physics Reports 396 (2004) 115 – 195
161
8 7 6
T/T0
5 4 3 2 1 -0.003
-0.002
-0.001
0
0.001
0.002
4 π2 −h6 δ/g4 m3
Fig. 19. Temperature of the gas as a function of the detuning for a sweep through the resonance from positive to negative detuning, for two initial temperatures. The total atomic density is equal to n = 1013 cm−3 .
1.1 1 0.9
TBEC /T0
0.8 0.7 0.6 0.5 0.4 0.3 0.2 -0.003
-0.002
-0.001
0
0.001
0.002
0.003
0.004
4 π2 −h6 δ/g4 m3
Fig. 20. Critical temperature for Bose–Einstein condensation as a function of detuning. The total density is equal to n = 1013 cm−3 . The solid line shows the result of the exact calculations. The dashed line shows the result of treating the gas as an ideal-gas mixture.
8rst pointed out by Sachdev [108]. This should be contrasted with the situation of an atomic Fermi gas near a Feshbach resonance, where only a BCS-BEC crossover exists [63]. The calculation of the full detuning-temperature phase diagram is work in progress and will be reported in a future publication [106].
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4.4. Many-body e8ects on the bound-state energy In this section we determine the e4ects of the atomic gas on the molecular binding energy. The 8rst step in an examination of these many-body e4ects is the calculation of the molecular self-energy given in Eq. (104). For simplicity, we neglect the energy dependence of the atom– molecule coupling constant and the many-body e4ects on this coupling constant. After subtraction of the energy-independent shift, the retarded molecular self-energy that includes many-body e4ects is given by the expression [1 + N ( K=2+k − ' ) + N ( K=2−k − ' )] dk 1 (+) 2 ˝7m (K; !) = 2g : (170) + (2)3 ˝!+ − K =2 − 2( k − ' ) 2 k Here, we have treated the atoms in the Hartree–Fock approximation which e4ectively implies that the chemical potential is shifted according to 8a(B)˝2 na (171) ≡ ' − 2T 2B na ; m where na is the density of the atoms. In this expression for the Hartree–Fock self-energy correction to the chemical potential we have neglected the energy-dependence of the interactions, which is justi8ed as long as the scattering length is much smaller than the thermal deBroglie wavelength of the atoms. From now on we restrict ourselves to the regime just above the critical temperature, where we are able to calculate various properties analytically. Since the chemical potential approaches zero from below in this regime, we are allowed to approximate the Bose distribution function of the atoms by 1 N (x) : (172) -x ' = ' −
Within this approximation, the self-energy of the molecules is given by 1 dk 1 (+) 2 ˝7m (K; !) = 4g ; 3 + (2) ˝! − K =2 − 2( k − ' ) -( K =4 + k − ' )
(173)
and we are allowed to also neglect the square-root term that results from the 8rst and last terms in the integrand in Eq. (170), and is due to two-atom physics. This integral is performed analytically. For ˝! ¡ K =2 − 2' the self-energy is real and given by
2 3=2 − ˝! −
=2 − 2'
=2 − 2' m 2g K K (+) : (174) ˝7m (K; !) = ˝3 ˝! For ˝! ¿ K =2 − 2' the self-energy contains an imaginary part and is given by
2 3=2 + i ˝! − =2 + 2'
=2 − 2' m 2g K K (+) : ˝7m (K; !) = − ˝3 ˝!
(175)
To 8nd the energy of the molecular state we have to solve for ˝! in the equation (+) ˝! − K =2 − (B) + 2' − ˝7m (K; !) = 0 :
(176)
A great deal of insight is gained by the graphic representation of this equation which is shown in Fig. 21. The solid line represents the real part of the molecular self-energy as a function of the
R.A. Duine, H.T.C. Stoof / Physics Reports 396 (2004) 115 – 195 εK/2-2µ′
−h
163
ω
hω-εK/2-δ+2µ
−
-εc (K)
Fig. 21. Graphical solution of the equation for the molecular bound-state energy. The solid line indicates the real part of the molecular self-energy as a function of ˝!. The dashed and dotted lines indicates the function ˝! − K =2 − (B) + 2' for di4erent values of the detuning (B). For ˝! ¡ K =2 − 2' , the value of ˝! at the intersections of the dashed and dotted lines with the solid line corresponds to the bound-state energy. For ˝! ¿ K =2 − 2' it corresponds to the energy of resonant states.
energy ˝!. The straight dashed and dotted lines correspond to ˝! − K =2 − (B) + 2', for two di4erent values of (B). From this 8gure it is clear that there is a real solution, i.e., a true bound state, if the detuning is such that (B) ¡ 4T 2B na +
2g2 m3=2 ≡ 4T 2B na + c (K) ≡ max (K) : ˝3 - K =2 − 2'
(177)
Note that this also implies that the position of the resonance in the magnetic 8eld is shifted according to 1 2g2 m3=2 2B 4T na + 3 √ ; (178) B 0 → B0 + T' ˝ - −2' due to many-body e4ects. For a magnetic 8eld such that the detuning is just below the maximum value max (K) given in Eq. (177), the bound-state energy is given by
2B 2 4T ˝2 K 2 n − (B) a + (179) ˝!K −2' 1 − +1
c (0) 2me4 with an e4ective mass given by −1 2 ((B) − 4T 2B na ) 3((B) − 4T 2B na ) me4 = 2m 1− :
c (0) 3
c (0)
(180)
This e4ective mass has a minimum value of 4m=3 at detuning (B) = 4T 2B na + 3 c (0)=4, and diverges for smaller detunings close to 4T 2B na . In the limit of the detuning (B) → −∞ we have to recover the two-body bound state with mass 2m, which shows that this divergence is due to the approximations we have adopted. As already discussed, we have in particular neglected the
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8rst and last terms in the integrand in Eq. (170) that result from two-atom physics. Nevertheless, the fact that the e4ective mass is smaller than the mass of a molecule close to resonance indicates that the molecule crosses over to a more complex many-body bound state of the system. Precisely at the shifted resonance at (B) = 4T 2B na + c (0) the e4ective mass is again equal to 2m. Another interesting feature of the excitation is that for a given detuning it only exist at small momenta such that Eq. (177) is obeyed. The intersections at energies ˝! ¿ K =2 − 2' in Fig. 21, as for example shown by the dotted line, correspond to resonant states since the self-energy contains an imaginary part at these energies. The energies of these resonant states is determined by solving for ˝! in the equation 2g2 m3=2 K =2 − 2' ˝! − K =2 + 2' − (B) + =0 : (181) ˝3 ˝! For a detuning that obeys the condition in Eq. (177) and such that (B) ¿ 2' − K =2 +
8g2 m3=2
K =2 − 2' ≡ min (K) ; ˝3 -
there are two solutions of this equation. They are given by 2 3=2
=2 − 2' 8g m K : ˝!± = 12 ( K =2 + (B) − 2') 1 ± 1 − ˝3 - ( K =2 + (B) − 2')2
(182)
(183)
For large detuning we have that ˝!+ K =2 + (B) − 2', from which we see that this resonant state physically corresponds to the bare molecular state, which has obtained a 8nite lifetime due to the interaction with the atomic continuum. The resonant state at energy ˝!− is not present in the two-atom case but arises purely due to many-body e4ects. This situation is somewhat similar to the Kondo-resonant state that arises in a Fermi gas near a Feshbach resonance [70]. An illustration of the many-body e4ects on the molecular bound-state energy is shown in Fig. 22. The dashed line indicates the situation in vacuum. For negative detuning there is a true molecular state whose energy depends quadratically on the detuning, as given in Eq. (126). For positive detuning the molecule has a 8nite lifetime and therefore corresponds to a resonant state, whose energy is for large detuning equal to the detuning. Due to many-body e4ects, the position of the Feshbach resonance is shifted. Nevertheless, there is still a molecular state with an energy dependence that is quadratic on the many-body renormalized detuning. However, for a detuning larger than min but less than max this molecular state coexists with two resonant states, one close to the detuning and one just above the continuum threshold. The molecular density of states for the latter situation is shown in Fig. 23. The delta function corresponds to the molecular bound state. The dashed lines indicate the position of the resonances. For large positive and large negative detuning the many-body e4ects are negligible and the result reduces to the two-atom result. Finally, we remark that the resonant state at energy ˝!− , that arises solely due to many-body e4ects, leads to a nonzero number of bare molecules, even if the temperature is much smaller than the detuning. This e4ect can be measured by directly measuring the number of bare molecules, as achieved recently by Chin et al. [109]. The investigation of the magnitude and temperature dependence of this e4ect is intended for future work.
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165
εm(B)
δmin δmax
δ(B)
Fig. 22. Molecular bound-state energy as a function of detuning. The dashed line show de molecular bound-state energy in vacuum as a function of detuning. The solid line shows the many-body e4ects on the bound-state energy.
ρm (k,ω)
Z δ (−hω-−hωK)
hωK
−
hω−
−
hω+
−
Fig. 23. Molecular density of states with many-body e4ects. Apart from the delta function that corresponds to the bound state there are two resonant states, indicated by the dashed lines.
5. Mean-*eld theories for the Bose–Einstein condensed phase In the 8rst part of this section we derive the mean-8eld theory that results from our e4ective quantum 8eld theory. This mean-8eld theory is appropriate for the description of the Bose–Einstein condensed phase of the gas. In Section 2 we discuss other possible mean-8eld theories and discuss the similarities and di4erences between them and our mean-8eld theory.
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5.1. Popov theory In this section we derive the mean-8eld equations for the atomic and molecular condensate wave functions. In the 8rst part of this section we derive the time-independent equations and discuss the excitation spectrum. In the second part we derive the time-dependent mean-8eld equations. 5.1.1. Time-independent mean-;eld equations The mean-8eld equations for the atomic and molecular condensate wave functions are derived most easily by varying the e4ective action in Eq. (112) with respect to a∗k; n and b∗k; n , respectively. Before doing so, however, we remark that an important property of this e4ective action is its invariance under global U (1) transformations. Namely, any transformation of the form ak; n → ak; n ei ; bk; n → bk; n e2i ;
(184)
with a real parameter, leaves the action unchanged. The conserved quantity, the so-called Noether charge, associated with this invariance is the total number of atoms. The appearance of the atomic and the molecular condensates breaks the U (1) invariance since the wave functions of these condensates have a certain phase. According to Goldstone’s theorem, an exact property of a system with a broken continuous symmetry is that its excitation spectrum is gapless [110]. Since our mean-8eld theory is derived by varying a U (1)-invariant action, this property is automically incorporated in the mean-8eld theory. To derive the time-independent mean-8eld equations, that describe the equilibrium values of the atomic and molecular condensatewave functions, we substitute into the e4ective action a0; 0 → *a -˝V + a0; 0 and b0; 0 → *m -˝V + b0; 0 . Here, *a and *m correspond to the atomic and molecular condensate wave functions, respectively. Requiring that the terms linear in a0; 0 and b0; 0 vanish from the e4ective action leads to the equations ! " 2B 2' − 2˝7HF |*a |2 *a + 2[g2B (2' − 2˝7HF )]∗ *∗a *m ; '*a = Tbg 2B (2' − 2˝7HF )]*m + g2B (2' − 2˝7HF )*2a : 2'*m = [(B) + ˝7m
(185)
A crucial ingredient in these equations is the Hartree–Fock self-energy of the noncondensed atoms. This self-energy is the mean-8eld energy felt by the noncondensed atoms due to the presence of the atomic condensate. Taking into account the energy dependence of the interactions, it is determined by the expression 2|g2B (' − ˝7HF )|2 HF 2B HF ˝7 = 2na + Tbg (' − ˝7 ) ; (186) 2B (' − ˝7HF ) ˝7HF + ' − (B) − ˝7m with na = |*a |2 the density of the atomic condensate. Its diagrammatic representation is given in Fig. 24. The overall factor of two comes from the constructive interference of the direct and exchange contributions. Far o4 resonance we are allowed to neglect the energy-dependence of the e4ective atom–atom interactions, and the Hartree–Fock self-energy of the atoms is given by 8a(B)˝2 na =m, as expected. The Hartree–Fock self-energy is essential for a correct description of the equilibrium properties of the system. The physical reason for this is understood as follows. In the condensed phase the chemical potential is positive. The energy of a condensate molecule is equal to 2', which
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+
167
2B
Tbg
Fig. 24. Hartree–Fock self-energy of the atoms. The dotted lines correspond to condensate atoms. The dashed line corresponds to the full molecular propagator.
is therefore larger than the continuum threshold of two atoms in vacuum. Without the incorporation of the Hartree–Fock self-energy, the molecular condensate would therefore always decay and an equilibrium solution of the mean-8eld equations would not exist. However, due to the presence of the atomic condensate the continuum threshold shifts by an amount 2˝7HF , and the molecular condensate is stable. To study the collective excitation spectrum over the ground state determined by Eq. (185), we consider the e4ective action up to second order in the Juctuations, which is known as the Bogliubov approximation [111]. To facilitate the notation we introduce the vector uk; n by means of ak; n a∗ −k; −n (187) uk; n ≡ : bk; n b∗−k; −n With this de8nition, the quadratic part of the e4ective action is given by ˝ † SB [u† ; u] = − uk; n · GB−1 (k; i!n ) · uk; n ; 2
(188)
k; n
where the Green’s function of the Juctuations is determined by −1 −1 G G a coup GB−1 = : −1 ∗ −1 [Gcoup ] Gm
(189)
The atomic part of this Green’s function is found from −1 −˝G (k; i! ) 0 n 0; a − ˝Ga−1 (k; i!n ) = 0 −˝G0;−a1 (k; −i!n ) 2B 2B (i˝!n − k =2 + 2' )na Tbg (2' )*2a + 2[g2B (2' )]∗ *m 2Tbg + 2B 2B Tbg (2' )(*∗a )2 + 2g2B (2' )*∗m 2Tbg (i˝!n − k =2 + 2' )na (190)
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where ' ≡ ' − ˝7HF . Note that in the absence of the coupling to the molecular condensate, this result reduces to the well-known result for the Green’s function that describes phonon propagation in a weakly-interacting Bose condensate. We have in this case, however, also explicitly taken into account the energy dependence of the coupling constants. Therefore we know that in the limit of vanishing coupling g2B the propagator in Eq. (190) has a pole that determines the gapless dispersion relation for the phonons. For energy-independent interactions this so-called Bogoliubov dispersion is given by ˝!k =
k2 +
8abg ˝2 na
k : m
The molecular part of the Green’s function GB (k; i!n ) is determined by −1 0 Gm (k; i!n ) −1 ; Gm (k; i!n ) = 0 Gm−1 (k; −i!n )
(191)
(192)
where the single-molecule propagator is given by − ˝Gm−1 (k; i!n ) = −i˝!n + k =2 + (B) − 2' 2B +˝7m (i˝!n − k =2 + 2' − 2˝7HF ) :
(193)
From the previous section we know that the Green’s function in Eq. (193) for negative detuning has a pole at the molecular binding energy. There are now, however, mean-8eld e4ects on this binding energy due to the presence of the atomic condensate, incorporated by the Hartree–Fock self-energy ˝7HF [87]. Finally, the Green’s function that describes the coupling between the atomic and molecular Juctuations is given by 0 2[g2B (i˝!n − k =2 + 2' )]∗ *∗a −1 : (194) − ˝Gcoup (k; i!n ) = 0 2g2B (i˝!n − k =2 + 2' )*a The spectrum of the collective excitations is determined by the poles of the retarded Green’s function for the Juctuations GB (k; !+ ). This implies that we have to solve for ˝! in the equation det GB−1 (k; !+ ) = 0 :
(195)
This is achieved numerically in the next section to determine the frequency of the Josephson oscillations between the atomic and the molecular condensate. However, we are already able to infer some general features of the excitation spectrum of the collective modes. We have seen that in the absence of the coupling between the atomic and molecular condensate, we have that one dispersion is equal to the gapless Bogoliubov dispersion with scattering length abg . In the presence of the coupling this branch corresponds again to phonons, but the dispersion is now approximately equal to the Bogoliubov dispersion for the full scattering length a(B). There is a second dispersion branch that for small coupling g2B lies close to the molecular binding energy. At nonzero coupling this branch corresponds to coherent atom–molecule oscillations, i.e., pairs of atoms oscillating back and forth between the atomic and molecular condensate. Physically, the di4erence between the two branches is understood by realizing that for the phonon modes the phases of the atomic and the molecular condensate are locked to each other and oscillate in phase. Since the action is invariant under the transformations in Eq. (184) we conclude that the phonons are indeed gapless, and,
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169
in fact, correspond to the Goldstone mode associated with the breaking of the U (1) symmetry by the condensates. For the coherent atom–molecule oscillations the phases of the atomic and molecular condensate oscillate out of phase and hence the associated dispersion is gapped. As a 8nal remark we note that we indeed have that det GB−1 (0; 0) = 0 ;
(196)
which shows that there is indeed a gapless excitation, in agreement with Goldstone’s theorem. 5.1.2. Time-dependent mean-;eld equations The time-dependent mean-8eld equations are found most easily by taking the expectation value of the Heisenberg equations of motion in Eq. (117). For notational convenience we restrict ourselves to the situation that we are close to resonance and hence neglect the energy dependence of the various couplings. Moreover, we only take into account the leading-order energy dependence of the molecular self-energy, as given in Eq. (116). Furthermore, we assume that we are at such low temperatures that the e4ects of the thermal cloud may be neglected. Within these approximations, the mean-8eld equations are given by 2 2 4abg ˝2 9*a (x; t) ˝∇ i˝ = − + |*a (x; t)|2 *a (x; t) + 2g*∗a (x; t)*m (x; t) ; m 9t 2m 2 2 9*m (x; t) ˝∇ i˝ = − + (B(t)) *m (x; t) + g*2a (x; t) 9t 4m 3=2 9 ˝ 2 ∇2 2 m −g i i˝ (197) + − 2˝7HF *m (x; t) : 2˝3 9t 4m Note that, since we use renormalized coupling constants in these equations, we should not explicitly include also the so-called anomalous averages because this leads to double-counting of the interatomic interactions. This is explained in detail in the next section. The equilibrium solutions of these mean-8eld equations are space independent and of the form *a (x; t) = *a e−i't=˝ ; *m (x; t) = *m e−2i't=˝ :
(198)
Substitution in Eq. (197) reproduces the time-independent equations for *a and *m within the above approximations. Moreover, by linearizing around these equilibrium solutions we 8nd again the collective-mode spectrum discussed in the previous subsection. We now discuss the solution of the homogeneous version of the time-dependent mean-8eld equations in Eq. (197). These equations are given by
3=2 9*m (t) 9 2 m i˝ = (B(t)) − g i i˝ − 2˝7HF *m (t) + g*2a (t) ; 9t 2˝3 9t 9*a (t) 4abg ˝2 (199) |*a (t)|2 *a (t) + 2g*∗a (t)*m (t) : = m 9t Two di4erent situations can occur, that of time-independent detuning and that of time-dependent detuning. Let us 8rst discuss the case of time-independent detuning. In this case we are able to i˝
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solve the equation for the molecular condensate wave function by introducing the Fourier transform of the zero-momentum part of the retarded molecular Green’s function. This Fourier transform is, for the most interesting case of negative detuning, given by d! (+) G (0; !)e−i!(t −t ) Gm(+) (t − t ) ≡ 2 m √ HF ˝!e−i(!+27 )(t −t ) i(t − t )g2 m3=2 ∞ d! =− ˝2 2 [˝! + 2˝7HF − (B)]2 + (g4 m3 =42 ˝6 )˝! 0 i −i(t − t )Z(B) exp − m (B)(t − t ) ; (200) ˝ where m (B) is the molecular binding energy that includes also the e4ects of the Hartree–Fock self-energy. The molecular condensate wave function is, in terms of this Green’s function, given by g ∞ (+) dt Gm (t − t )*2a (t ) + *m (0)e−i m (B)t=˝ ; (201) *m (t) = ˝ 0 for t ¿ 0. This result is substituted in the equation for the atomic condensate wave function, which can subsequently be solved numerically. The second situation we can have is that of a time-dependent detuning. To take into account the fractional derivative acting on the molecular wave function in the second equation in Eq. (199), we use its de8nition in frequency space. Hence we have that 9 9 ∞ ∞ d! −i!(t −t ) i˝ *m (t) = i˝ dt *m (t ) e 9t 9t −∞ 2 −∞ ∞ ∞ d! √ dt ˝!e−i!(t −t ) *m (t ) : (202) ≡ −∞ −∞ 2 This speci8c de8nition is referred to in the literature as the Weyl de8nition of a fractional derivative [112]. Unfortunately, the integral over ! in the above expression does not converge. This problem is overcome by considering also the next-order energy-dependence of the molecular self-energy. Therefore, we take for the molecular self-energy the expression in Eq. (124), i.e., the molecular self-energy with the e4ective range rbg = 0. The equation for the molecular mean 8eld is then given by
ig2 m3=2 =2˝3 i˝(9=9t) − 2˝7HF 9 i˝ − (B(t)) + *m (t) = g*2a (t) : (203) √ 9t 1 − i|abg | m=˝ i˝(9=9t) − 2˝7HF The term that involves the fractional derivatives is now rewritten as i(g2 m3=2 =2˝3 ) i˝(9=9t) − 2˝7HF *m (t) √ 1 − i(|abg | m=˝) i˝(9=9t) − 2˝7HF √ ∞ d! i(g2 m3=2 =2˝3 ) ˝! − 2˝7HF e−i!(t −t ) *m (t ) √ = dt : √ 2 1 − i(|abg | m=˝) ˝! − 2˝7HF −∞
(204)
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For large ! the integrand becomes equal to a constant which gives rise to a delta function (t − t ). Taking this into account, the 8nal result for this term is given by i(g2 m3=2 =2˝3 ) i˝(9=9t) − 2˝7HF *m (t) √ 1 − i(|abg | m=˝) i˝(9=9t) − 2˝7HF ∞ g2 *m (t) − i =− d x*m (t − x+) 2˝2 |abg |m 0 √ 1 −2ix7HF + ix √ ×e ; (205) − e Erfc( ix) ix where the characteristic time + ≡ ma2bg =˝ and the complementary error function is de8ned by means of ∞ 2 2 dwe−w ≡ 1 − Erf (z) : (206) Erfc(z) ≡ √ z This 8nal result shows that the term involving the fractional derivatives may be dealt with numerically as a term that is nonlocal in time. In the next section we present results of numerical solutions of the time-dependent mean-8eld equations using the Green’s function method. 5.2. Hartree–Fock–Bogoliubov theory A completely di4erent approach to arrive at mean-8eld equations that describe the Bose– Einstein condensed phase of a system with Feshbach-resonant interactions has been put forward by Kokkelmans and Holland [81] and Mackie et al. [80]. Their treatments are physically similar but di4er in the way the renormalization of the bare couplings and detuning is carried out. In 8rst instance, we discuss here the approach of Kokkelmans and Holland. Since the Hartree–Fock–Bogoliubov theory is usually derived by make use of the operator formalism, we abandon for a moment our functional approach and switch in this section to the formulation of quantum mechanics in terms of second-quantized operators. The starting point of Kokkelmans and Holland is the microscopic atom–molecule hamiltonian in Eq. (87). The 8rst step is to approximate the interatomic potential and the atom–molecule coupling as contact interactions, according to V↑↑ (x − x ) V0 (x − x ) ; g↑↓ (x − x ) g0 (x − x ) :
(207)
Roughly speaking, this approximation is validated by the fact that the deBroglie wavelength of the atoms and molecules is much larger than the range of the interactions. However, the use of contact interactions leads to ultraviolet divergencies in the theory which have to be regularized by introducing a ultraviolet cut-o4 k> in momentum space. The unknown microscopic interaction parameters V0 and g0 are then expressed in terms of the experimentally known parameters g, T', and abg , and the cut-o4 k> , in such a way that the 8nal equations correctly describe the two-atom physics and are cut-o4 independent in the limit of a large cut-o4. This renormalization procedure is discussed in detail below.
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First we derive the so-called Hartree–Fock–Bogoliubov equations of motion. Within the above approximation, the hamiltonian for the system is given by 2 2 V0 ˆ † ˝∇ † ˆ ˆ ˆ H = dx a (x) − + (x) a (x) ˆ a (x) 2m 2 a 2 2 ˝∇ ˆ + dx m (x) − + ?(B) ˆ m (x) 4m +g0 dx[ ˆ †m (x) ˆ a (x) ˆ a (x) + h:c:] ; (208) where ?(B) is a bare and also cut-o4 dependent detuning for the molecular state. In this hamiltonian, the SchrPodinger operators that annihilate an atom and a molecule are denoted by ˆ a (x) and ˆ m (x), respectively. Their hermitian conjugates are the creation operators. The starting point in the derivation of the Hartree–Fock–Bogoliubov equations of motion are the equations of motion for the Heisenberg operators ˆ a (x; t) and ˆ m (x; t), that follow from the hamiltonian in Eq. (208). They are given by 2 2 ˝∇ 9 ˆ a (x; t) † ˆ ˆ = − + V0 a (x; t) a (x; t) ˆ a (x; t) + 2g0 ˆ †a (x; t) ˆ m (x; t) ; i˝ 9t 2m 2 2 ˝∇ 9 ˆ m (x; t) = − + ?(B) ˆ m (x; t) + g0 ˆ 2a (x; t) : i˝ (209) 9t 4m The next step is to separate out the expectation value of the Heisenberg operators. These expectation values are constant in space since we are dealing with a homogeneous system. We write the Heisenberg operators as a sum of their expectation values and an operator for the Juctuations according to ˆ a (x; t) = ˆ a (x; t) + 2ˆa (x; t) ≡ *a (t) + 2ˆa (x; t) ; ˆ m (x; t) = ˆ m (x; t) + 2ˆm (x; t) ≡ *m (t) + 2ˆm (x; t) :
(210)
We substitute this result into the Heisenberg equations of motion and take the expectation values of these equations. These expectation values are then decoupled in a manner that is similar to Wick theorem. This is, of course, an approximation in this case since we are dealing with an interacting system. In detail, we only take into account the expectation values ˆ a , ˆ m , 2ˆa 2ˆa , and 2ˆ†a 2ˆa . This leads to four coupled equations of motion for these expectation values. We de8ne the so-called normal and anomalous expectation values according to GN (r; t) ≡ 2ˆ†a (x; t)2ˆa (x ; t) ; GA (r; t) ≡ 2ˆa (x; t)2ˆa (x ; t) ;
(211)
which only depend on the di4erence r = x − x due to translational invariance of the system. Note that the normal average yields the density of non-condensed atoms according to n (t) = GN (0; t). Including the normal average does not alter the conclusions of the following discussion. Therefore, we assume from now on that we are at such low temperatures that there is essentially no thermal cloud present, and therefore take GN (r; t) = 0.
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173
The Hartree–Fock–Bogoliubov equations of motion are given by 9*a (t) = V0 |*a (t)|2 *a (t) + [V0 GA (0; t) + 2g0 *m (t)]*∗a (t) ; i˝ 9t 9*m (t) = ?(B)*m (t) + g0 [*2a (t) + GA (0; t)] ; i˝ 9t 2 2 ˝∇ 9 2 i˝ GA (r; t) = − + 4V0 |*a (t)| GA (r; t) 9t m +[V0 *2a (t) + V0 GA (0; t) + 2g0 *m (t)](r) :
(212)
Note that, as they stand, these equations cannot be derived by varying a U (1)-invariant action. However, we have seen that this U (1) invariance is an exact property of the theory. This problem is overcome by realizing that the anomalous average GA is in fact proportional to the atomic condensate wave function, since it is zero in the normal phase of the gas. More precisely, we have that GA ˙ *2a which renders the equations for the atomic and molecular condensate wave function U (1)-invariant. Moreover, elimination of the anomalous average for the Hartree–Fock–Bogoliubov equations of motion in Eq. (212) leads to renormalization of the bare couplings V0 and g0 . We have already seen in Section 3.1 that introducing a pairing 8eld into the theory leads to a summation of the ladder Feynman diagrams. We expect something similar to occur in this case [113,114]. To study how this renormalization works in detail we study the equilibrium solutions of the Hartree–Fock–Bogliubov equations. Therefore, we substitute *a (t) = *a e−i't=˝ ; *m (t) = *m e−2i't=˝ ; GA (r; t) = GA (r)e−2i't=˝ ;
(213)
from which we 8nd the time-independent Hartree–Fock–Bogoliubov equations '*a = V0 |*a |2 *a + [V0 GA (0) + 2g0 *m ]*∗a ; 2'*m = ?(B)*m + g0 [*2a + GA (0)] ; 2 2 ˝∇ + 4V0 |*a |2 GA (r) + [V0 *2a + V0 GA (0) + 2g0 *m ](r) : (214) 2'GA (r) = − m The equation for the anomalous average GA (r) is solved by Fourier transformation. This gives the result # V0 =V |k|¡k> 1=(2'+ − 2 k − 4V0 |*a |2 ) g0 2 # *a + 2 * m ; GA (0) = (215) 1 − (V0 =V ) |k|¡k> 1=(2'+ − 2 k − 4V0 |*a |2 ) V0 which explicitly shows that the anomalous average is proportional to the atomic condensate wave function. Note also that we have to regularize this expression by using the ultraviolet cut-o4 k> , since it would be ultraviolet divergent otherwise. Converting the sum over momenta to an integral, we 8nd the 8nal result for the anomalous average (V0 m3=2 =2˝3 )i 2' − 4V0 |*a |2 − (V0 mk> =22 ˝2 ) g0 GA (0) = (216) *2a + 2 *m : V0 1 − (V0 m3=2 =2˝3 )i 2' − 4V0 |*a |2 + (V0 mk> =22 ˝2 )
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Substitution of this result into the equations of motion for the atomic and molecular condensate wave functions gives in 8rst instance '*a = Vr |*a |2 *a + 2gr *∗a *m ; 2'*m = ?r (B)*m + gr *2a ;
(217)
where the renormalized interaction and atom–molecule coupling are given by (V02 m3=2 =2˝3 )i 2' − 4V0 |*a |2 − (V0 mk> =22 ˝2 ) Vr = + V0 ; 1 − (V0 m3=2 =2˝3 )i 2' − 4V0 |*a |2 + (V0 mk> =22 ˝2 ) (g0 V0 m3=2 =2˝3 )i 2' − 4V0 |*a |2 − (V0 mk> =22 ˝2 ) gr = + g0 ; 1 − (V0 m3=2 =2˝3 )i 2' − 4V0 |*a |2 + (V0 mk> =22 ˝2 ) and the renormalized detuning is given by (V0 m3=2 =2˝3 )i 2' − 4V0 |*a |2 − (V0 mk> =22 ˝2 ) g02 + ?(B) : ?r (B) = 2 V0 1 − (V0 m3=2 =2˝3 )i 2' − 4V0 |*a |2 + (V0 mk> =22 ˝2 )
(218)
(219)
Finally, we have to express these renormalized quantities in terms of the experimentally known parameters abg , g, and (B). Moreover, this has to be performed in a manner that does not depend on the cut-o4 in the limit k> → ∞. The renormalization procedure used by Kokkelmans and Holland is given by 4abg ˝2 =m V0 = ; 1 − (mk> =22 ˝2 )(4abg ˝2 =m) g0 =
g 1 − (mk>
=22 ˝2 )(4a
bg ˝
2 =m)
;
mk> g0 g : (220) 42 ˝2 Eliminating the microscopic parameters V0 , g0 , and ?(B) in favor of abg , g, and (B) 8nally yields the renormalized mean-8eld equations for the atomic and molecular wave functions 4abg ˝2 =m 2g '*a = |*a |2 *a + *∗a *m ; 1 + iabg m=˝2 (2' − 4V0 |*a |2 ) 1 + iabg m=˝2 (2' − 4V0 |*a |2 )
2' − 4V0 |*a |2 g2 m3=2 2'*m = (B) − i *m 2˝3 1 − i|abg | m=˝2 (2' − 4V0 |*a |2 ) ?(B) = (B) +
+
1 + iabg
g m=˝2 (2'
− 4V0 |*a |2 )
*2a ;
(221)
where we have retained the term 4V0 |*a |2 in the energy arguments of the coupling constants. In the limit k> → ∞ this term vanishes and the above renormalized equations no longer depend on the microscopic parameters and the cut-o4. The above equations are very similar to the mean-8eld equations of our e4ective 8eld theory in Eq. (185), if we neglect the e4ective range of the interactions in the couplings and the self-energy of
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the molecules in the latter equations. There is, however, another and much more important di4erence between the two mean-8eld theories. In the mean-8eld theory that we have derived from our e4ective quantum 8eld theory we have included the Hartree–Fock self-energy that is due to the mean-8eld interactions of the condensate on the thermal atoms. This Hartree–Fock self-energy is crucial for a correct description of the equilibrium properties of the system. In the Hartree–Fock–Bogoliubov equations the Hartree–Fock self-energy is replaced by the energy 4V0 |*a |2 , which corresponds to the mean-8eld energy resulting from the unrenormalized interaction. The fact that the interaction between the condensed and noncondensed atoms is not renormalized is a well-known problem of the Hartree–Fock–Bogoliubov theory [115]. Note also that for a nonzero e4ective range rbg the two-atom physics is not incorporated exactly, and this will lead to a discrepancy with experiment as shown in the following section. Although the renormalization of the interactions between condensate atoms is, for rbg = 0, correctly achieved, the interactions between condensate atoms and thermal atoms is not correctly incorporated. In the limit where the cut-o4 k> goes to in8nity this mean-8eld energy actually vanishes and we conclude from our previous discussion that the Hartree–Fock– Bogoliubov equations in Eq. (214) have no equilibrium solution. As a result also a linear-response analysis, similar to the one carried out in Section 6, is not possible with this approach. Moreover, the above renormalization procedure relies on the presence of the anomalous average GA (r) which makes the theory inapplicable above the critical temperature for Bose–Einstein condensation. Hence also a description of the thermal cloud of a Bose–Einstein condensed gas cannot be obtained in this manner. Note also that the above result explicitly shows that the inclusion of the pairing 8eld GA (r) indeed leads to the summation of the ladder diagrams. This is the reason why it is exact not to include anomalous averages in our mean-8eld equations. Their e4ect is already incorporated by using properly renormalized coupling constants. We are now in the position to discuss also the renormalization procedure used by Mackie et al. [80]. First of all, the nonresonant interactions between the atoms are neglected in their approach, i.e., V0 is taken equal to zero in the Hartree–Fock–Bogoliubov equations in Eq. (212). This simpli8cation is justi8ed for magnetic 8elds suKciently close to resonance. Subsequently, in the absence of an atomic condensate, i.e., *a = 0, and for a given cut-o4 k> , these authors solve the time-independent version of these equations to determine the molecular binding energy. To 8x the unknown bare detuning ?(B) and bare atom–molecule coupling constant g0 two constraints are needed. The 8rst is that the Feshbach resonance is at its experimentally observed value of the magnetic 8eld. The second is to take the calculated molecular binding energy equal to the experimentally observed oscillation frequency in the number of atoms in the atomic condensate [71]. This approach neglects the possibility of many-body e4ects, in that this observed frequency is not precisely equal to the molecular binding energy but is shifted due to the presence of the atomic condensate, as we shall see in the next section. The above procedure is then carried out for a suKciently large cut-o4 which renders the results independent of the precise value of this cut-o4. Finally, we make some remarks about the theory put forward by KPohler et al. [86]. These authors do not explicitly include the molecular 8eld responsible for the Feshbach resonance into their theory, but instead use a separable pseudopotential for the interaction between the atoms that, when inserted in the Lippmann–Schwinger equation, reproduces the energy-dependent T -matrix. Subsequently, they use the single-channel version of the above-described Hartree–Fock–Bogoliubov theory to arrive at their mean-8eld equations. The theory of KPohler et al. is derived from our e4ective atom–molecule approach by neglecting the e4ect of the molecular condensate on the atoms. The molecular 8eld can
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then be integrated out, which leads to an energy-dependent T -matrix for the atoms. We have seen in Eq. (128) that close to resonance the energy dependence of this T -matrix is equivalent to the energy dependence of the T -matrix in the single-channel case. Close to resonance, therefore, the mean-8eld theory of KPohler et al. incorporates the correct two-atom physics. However, their approach cannot fully recover all the properties of the molecules, which have been integrated out of the problem. This can for instance be seen from the fact that the theory contains only the ratio g2 =T' instead of the independent quantities g and T', seperately. Their theory also does not incorporate the mean-8eld shift on the noncondensed atoms due to the atomic condensate, as we have seen explicitly above. The latter feature again disables a linear-response analysis of the beautiful experiments we are going to discuss next.
6. Coherent atom–molecule oscillations In this section we discuss the experimental observation of atom–molecule coherence in a Bose– Einstein condensate [71,82], and its theoretical description in terms of the mean-8eld theory derived in the previous section. In the 8rst section we discuss the experimental results. In the next section we calculate the magnetic-8eld dependence of the frequency of the coherent atom–molecule oscillations in linear-response theory. In the 8nal section we present the results of calculations that go beyond this linear approximation. 6.1. Experiments In the experiments of Donley et al. [71] and Claussen et al. [82], performed both in Wieman’s group at JILA, one makes use of the Feshbach resonance at B0 = 155:041(18) G(auss) in the |f = 2; mf = −2 hyper8ne state of 85 Rb. The width of this resonance is equal to TB = 11:0(4) G and the o4-resonant background scattering length is given by abg = −443a0 , with a0 the Bohr radius. The di4erence in the magnetic moment between the open channel and the closed channel is given by T' = −2:23'B , with 'B the Bohr magneton [81]. In both experiments, one starts from a stable and essentially pure condensate of about Nc = 10 000 atoms at a magnetic 8eld such that the e4ective scattering length is close to zero. This implies that, since the condensate is in the noninteracting limit, its density pro8le is determined by the harmonic-oscillator groundstate wave function. The harmonic external trapping potential is axially symmetric, with trapping frequencies ?r = 17:4 and ?z = 6:8 Hz in the radial and axial directions, respectively. Starting from this situation, one quickly ramps the magnetic 8eld to a value Bhold close to the resonant value and keeps it there for a short time thold before ramping to a value Bevolve . The magnetic 8eld is kept at this last value for a time tevolve before performing a similar pulse to go back to the initial situation. The duration of all four magnetic-8eld ramps is given by tramp . A typical pulse is illustrated in Fig. 25. Both the ramp time tramp and the hold time thold are kept 8xed at values of 10–15 s. The time tevolve between the pulses is variable. Such a double-pulse experiment is generally called a Ramsey experiment. Its signi8cance is most easily understood from a simple system of two coupled harmonic oscillators. Consider therefore
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177
166
164
B (G)
162
tevolve
160
158
156
B0 154 0
20
40
60
80
100
120
time (µs)
Fig. 25. Typical magnetic-8eld pulse sequence as used in the experiments of Donley et al. [71] and Claussen et al. [82].
the hamiltonian Hˆ = 12 (aˆ†
bˆ† ) ·
(t) /
aˆ · ; −(t) bˆ /
(222)
where aˆ† and bˆ† create a quantum in the oscillators a and b, respectively, and / denotes the coupling between the two oscillators. We consider 8rst the situation that the detuning (t) is time independent. The exact solution is found easily by diagonalizing the hamiltonian. We assume that initially there are only quanta in ˆ oscillator a and none in b, so that we have that bˆ† b (0) = 0. The number of quanta in oscillator a as a function of time is then given by /2 2 aˆ† a (t) ˆ = 1− sin ($t=2) aˆ† a (0) ˆ ; (223) (˝$)2 with the frequency $ given by ˝$ = 2 + /2 :
(224)
We see that the number of quanta in the oscillator a oscillates in time with frequency $. Such oscillations are called Rabi oscillations. Note that the number of quanta in oscillator b is determined by ˆ =− bˆ† b (t)
/2 sin2 ($t=2) aˆ† a (0) ˆ ; (˝$)2
(225)
so that the total number of quanta is indeed conserved. Suppose now that we start from the situation with all quanta in the oscillator a and none in b and that the detuning is such that (t)/. Then we have from Eq. (223) that aˆ† a (t) ˆ aˆ† a (0) ˆ † ˆ ˆ and b b (t) 0. Starting from this situation, we change the detuning instantaneously to a value
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(t) 0 and keep it at this value for a time thold . During this hold time quanta in oscillator a will go to oscillator b. Moreover, if thold is such that ˝ ; (226) 2 / on average half of the quanta in oscillator a will go to oscillator b. Such a pulse is called a =2-pulse. The de8ning property of a =2-pulse is that it creates a superposition of the oscillators a and b, such that the probabilities to be in oscillators a and b are equal, and therefore equal to 1=2. This ˆ is indicated by the average aˆ† b (t). At t = 0 this average is equal to zero because there is no ˆ superposition at that time. We can show that after the above =2-pulse the average aˆ† b (t) reaches its maximum value. In detail, the state after the =2-pulse is equal to
N 1 aˆ† + bˆ† √ √ |0 ; (227) N! 2 thold
where the ground state is denoted by |0 , and N = aˆ† a (0). ˆ We can now imagine the following experiment. Starting from the situation (t)/, we perform a =2-pulse. Then jump to a certain value evolve for a time tevolve , and after this perform another =2-pulse and jump back to the initial situation. The number of quanta in the oscillator a, a measurable quantity, then oscillates as a function of tevolve with the oscillation frequency determined by Eq. (224) evaluated at the detuning evolve . The second =2-pulse enhances the contrast of the measurement thus providing a method of measuring the frequency $ as a function of the detuning with high precision. This is basically the idea of the Ramsey experiments performed by Donley et al. [71] and Claussen et al. [82]. Roughly speaking, the atomic condensate corresponds to oscillator a and the molecular condensate to oscillator b. Therefore, after performing the double-pulse sequence in the magnetic 8eld one makes a light-absorption image of the atomic density from which one extracts the number of condensate and noncondensed atoms. Since this imaging technique is sensitive to a speci8c absorption line of the atoms it does not measure the number of molecules. From the above discussion we expect to observe oscillations in the number of condensate atoms. Moreover, if the situation is such that the detuning between the pulses is relatively large the e4ect of the coupling can be neglected and the frequency of the observed oscillations corresponds to the energy di4erence between the atoms and the molecules, i.e., the molecular binding energy. This is indeed what is observed, thereby providing compelling evidence for the existence of coherence between atoms and molecules. In Fig. 26 the experimental results of Claussen et al. [82] are presented. Fig. 26(a) and (b) show the number of atoms in the atomic Bose–Einstein condensate as a function of tevolve after a double-pulse sequence. Clearly, there is an oscillation in the number of atoms in both cases. In Fig. 26(a) the magnetic 8eld between the pulses is Bevolve = 156:840(25) G. In Fig. 26(b) we have Bevolve = 159:527(19) G which is further from resonance. This explains also the increase in frequency from (a) to (b) since further from resonance the molecular binding energy is larger. What is also observed is that there is a damping of the oscillations and an overall loss of condensate atoms. Experimentally, the number of atoms in the condensate is 8t to the formula Nc (t) = Naverage − &t + A exp(−-t) sin(!e t + *) ;
(228)
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binding energy (kHz)
1000
BEC number
10000
8000 8000
6000 6000
4000
0
100
(a)
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t evolve (µs)
(c)
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25
12 8
800
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4 0 156.0
156.5
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0 156 157 158 159 160 161 162
30
(d)
magnetic field (G)
Fig. 26. Experimental observation of coherent atom–molecule oscillations. The 8gures are taken from Ref. [82]. Figures (a) and (b) show the number of atoms in the atomic condensate as a function of the time between the two pulses in the magnetic 8eld. The solid line indicates the 8t in Eq. (228). For (a) we have that Bevolve = 156:840(25) G. The frequency and damping rates are respectively given by ?e = 2 × 0:58(12) kHz, & = 7:9(4) atom=s, and - = 2 × 0:58(12) kHz. For (b) the magnetic 8eld Bevolve = 159:527(19) G and ?e = 157:8(17) kHz. The damping is negligible for the time that is used to determine the frequency. Note that the frequency has increased for the magnetic 8eld further from resonance. Figures (c) and (d) show the observed frequency of the coherent atom–molecule oscillations as a function of the magnetic 8eld. The solid line is the result for the molecular binding energy found from a two-body coupled-channels calculation using the experimental results for the frequency to accurately determine the interatomic potential [82]. Only the black points were included in the 8t. The inset shows that, close to resonance, the observed frequency deviates from the two-body result. Reprinted 8gure with permission from N.R. Claussen, S.J.J.M.F. Kokkelmans, E.A. Donley, C.E. Wieman, Phys. Rev. A 67 (2003) 060701(R). ? 2003 American Physical Society.
where Naverage is the average number of condensate atoms, A and * are the oscillation amplitude and phase, respectively, and - is the damping rate of the oscillations. The overall atom loss is characterized by a rate constant &. The experimentally observed frequency is equal to !e = 2 ?2e − [-=2]2 . By de8ning the frequency of the coherent atom–molecule oscillation in this way one compensates for the e4ects of the damping on the frequency. For the results in Fig. 26(a) we have that - = 2 × 0:58(12) kHz and & = 7:9(4) atom=s. The frequency is equal to ?e = 9:77(12) kHz. For Fig. 26(b) the frequency is equal to ?e = 157:8(17) kHz. The damping and loss rate are negligible for the short time used to determine the frequency. It is found experimentally that both the damping rate and the loss rate increase as Bevolve approaches the resonant value. In Fig. 26(c) and (d) the results for the frequency as a function of Bevolve are presented. The solid line shows the result of a two-body coupled-channels calculation of the molecular binding energy [82]. The parameters of the interatomic potentials are 8t to the experimental results for the
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frequency. Clearly, the frequency of the coherent atom–molecule oscillations agrees very well with the molecular binding energy in vacuum over a large range of the magnetic 8eld. Moreover, in the magnetic-8eld range Bevolve 157–159 G the frequency of the oscillations is well described by the formula | m (B)| = ˝2 =ma2 (B) for the binding energy, derived in Section 4.1.2. Close to resonance, however, the measured frequency deviates from the two-body result. The deviating experimental points are shown by open circles and are not taken into account in the determination of the interatomic potential. This deviation is due to many-body e4ects [83]. Although some of the physics of these coherent atom–molecule oscillations can roughly be understood by a simple two-level picture, it is worth noting that the physics of a Feshbach resonance is much richer. First of all, during Rabi oscillations in a simple two-level system one quantum in a state oscillates to the other state. In the case of a Feshbach resonance pairs of atoms oscillate back and forth between the dressed-molecular condensate and the atomic condensate. Therefore, the hamiltonian is not quadratic in the annihilation and creation operators and the physics is more complicated. In particular the dressed molecule may decay into two noncondensed atoms instead of forming two condensate atoms. This process is discuss in detail below. Second, the observed atom–molecule oscillations are oscillations between an atomic condensate and a dressed molecular condensate. The fact that one of the levels is a dressed molecule implies that by changing the magnetic 8eld not only the detuning is altered, but also the internal state of the molecule itself. This is seen most easily by considering the linearized version of the time-dependent mean-8eld equation in Eq. (199). Writing *a (t) = *a e−i't=˝ + *a (t) and *m (t) = *m e−2i't=˝ + *m (t), we have that
3=2 9*m (t) 9 2 m HF i˝ *m (t) + 2g*a *a (t) ; i i˝ − 2˝7 = (B) − g 9t 2˝3 9t 9*a (t) (229) = 2g*∗a *m (t) ; 9t where we neglected the o4-resonant part of the interatomic interactions. This is justi8ed suKciently close to resonance, where we are also allowed to neglect the energy dependence of the atom–molecule coupling constant. Consider 8rst the situation that the fractional derivative is absent in the linearized mean-8eld equations in Eq. (229), i.e., we are dealing with the model of Drummond et. al. [72], and Timmermans et al. [73,85]. These coupled equations describe exactly the same Rabi oscillations as the coupled harmonic oscillators in Eq. (222), with the coupling equal to / = |4g*a |. In the context of particle-number oscillations between condensates, Rabi oscillations are referred to as Josephson oscillations and the associated frequency is called the Josephson frequency. The Josephson frequency in the absence of the fractional derivative term in Eq. (229) is given by ˝!Jbare = 2 (B) + 16g2 na ; (230) i˝
which reduces to ˝!Jbare |(B)| suKciently far o4 resonance where the coupling may be neglected. This result does not agree with the experimental result because, by neglecting the fractional derivative, which corresponds to the molecular self-energy, we are describing Josephson oscillations between an atomic condensate and a condensate of bare molecules instead of dressed molecules. Furthermore, using the result in Eq. (223) we have that the amplitude of these oscillations
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181
is given by 16g2 na : (231) [(B)]2 In 8rst approximation we take the dressing of the molecules into account as follows. If we are in the magnetic-8eld range where the Josephson frequency deviates not too much from the molecular binding energy, we are allowed to expand the propagator of the molecules around the pole at the bound-state energy. As we have seen in Section 4.1.3 this corresponds to introducing the dressed molecular 8eld and leads to the Heisenberg equations of motion in Eq. (148). The linearized mean-8eld equations that describe the Josephson oscillations of a atomic and a dressed-molecular condensate are therefore given by 9*m (t) i˝ = m (B)*m (t) + 2g Z(B)*a *a (t) ; 9t 9*a (t) (232) i˝ = 2g Z(B)*∗a *m (t) ; 9t and lead to the Josephson frequency 2 (B) + 16g2 Z(B)n ; ˝!J = m (233) a Abare = J
which reduces to ˝!J | m (B)| in the situation where the coupling is much smaller than the binding energy. This result agrees with the experimental fact that the measured frequency is, suKciently far from resonance, equal to the molecular binding energy. Moreover, the initial deviation from the two-body result in the measured frequency is approximately described by the equation for the Josephson frequency in Eq. (233). The amplitude of the oscillations is in this case given by 16g2 Z(B)na AJ = ; (234) [ m (B)]2 which close to resonance is much larger than the result in Eq. (231). To get more quantitative understanding of the magnetic-8eld dependence of the Josephson frequency over the entire experimentally investigated range of magnetic 8eld we calculate this frequency in a linear-response approximation, including the energy dependence of the atom–molecule coupling and the atom–atom interactions. Before doing so, we make some remarks about the origin of the damping of the coherent atom–molecule oscillations and the overall loss of atoms that is observed in the experiments. One contribution to the damping is expected to be due to rogue dissociation [80]. Physically, this process corresponds to a pair of condensate atoms forming a dressed condensate molecule that then breaks up into two noncondensed atoms instead of oscillating back to the atomic condensate. This process is incorporated into our theory by the imaginary part of the molecular self-energy. As explained in Section 4.1.3 in the derivation of the Heisenberg equations of motion in Eq. (148), that involve the dressed molecules, we have neglected such a process. It is, however, incorporated in the full solution of the mean-8eld equation in Eq. (199). In the last section we present the results of numerical solutions of these equations. The overall loss of atoms from the atomic condensate is also partially due to the rogue-dissociation process. The experimental fact that a signi8cant thermal component is formed during the double-pulse sequence supports this idea. Apart from this process, it may also be that conventional loss processes,
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such as dipolar decay and three-body recombination play a role. Although such processes are expected to become more important near a Feshbach resonance, they are, however, not included in our simulations since there is no detailed knowledge about the precise magnetic-8eld dependence near the resonance. In principle, however, these loss processes could be straightforwardly included in our calculations, by adding the appropriate imaginary terms to the mean-8eld equations. Another possible mechanism is the loss of atoms due to elastic collisions, the so-called quantum evaporation process [42]. This process is also not included in our present calculations. 6.2. Josephson frequency With the mean-8eld theory derived in the previous sections we now calculate the magnetic-8eld and density dependence of the Josephson frequency of the coherent atom–molecule oscillations, in a linear approximation. The only parameter that has not been determined yet is the e4ective range of the interatomic interactions rbg . All other parameters are known for 85 Rb. The e4ective range is determined by calculating the molecular binding energy in vacuum and comparing the result with the experimental data. We have seen that far o4 resonance the Josephson frequency is essentially equal to the molecular binding energy. Since the e4ect of a nonzero e4ective range only plays a role for large energies, and thus is important far o4 resonance, this comparison uniquely determines the e4ective range. As explained in detail in Section 4.1.2, the molecular binding energy is determined by solving for E in the equation (+) E − (B) − ˝7m (E) = 0 :
(235)
For 85 Rb the background scattering length is negative and the e4ective range turns out to be positive. The retarded molecular self-energy is therefore given by (+) ˝7m (E)
g2 m =− r 2˝2 1 − 2 abgbg
(1 − 2(rbg =abg ))mE=˝2 − (rbg mE=2˝2 ) : 1 + iabg (1 − 2(rbg =abg ))mE=˝2 − (rbg abg mE=2˝2 ) i
(236)
In Fig. 27 the result of the numerical solution of Eq. (235) is shown for rbg = 185a0 . Also shown in this 8gure are the experimental data points. Clearly, far o4 resonance there is good agreement between our results and the experimental data points. Therefore, we use this value for the e4ective range from now on in all our calculations. The absolute value of the detuning is shown by the dotted line, and deviates signi8cantly from the binding energy. The dashed line in Fig. 27 indicates the formula | m | = ˝2 =ma2 . As we have derived in Section 4.1.2 this formula should accurately describe the magnetic-8eld dependence of the binding energy close to resonance. Clearly, the solid line that indicates the result that includes the nonzero e4ective range becomes closer to the dashed line as we approach resonance. However, there is a signi8cant range of magnetic 8eld where we need to include the e4ective range in our calculations. Closer to the resonance, the experimental points start to deviate from the two-atom binding energy. This deviation is taken into account by considering many-body e4ects. Note, therefore, that the expected oscillation frequency ˝2 =ma2 never leads to a quantitative agreement with experiment. As mentioned previously, we calculate the many-body e4ects on the frequency of the coherent atom–molecule oscillations in linear approximation. Therefore, we 8rst need to determine the
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183
100000
frequency (kHz)
10000
1000
100
10
1 156
157
158
159
160
161
162
B (G)
Fig. 27. Molecular binding energy in vacuum. The solid line shows the result of a calculation with rbg =185a0 . The dashed line shows | (B)| = ˝2 =ma2 . The experimental points are taken from [82]. The dotted line shows the detuning |(B)|.
equilibrium around which to linearize. In detail, the equilibrium values of the atomic and molecular condensate wave functions are determined by solving the time-independent mean-8eld equations in Eq. (185) together with the equation for the Hartree–Fock self-energy in Eq. (186) at a 8xed chemical potential '. To compare with the experimental results it is more convenient to solve these equations at a 8xed condensate density. The chemical potential is then determined from these equations. In Fig. 28 we show the result of this calculation for an atomic condensate density of na = 2 × 1012 cm−3 . The solid line shows the Hartree–Fock self-energy ˝7HF and the dashed line the chemical potential as a function of the magnetic 8eld, both in units of the energy 4a(B)˝2 na =m. Note that far o4 resonance, where the energy dependence of the interaction may be neglected, we have that ' =4a(B)˝2 na =m and ˝7HF =2'. This is the expected result. The inset of Fig. 28 shows the fraction of bare molecules |*m |2 =na . Note that this fraction is always very small. This justi8es neglecting the atom–molecule and molecule–molecule interactions since from this 8gure we see that the mean-8eld energies associated with these interactions are at least three orders of magnitude smaller. A posteriori this observation justi8es neglecting the e4ect of the presence of the molecular condensate on the atoms in the approach of KPohler et al. [86]. Since the coherent atom–molecule oscillations are a collective mode where the amplitude of the atomic and molecular condensate wave functions oscillate out-of-phase, we study the collective modes of the system. As explained in detail in the previous section, the frequencies of the collective modes are determined by Eq. (195). This equation is solved numerically and yields a dispersion relation with two branches. The result of this calculation is shown in Fig. 29 for an atomic condensate density of na = 2 × 1012 cm−3 and a magnetic 8eld of B = 157 G. The momentum is indicated in units of the inverse coherence length A−1 = 16a(B)na . The upper branch corresponds to the gapless phonon excitations. For small momenta this branch has a linear momentum dependence. The upper dashed line indicates the Bogoliubov dispersion in Eq. (191) evaluated at the scattering length a(B). For small momentum
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fraction of molecules
6
5
3
0.0006 0.0004 0.0002 0 -0.0002 156 157 158 159 160 161 162
−
hΣHF and µ
4
0.001 0.0008
2
1
0 156
157
158
159
160
161
162
B(G)
frequency (kHz)
Fig. 28. Hartree–Fock self-energy (solid line) and chemical potential (dashed line) as a function of the magnetic 8eld for an atomic condensate density of na = 2 × 1012 cm−3 . Both quantities are shown in units of 4a(B)˝2 na =m. Far o4 resonance, where the energy dependence of the interactions can be safely neglected we have that ˝7HF = 8a(B)˝2 na =m and ' = 4a(B)˝2 na =m, as expected. The inset shows the fraction of bare molecules as a function of the magnetic 8eld.
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 0
0.2
0.4
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0.8
1
0.6
0.8
1
-12.15 -12.2 -12.25 -12.3 -12.35 -12.4 -12.45 -12.5 -12.55
k Fig. 29. The dispersion relation for the collective modes of an atom–molecule system for a condensate density of na = 2 × 1012 cm−3 at a magnetic 8eld of B = 157 G. The momentum is measured in units of the inverse coherence −1 length A = 16a(B)na . The upper branch corresponds to the gapless dispersion for phonons. The solid line is the result of the full calculation, the dashed line shows the Bogoliubov dispersion for the scattering length a(B). The lower branch corresponds to the coherent atom–molecule oscillations. The solid line is the result of the full calculation whereas the dashed line shows the result with the same zero-momentum part, but with the momentum dependence determined by ˝2 k2 =4m.
the solid and the dashed line are almost identical. For larger momenta the numerically exact result is smaller, due to the energy dependence of the interactions that e4ectively reduce the scattering length.
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162
B (G)
Fig. 30. Josephson frequency of coherent atom–molecule oscillations for various values of the condensate density. The solid lines are the results of calculations for nonzero condensate density. The di4erent lines correspond from top to bottom to the decreasing condensate densities na = 5 × 1012 cm−3 , na = 2 × 1012 cm−3 , and na = 1012 cm−3 . The dashed line corresponds to the molecular binding energy in vacuum, i.e., na = 0. The experimental data points, taken from Ref. [82], are also shown.
The lower branch corresponds to the coherent atom–molecule oscillations and is gapped. The solid line indicates the result of the full calculations. For small momenta it is well described by ˝!k −˝!J + k =2 ;
(237)
where !J is the Josephson frequency. The dispersion resulting from this last equation is shown in the lower part Fig. 29 by the dashed line. This momentum dependence is to be expected since suKciently far from resonance the atom–molecule oscillations reduce to a two-body excitation. The fact that the dispersion is negative is due to the fact that we are linearizing around a metastable situation with more atoms than molecules. Although this is the experimentally relevant situation, the true equilibrium situation for negative detuning corresponds to almost all atoms in the molecular state [85]. In Fig. 30 we present the results for the Josephson frequency as a function of the magnetic 8eld, for di4erent values of the condensate density. The solid lines in this 8gure show, from top to bottom, the results for an decreasing nonzero condensate density. The respective condensate densities are given by na = 5 × 1012 cm−3 , na = 2 × 1012 cm−3 , and na = 1012 cm−3 . The dashed line shows the molecular binding energy in vacuum. The Josephson frequency reduces to the molecular binding energy for all values of the condensate density, in agreement with previous remarks. Nevertheless, suKciently close to resonance there is a deviation from the two-body result due to many-body e4ects. This deviation becomes larger with increasing condensate density. In order to confront our results with the experimental data we have to realize that the experiments are performed in a magnetic trap. Taking only the ground states *a (x) and *m (x) into account for both the atomic and the molecular condensates, respectively, this implies e4ectively that the atom– molecule coupling g is reduced by an overlap integral. Hence we√ de8ne the e4ective homogeneous √ condensate density by means of na = Na [ dx*2a (x)*m (x)]2 = 16 2Na m3=2 ?r ?z =(1253 ˝3=2 ), where
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relative frequency (kHz)
8 B=156.5 G
6
B=157 G
4
2
0
1011
two-body binding energy 1012
1013 -3
1014
na (cm )
Fig. 31. Josephson frequency of coherent atom–molecule oscillations as a function of the condensate density, for 8xed magnetic 8eld. We have subtracted the molecular binding energy.
Na denotes the number of condensed atoms and ?r and ?z the radial and axial trapping frequencies, respectively. For the experiments of Claussen et al. we have that Na 8000 during the oscillations close to resonance as seen from Fig. 26, which results in an e4ective density of na 2 × 1012 cm−3 . This agrees also with the e4ective homogeneous density quoted by Claussen et al. [82]. The solid curve in Fig. 30 clearly shows an excellent agreement with the experimentally observed frequency for this density. It is important to note that there are two hidden assumptions in the above comparison. First, we have used that the dressed molecules are trapped in the same external potential as the atoms. This is not obvious because the bare molecular state involved in the Feshbach resonance is high-8eld seeking and therefore not trapped. However, Eq. (133) shows that near resonance almost all the amplitude of the dressed molecule is in the low-8eld seeking open channel and its magnetic moment is therefore almost equal to twice the atomic magnetic moment. Second, we have determined the frequency of the coherent atom–molecule oscillations in equilibrium. In contrast, the observed oscillations in the number of condensate atoms is clearly a nonequilibrium phenomenon. This is, however, expected not to play an important role because the Ramsey-pulse sequence is performed on such a fast time scale that the response of the condensate wave function can be neglected. By variationally solving the Gross–Pitaevskii equation for the atomic condensate wave function, we have explicitly checked that after a typical pulse sequence its width is only a few percent larger than the harmonic oscillator ground state. Finally, we calculate the Josephson frequency as a function of the condensate density. The results of this calculation are presented in Fig. 31, for various values of the magnetic 8eld which is kept 8xed in these calculations. In the presentation of the results we have subtracted the molecular binding energy to bring out the many-body e4ects more clearly. As expected, the di4erence between the Josephson frequency and the molecular binding energy increases with increasing condensate density. Moreover, for values of the magnetic 8eld closer to resonance the di4erence is also larger.
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187
The above calculations in the linear approximation give already a great deal of insight in the coherent atom–molecule oscillations, and, in particular, in their many-body aspects. In the next section we aim at achieving also insight in the nonlinear dynamics and damping resulting from the time-dependent mean-8eld equations for the double-pulse experiments. In particular, we also discuss the rogue-dissociation process. The nonlinear e4ects in these experiments has 8rst been discussed by Kokkelmans and Holland [81], Mackie et al. [80], and KPohler et al. [86], on the basis of their mean-8eld approaches summarized in Section 5.2. The numerical solutions of the Hartree–Fock–Bogoliubov equations in Eq. (212) calculated by Kokkelmans and Holland indeed show an oscillatory behavior in the number of condensate atoms with a frequency that is in reasonably good agreement with the experiments of Ref. [71]. These authors also consider the normal component of the atomic gas, and in particular calculate the energy of the noncondensed atoms after the magnetic-8eld pulse sequence. The results of these calculation for the energy of the thermal atoms is of the same order of magnitude as the observed energy of the burst atoms. Moreover, Kokkelmans and Holland also 8nd a phase shift between the oscillations in the number of condensate atoms, and the number of noncondensed atoms. Such a shift is indeed observed experimentally. The calculations carried out by Mackie et al. also show an oscillatory behavior in the number of condensate atoms. However, these authors do not make a detailed comparison with experiment, as their aim is mostly to extract the qualitative physics of their mean-8eld equations. The solutions of the mean-8eld equations by KPohler et al. have also an oscillatory behavior in the number of condensate atoms as a function of the time between the two pulses in the Ramsey experiment. These authors also calculate the energy of the burst atoms, and 8nd results very similar to those of Kokkelmans and Holland. None of the above calculations discusses the experimentally observed damping of the oscillations, and in particular its magnetic-8eld dependence. This damping will be discussed in the next section. 6.3. Beyond linear response In this section we discuss the numerical solution of the time-dependent mean-8eld equations using the methods described in Section 5.1.2. We focus here on the situation where the detuning is only changed instantaneously, so that we are allowed to use the Green’s function method discussed in this section. After the elimination of the molecular condensate wave function from the mean-8eld equations, the e4ective equation for the atomic condensate wave function is then given by i˝
9*a (t) 4abg ˝2 |*a (t)|2 *a (t) + 2g*∗a (t)*m (0)e−i m (B)t=˝ = 9t m 2ig2 *∗a (t) t − dt Z(B)e−i=˝ m (B)(t −t ) *2a (t ) ˝ 0 g2 m3=2 + ˝2
0
∞
√ HF d! ˝!e−i(!+27 )(t −t ) *2a (t ) : 2 [˝! + 2˝7HF − (B)]2 + (g4 m3 =42 ˝6 )˝!
(238)
In this equation, the term that involves the integral over frequencies describes the fact that a pair of condensate atoms that forms a molecule can decay into a pair of noncondensed atoms with opposite
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fraction of atoms in condensate
1
0.95
0.9
0.85
0.8
0.75 0
50
100
150
200
250
300
time (µs)
Fig. 32. Fraction of atoms in the atomic condensate. The solid line shows the result of the inclusion of the rogue-dissociation process into the calculations. The dashed line shows the result of a calculation without this process. The dotted line shows the result for a calculation that includes the estimate in Eq. (242). We have taken the parameters Binit = 162 G, Bevolve = 158 G, and na = 2 × 1012 cm−3 .
momenta, i.e., the rogue-dissociation process. In the absence of this term the equation e4ectively takes into account the dressing of molecules in an adiabatic manner, and describes Josephson oscillations between a condensate of atoms and dressed molecules. As we have discussed in the previous section, the above equation is only applicable to the situation of a sudden change in magnetic 8eld. Therefore, we perform the following calculation. For a given magnetic 8eld Binit and atomic condensate density we calculate the equilibrium values of the molecular wave functions and the Hartree–Fock self-energy, using the time-independent mean-8eld equations in Eqs. (185) and (186). Then we change the magnetic 8eld instantaneously to the value Bevolve and keep it at this value. In Fig. 32 the results of the calculations for this situation are shown, with Binit = 162 G and Bevolve = 158 G. The atomic condensate density is taken equal to na = 2 × 1012 cm−3 . The dashed line shows the result for a calculation without the rogue-dissociation process and shows oscillations where a fraction of the atoms is converted into molecules and oscillates back and forth between the atomic and dressed molecular condensate. Since there is no decay mechanism, all of the atoms come back into the atomic condensate at times equal to a multiple of the oscillation period. The solid line shows the result of a calculation that includes the rogue-dissociation process. Clearly, the number of condensate atoms oscillates in this case as well. However, not all of the atoms come back into the atomic condensate and there is a decay of the number of atoms in the atomic condensate. This is precisely due to the above-mentioned rogue-dissociation process. Although the preliminary calculations presented in this section are limited to the case of a step in the magnetic 8eld, they nevertheless present some insight in the e4ects of the rogue-dissociation process on the coherent atom–molecule oscillations in a Ramsey experiment. In future work we intend to study also the case of time-dependent magnetic 8elds, by an exact numerical treatment of the fractional derivative in our time-dependent mean-8eld equations. In particular, we are interested in the magnetic-8eld dependence of the damping that is caused by the rogue-dissociation process.
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We can estimate this dependence as follows. The Green’s function associated with the roguedissociation process, i(t − t )g2 m3=2 ˝2 √ ∞ HF d! ˝!e−i(!+27 )(t −t ) × ; 2 [˝! + 2˝7HF − (B)]2 + (g4 m3 =42 ˝6 )˝! 0 is sharply peaked in time. Hence we approximate this Green’s function by (+) (t − t ) = − Grog
(+) (+) (t − t ) +(B)Grog (0)(t − t ) ; Grog
with the time scale +(B) given by tc (+) +(B) = dt Grog (t) ; −∞
(239)
(240)
(241)
with tc a positive cut-o4 that is determined such that the result for +(B) depends only very weakly (+) on tc . The Green’s function evaluated at zero time equals Grog (0) = 1 − Z(B), a result which follows from the sum rule for the molecular density of states in Eq. (132). This gives the contribution −2i[1 − Z(B)]g2 +(B) |*a (t)|2 *a (t) ; (242) ˝ to the right-hand side of Eq. (238). The rate equation for the atomic density that follows from this term is given by
4[1 − Z(B)]g2 +(B) 2 dna − na (t) ; (243) dt ˝2 which after linearization leads to the following equation for the number of condensate atoms: dNa (t) (244) −-Na (t) ; dt with the rate - given by 8[1 − Z(B)]g2 +(B)na : (245) ˝2 We observe from this equation that the loss rate of atoms from the atomic condensate due to the rogue-dissociation process increases as the magnetic 8eld approaches its resonant value. This is indeed what is observed experimentally [82]. Far o4 resonance the loss rate vanishes since the wave function renormalization factor Z(B) → 1 in this limit. For the parameters of Fig. 26(a) at the e4ective homogeneous density na = 2 × 1012 cm−3 , we have that +(B) 1:28 × 10−9 s, which leads to - 0:45 kHz. The dotted line in Fig. 32 shows the result of a calculation that includes the term in Eq. (242). The exact result, shown by the solid line, and this approximate result show the same overall damping rate. This justi8es the approximation for the Green’s function in Eq. (240). The result for the damping rate - is about a factor of eight smaller than the experimental result. To further investigate the magnetic-8eld dependence of the damping of the coherent atom– molecule oscillations, we have calculated the numerical solution of the e4ective equation of motion for the atomic condensate wave function for a step in the magnetic 8eld, for three di4erent 8nal magnetic 8elds. The results of these calculations are shown in Fig. 33. The solid, dashed, -
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fraction of atoms in condensate
1
0.9
0.8
0.7
0.6
0.5
0.4 0
50
100
150
200
250
300
350
400
time (µs)
Fig. 33. Fraction of atoms in the atomic condensate after a step in the magnetic 8eld. The solid line corresponds to Bevolve = 156:1 G. The dashed and dotted line correspond to a magnetic 8eld of Bevolve = 156:5 G and Bevolve = 156:9 G, respectively. The initial magnetic 8eld is Binit = 162 G and the density of the atomic condensate is na = 2 × 1012 cm−3 .
frequency (kHz)
12 10 8
β (kHz)
14
1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 156
156.5
157
6 4 2 0 156 156.1 156.2 156.3 156.4 156.5 156.6 156.7 156.8 156.9 157
B (G)
Fig. 34. Frequency and damping as a function of the magnetic 8eld. The solid line corresponds to the frequency found by means of linear-response theory.
and dotted lines corresponds to a magnetic 8eld of Bevolve = 156:1 G, Bevolve = 156:5 G, and Bevolve = 156:9 G, respectively. The initial equilibrium corresponds to an atomic condensate density of na = 2 × 1012 cm−3 at a magnetic 8eld of Binit = 162 G. Note the increase in the frequency with increasing magnetic 8eld. The magnetic-8eld dependence of the frequency and damping of the coherent atom–molecule oscillations is found from these numerical results by 8tting with the equation in Eq. (228). The results are presented in Fig. 34. The solid line corresponds to the Josephson frequency of the coherent atom–molecule oscillations that was found by means of the linear-response
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191
calculation of the previous section. The deviation for large magnetic 8elds is understood because we have, in our numerical solution of the e4ective mean-8eld equation, not taken into account the higher-order energy-dependences of the molecular self-energy that are fully taken into account in the linear-response theory. The inset shows the damping as a function of the magnetic 8eld. Note the increase of the damping as the magnetic 8eld approaches its resonant value. This is expected from the estimate in Eq. (245). The above analysis indicates that the rogue-dissociation process gives possibly a contribution to the experimentally observed damping of the coherent atom–molecule oscillations. Presumably, however, also other mechanisms contribute to the observed damping. In particular, we mention here the quantum evaporation process, that was shown to be important in the single-pulse experiments [42]. The detailed investigation of the damping of the coherent atom–molecule oscillation is a subject for further study. 7. Conclusions and outlook In this review paper we have presented the derivation of an e4ective quantum 8eld theory suitable for the description of a Bose gas near a Feshbach resonance, since it incorporates the two-atom physics exactly. We have presented several applications of this theory, both above and below the critical temperature for Bose–Einstein condensation. In the last part of this paper we have studied in detail the magnetic-8eld dependence of the frequency of the coherent atom–molecule oscillations and have obtained excellent agreement with the experimental results. In particular, we have been able to quantitatively explain the many-body e4ects on this frequency by making use of a linear-response approximation to our mean-8eld equations. Although we have already presented some numerical solutions of the mean-8eld equations that improve on this approximation, a great deal of work still has to be done. The numerical solution of these equations for the situation of time-dependent detuning is rather involved. Nevertheless, work in this direction is in progress and will be reported in a future publication. As already mentioned, we have also discussed the properties of the gas above the critical temperature. This discussion was mainly concerned with the equilibrium properties of the gas and we studied the many-body e4ects on the bound-state energy of the molecular state. An important conclusion of this study is that, for certain values of the parameters, there exists a many-body induced resonant state with a relatively small energy. In future work we intend to study the e4ects of the appearance of this resonant state in the molecular density of states on the properties of the gas. In particular we expect that due to this e4ect the number of molecules in the gas will be large even at relatively large detuning, which can not be explained on the basis of two-atom physics. Furthermore, to study the normal state also in an out-of-equilibrium situation, we should derive a quantum kinetic theory that describes the evolution of the local occupation numbers of the atoms and molecules. Moreover, the description of the Bose–Einstein condensed phase of the gas at nonzero temperatures requires a modi8cation of the mean-8eld equations such that they include the e4ects of the thermal clouds of atoms and molecules, and we need equations for the evolution of the local occupation numbers of the latter. The extension of the theory presented in this paper to these situations can be derived in a unifying manner by using a functional formulation of the Schwinger– Keldysh nonequilibrium theory [116], and is especially important in view of the ongoing e4ort
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to produce ultracold molecules by means of a sweep in the magnetic 8eld through the Feshbach resonance [64]. The theory presented in this paper is generalized to a gas of fermionic atoms in a straightforward manner [70,87]. One modi8cation is that to have s-wave scattering between fermionic atoms we have to have a mixture of atoms with two hyper8ne states, since the Pauli principle forbids s-wave scattering between identical fermions. Furthermore, the properties of the dressed molecular state is altered due to the presence of the Fermi sphere. A molecule with zero momentum only decays if its energy is above twice the Fermi energy. If the molecular state lies below twice the Fermi energy, the equilibrium situation is a Bose–Einstein condensate of molecules. If we start from this situation and increase the detuning, the Bose–Einstein condensate of molecules crosses over to a Bose–Einstein condensate of Cooper pairs, i.e., a BCS–BEC crossover occurs [63,59]. In view of the ongoing experiments with atomic Fermi gases near a Feshbach resonance [64–69], it is particularly interesting to study the e4ects of nonadiabticity on the crossover from a Bose–Einstein condensate of molecules to a degenerate Fermi gas. In particular, the atomic distribution function after such a sweep, and its dependence on the duration of the sweep, is of great interest, since this will determine whether or not a BCS-state will form after equilibration. Determination of the atomic distribution function requires, in 8rst instance, knowledge of the solution of the mean-8eld equation for the molecular condensate for time-dependent detuning. Work in this direction is in progress. We also intend to study the equilibrium properties of this crossover, and in particular the behavior of the critical temperature, in detail in future work. Clearly, Feshbach resonances present an exciting opportunity for the experimental and theoretical study of the many-body properties of atomic and molecular Bose and Fermi gases. There is little doubt that these Feshbach resonances will 8nd many new applications in the years to come. Acknowledgements It is a great pleasure to thank Frieda van Belle, Eric Cornell, Neil Claussen, Gianmaria Falco, Behnam Farid, Randy Hulet, Niels de Keijzer, Wolfgang Ketterle, Mathijs Romans, Subir Sachdev, Kareljan Schoutens, Peter van der Straten, Bart Vlaar, and Carl Wieman for their contributions to this review paper. Furthermore, we would like to acknowledge the hospitality of the European Centre for Theoretical Studies in Nuclear Physics and Related Areas (ECT*) during the Summer Program on Bose–Einstein condensation. This work is supported by the Stichting voor Fundamenteel Onderzoek der Materie (FOM) and by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO). References [1] [2] [3] [4] [5] [6] [7] [8]
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Physics Reports 396 (2004) 197 – 403 www.elsevier.com/locate/physrep
Constructive methods of invariant manifolds for kinetic problems Alexander N. Gorbana; b; c;∗ , Iliya V. Karlina; b , Andrei Yu. Zinovyevb; c a
ETH-Zentrum, Department of Materials, Institute of Polymers, Sonneggstr. 3, ML J19, CH-8092 Z%urich, Switzerland b Institute of Computational Modeling SB RAS, Akademgorodok, Krasnoyarsk 660036, Russia c Institut des Hautes Etudes Scienti1ques, Le Bois-Marie, 35, route de Chartres, F-91440, Bures-sur-Yvette, France Accepted 13 March 2004 editor: I. Procaccia
Abstract The concept of the slow invariant manifold is recognized as the central idea underpinning a transition from micro to macro and model reduction in kinetic theories. We present the Constructive Methods of Invariant Manifolds for model reduction in physical and chemical kinetics, developed during last two decades. The physical problem of reduced description is studied in the most general form as a problem of constructing the slow invariant manifold. The invariance conditions are formulated as the di6erential equation for a manifold immersed in the phase space (the invariance equation). The equation of motion for immersed manifolds is obtained (the 1lm extension of the dynamics). Invariant manifolds are 8xed points for this equation, and slow invariant manifolds are Lyapunov stable 8xed points, thus slowness is presented as stability. A collection of methods to derive analytically and to compute numerically the slow invariant manifolds is presented. Among them, iteration methods based on incomplete linearization, relaxation method and the method of invariant grids are developed. The systematic use of thermodynamics structures and of the quasi-chemical representation allow to construct approximations which are in concordance with physical restrictions. The following examples of applications are presented: nonperturbative deviation of physically consistent hydrodynamics from the Boltzmann equation and from the reversible dynamics, for Knudsen numbers Kn ∼ 1; construction of the moment equations for nonequilibrium media and their dynamical correction (instead of extension of list of variables) to gain more accuracy in description of highly nonequilibrium =ows; determination of molecules dimension (as diameters of equivalent hard spheres) from experimental viscosity data;
∗ Corresponding author. ETH-Zentrum, Department of Materials, Institute of Polymers, Sonneggstr. 3, ML J19, CH-8092 ZDurich, Switzerland. E-mail addresses: [email protected] (A.N. Gorban), [email protected] (I.V. Karlin), [email protected] (A.Y. Zinovyev).
c 2004 Elsevier B.V. All rights reserved. 0370-1573/$ - see front matter doi:10.1016/j.physrep.2004.03.006
198
A.N. Gorban et al. / Physics Reports 396 (2004) 197 – 403
model reduction in chemical kinetics; derivation and numerical implementation of constitutive equations for polymeric =uids; the limits of macroscopic description for polymer molecules, etc. c 2004 Elsevier B.V. All rights reserved. PACS: 05.20.Dd; 02.30.Mv; 02.70.Dh; 05.70.Ln Keywords: Model reduction; Invariant manifold; Entropy; Kinetics; Boltzmann equation; Fokker–Planck equation; Navier–Stokes equation; Burnett equation; Quasi-chemical approximation; Oldroyd equation; Polymer dynamics; Molecular individualism; Accuracy estimation; Post-processing
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. The source of examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. The Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1. The equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2. The basic properties of the Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3. Linearized collision integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Phenomenology and Quasi-chemical representation of the Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Kinetic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Methods of reduced description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1. The Hilbert method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2. The Chapman–Enskog method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3. The Grad moment method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4. Special approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.5. The method of invariant manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.6. Quasi-equilibrium approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Discrete velocity models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6. Direct simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7. Lattice gas and Lattice Boltzmann models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8. Other kinetic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.1. The Enskog equation for hard spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.2. The Vlasov equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.3. The Fokker–Planck equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9. Equations of chemical kinetics and their reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.1. Outline of the dissipative reaction kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.2. The problem of reduced description in chemical kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.3. Partial equilibrium approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.4. Model equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.5. Quasi-steady state approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.6. Thermodynamic criteria for selection of important reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.7. Opening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Invariance equation in the di6erential form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Film extension of the dynamics: slowness as stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Equation for the 8lm motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Stability of analytical solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Entropy, quasi-equilibrium and projectors 8eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Moment parameterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
200 207 207 207 209 210 211 212 213 213 215 216 217 217 219 219 219 220 220 220 221 221 222 222 226 227 228 229 231 232 233 235 235 237 243 243
A.N. Gorban et al. / Physics Reports 396 (2004) 197 – 403 5.2. Entropy and quasi-equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Thermodynamic projector without a priori parameterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 1: Quasi-equilibrium projector and defect of invariance for the local Maxwellians manifold of the Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 2: Scattering rates versus moments: alternative Grad equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Newton method with incomplete linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 3: Nonperturbative correction of Local Maxwellian manifold and derivation of nonlinear hydrodynamics from Boltzmann equation (1D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 4: Nonperturbative derivation of linear hydrodynamics from Boltzmann equation (3D) . . . . . . . . . . . . . . . Example 5: Dynamic correction to moment approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Decomposition of motions, nonuniqueness of selection of fast motions, self-adjoint linearization, Onsager 8lter and quasi-chemical representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 6: Quasi-chemical representation and self-adjoint linearization of the Boltzmann collision operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Relaxation methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 7: Relaxation method for the Fokker–Planck equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Method of invariant grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1. Grid construction strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1. Growing lump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.2. Invariant =ag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.3. Boundaries check and the entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2. Instability of 8ne grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3. What space is the most appropriate for the grid construction? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4. Carleman’s formulae in the analytical invariant manifolds approximations. First bene8t of analyticity: superresolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 8: Two-step catalytic reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 9: Model hydrogen burning reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. Method of natural projector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 10: From reversible dynamics to Navier–Stokes and post-Navier–Stokes hydrodynamics by natural projector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 11: Natural projector for the McKean model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Slow invariant manifold for a closed system has been found. What next? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1. Slow dynamics in open systems. Zero-order approximation and the thermodynamic projector . . . . . . . . . . . . . 11.2. Slow dynamics in open systems. First-order approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3. Beyond the 8rst-order approximation: higher-order dynamical corrections, stability loss and invariant manifold explosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4. Lyapunov norms, 8nite-dimensional asymptotic and volume contraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 12: The universal limit in dynamics of dilute polymeric solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 13: Explosion of invariant manifold, limits of macroscopic description for polymer molecules, molecular individualism, and multimodal distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Accuracy estimation and postprocessing in invariant manifolds construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 14: Defect of invariance estimation and switching from the microscopic simulations to macroscopic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction In this review, we present a collection of constructive methods to study slow (stable) positively invariant manifolds of dynamic systems. The main objects of our study are dissipative dynamic systems (8nite or in8nite) which arise in various problems of kinetics. Some of the results and methods presented herein may have a more general applicability, and can be useful not only for dissipative systems but also, for example, for conservative systems. Nonequilibrium statistical physics is a collection of ideas and methods to extract slow invariant manifolds. Reduction of description for dissipative systems assumes (explicitly or implicitly) the following picture: There exists a manifold of slow motions in the phase space of the system. From the initial conditions the system goes quickly in a small neighborhood of the manifold, and after that moves slowly along this manifold (see, for example, [1]). The manifold of slow motion must be positively invariant: if the motion starts on the manifold at t0 , then it stays on the manifold at t ¿ t0 . Frequently used wording “invariant manifold” is not really exact: for the dissipative systems, the possibility of extending the solutions (in a meaningful way) backwards in time is limited. So, in nonequilibrium statistical physics we study positively invariant slow manifolds. The necessary invariance condition can be written explicitly as the di6erential equation for the manifold immersed into the phase space. 1 A dissipative system may have many closed positively invariant sets. For example, for every set of initial conditions K, uni8cation of all the trajectories {x(t); t ¿ 0} with initial conditions x(0) ∈ K is positively invariant. Thus, selection of the slow (stable) positively invariant manifolds becomes an important problem. 2 One of the diRculties in the problem of reducing the description is pertinent to the fact that there exists no commonly accepted formal de8nition of slow (and stable) positively invariant manifold. This diRculty is resolved in Section 4 of our review in the following way: First, we consider manifolds immersed into a phase space and study their motion along trajectories. Second, we subtract from this motion the motion of immersed manifolds along themselves, and obtain a new equation for dynamics of manifolds in phase space: the 1lm extension of the dynamics. Invariant manifolds are 8xed points for this extended dynamics, and slow invariant manifolds are Lyapunov stable 1xed points.
1 This picture is directly applicable to dissipative systems. Time separation for conservative systems and the way from the reversible mechanics (for example, from the Liouville equation) to dissipative systems (for example, to the Boltzmann equation) requires some additional ideas and steps. For any conservative system, a restriction of its dynamics onto any invariant manifold is conservative again. We should represent a dynamics of a large conservative system as a result of dynamics in its small subsystems, and it is necessary to take into account that a macroscopically small interval of time can be considered as an in8nitely large interval for a small subsystem, i.e. microscopically. It allows us to represent a relaxation of such a large systems as an ensemble of indivisible events (for example, collision) which happen to its small subsystems. The Bogolyubov–Born–Green–Kirkwood–Yvon (BBGKY) hierarchy and Bogolyubov method for derivation of the Boltzmann equation give us the unexcelled realization of this approach [2]. 2 Nevertheless, there exists a di6erent point of view: “nonuniqueness, when it arises, is irrelevant for modeling” [3], because the di6erences between the possible manifolds are of the same order as the di6erences we set out to ignore in establishing the low-dimensional model.
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The main body of this review is about how to actually compute the slow invariant manifold. Here we present three approaches to constructing slow (stable) positively invariant manifolds. • Iteration method (the Newton method subject to incomplete linearization); • Relaxation methods based on a 8lm extension of the original dynamic system; • The method of natural projector The Newton method (with incomplete linearization) is convenient for obtaining the explicit formulas—even one iteration can give a good approximation. Relaxation methods are oriented more at the numerical implementation. Nevertheless, several 8rst steps also can give appropriate analytical approximations, competitive with other methods. Finally, the method of natural projector constructs not the manifold itself but a projection of slow dynamics from the slow manifold onto some set of variables. The Newton method subject to incomplete linearization was developed for the construction of slow (stable) positively invariant manifolds for the following problems: • • • •
Derivation of the post-Navier–Stokes hydrodynamics from the Boltzmann equation [4,6,7]. Description of the dynamics of polymers solutions [8]. Correction of the moment equations [9]. Reduced description for the chemical kinetics [10,11,81].
Relaxation methods based on a 8lm extension of the original dynamic system were applied for the analysis of the Fokker–Planck equation [12]. Applications of these methods in the theory of the Boltzmann equation can bene8t from the estimations, obtained in the papers [14,15]. The method of natural projector was initially applied to derivation of the dissipative equations of macroscopic dynamics from the conservative equations of microscopic dynamics [16–21]. Using this method, new equations were obtained for the post-Navier–Stokes hydrodynamics, equations of plasma hydrodynamics and others [17,21]. This short-memory approximation is applied to the Wigner formulation of quantum mechanics [22]. The dissipative dynamics of a single quantum particle in a con8ning external potential is shown to take the form of a damped oscillator whose e6ective frequency and damping coeRcients depend on the shape of the quantum-mechanical potential [22]. The method of natural projector can also be applied e6ectively for the dissipative systems: instead of the Chapman–Enskog method in theory of the Boltzmann equation, etc. A natural initial approximation for the methods under consideration is a quasi-equilibrium manifold. It is the manifold of conditional maxima of the entropy. Most of the works on nonequilibrium thermodynamics deal with corrections to quasi-equilibrium approximations, or with applications of these approximations (with or without corrections). The construction of the quasi-equilibrium allows for the following generalization: Almost every manifold can be represented as a set of minimizers of the entropy under linear constrains. However, in contrast to the standard quasi-equilibrium, these linear constrains will depend on the point on the manifold. We describe the quasi-equilibrium manifold and a quasi-equilibrium projector on the tangent space of this manifold. This projector is orthogonal with respect to entropic scalar product (the bilinear form de8ned by the negative second di6erential of the entropy). We construct the thermodynamical projector, which transforms the arbitrary vector 8eld equipped with the given Lyapunov function (the entropy) into a vector 8eld with
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the same Lyapunov function for an arbitrary ansatz manifold which is not tangent to the level of the Lyapunov function. The uniqueness of this construction is demonstrated. Here, a comment on the status of the most of the statements in this text is in order. Just like the absolute majority of all claims concerning such things as general solutions of the Navier– Stokes and Boltzmann equations, etc., they have the status of being plausible. They can become theorems only if one restricts essentially the set of the objects under consideration. Among such restrictions we should mention cases of exact reduction, i.e. exact derivation of the hydrodynamics from the kinetics [23,24]. In these (still in8nite-dimensional) examples one can compare di6erent methods, for example, the Newton method with the methods of series summation in the perturbation theory [24,25]. Also, it is necessary to stress here, that even if in the limit all the methods lead to the same results, they can give rather di6erent approximations “on the way”. The rigorous grounds of the constructive methods of invariant manifolds should, in particular, include the theorems about persistence of invariant manifolds under perturbations. The most known result of this type is the Kolmogorov–Arnold–Moser theory about persistence of almost all invariant tori of completely integrable system under small perturbation [28–30]. Such theorems exist for some classes of in8nite dimensional dissipative systems too [31]. Unfortunately, it is not proven until now that many important systems (the Boltzmann equation, the 3D Navier–Stokes equations, the Grad equations, etc.) belong to these classes. So, it is necessary to act with these systems without a rigorous basis. Two approaches are widely known to the construction of the invariant manifolds: the Taylor series expansion [32,33] and the method of renormalization group [34–40]. The advantages and disadvantages of the Taylor expansion are well-known: constructivity against the absence of physical meaning for the high-order terms (often) and divergence in the most interesting cases (often). In the paper [37] a geometrical formulation of the renormalization group method for global analysis was given. It was shown that the renormalization group equation can be interpreted as an envelope equation. Recently [38] the renormalization group method was formulated in terms of the notion of invariant manifolds. This method was applied to derive kinetic and transport equations from the respective microscopic equations [39]. The derived equations include the Boltzmann equation in classical mechanics (see also the paper [36], where it was shown for the 8rst time that kinetic equations such as the Boltzmann equation can be understood naturally as renormalization group equations), Fokker–Planck equation, a rate equation in a quantum 8eld theoretical model, etc. The renormalization group approach was applied to the stochastic Navier–Stokes equation that describes fully developed =uid turbulence [41–43]. For the evaluation of the relevant degrees of freedom the renormalization group technique was revised for discrete systems in the recent paper Ref. [40]. The kinetic approach to subgrid modeling of =uid turbulence became more popular during the last decade [44–47]. A mean-8eld approach (8ltering out subgrid scales) is applied to the Boltzmann equation in order to derive a subgrid turbulence model based on kinetic theory. It is demonstrated [47] that the only Smagorinsky type model which survives in the hydrodynamic limit on the viscosity time scale is the so-called tensor-di6usivity model [48]. The new quantum 8eld theory formulation of the problem of persistence of invariant tori in perturbed completely integrable systems was obtained, and the new proof of the KAM theorem for analytic Hamiltonians based on the renormalization group method was given [49].
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From the authors of the paper Ref. [36] point of view, the relation of renormalization group theory and reductive perturbation theory has simultaneously been recognized: renormalization group equations are actually the slow-motion equations which are usually obtained by reductive perturbation methods. The 8rst systematic and (at least partially) successful method of constructing invariant manifolds for dissipative systems was the celebrated Chapman–Enskog method [51] for the Boltzmann kinetic equation. The Chapman–Enskog method results in a series development of the so-called normal solution (the notion introduced by Hilbert [52]) where the one-body distribution function depends on time and space through its locally conserved moments. To the 8rst approximation, the Chapman–Enskog series leads to hydrodynamic equations with transport coeRcients expressed in terms of molecular scattering cross-sections. However, next terms of the Chapman–Enskog bring in the “ultra-violet catastrophe” (noticed 8rst by Bobylev [53]) and negative viscosity. These drawbacks pertinent to the Taylor-series expansion disappear as soon as the Newton method is used to construct the invariant manifold [6]. The Chapman–Enskog method was generalized many times [54] and gave rise to a host of subsequent works and methods, such as the famous method of the quasi-steady state in chemical kinetics, pioneered by Bodenstein and Semenov and explored in considerable detail by many authors (see, for example, [55–59,10]), and the theory of singularly perturbed di6erential equations [55,60–65]. There exist a group of methods to construct an ansatz for the invariant manifold based on the spectral decomposition of the Jacobian. The idea to use the spectral decomposition of Jacobian 8elds in the problem of separating the motions into fast and slow originates from methods of analysis of sti6 systems [66], and from methods of sensitivity analysis in control theory [67,68]. One of the currently most popular methods based on the spectral decomposition of Jacobian 8elds is the construction of the so-called intrinsic low-dimensional manifold (ILDM) [69]. These methods were thoroughly analyzed in two papers [70,71]. It was shown that the successive applications of the computational singular perturbation algorithm (developed in [68]) generate, order by order, the asymptotic expansion of a slow manifold, and the manifold identi8ed by the ILDM technique (developed in [69]) agrees with the invariant manifold to some order. The theory of inertial manifold is based on the special linear dominance in higher dimensions. Let an in8nite-dimensional system have a form: u˙ + Au = R(u), where A is self-adjoint, and has discrete spectrum i → ∞ with suRciently big gaps between i , and R(u) is continuous. One can build the slow manifold as the graph over a root space of A [72]. The textbook [76] provides an exhaustive introduction to the scope of main ideas and methods of this theory. Systems with linear dominance have limited utility in kinetics. Often neither a big spectral gaps between i exists, no i → ∞ (for example, for the simplest model BGK equations, or for the Grad equations). Nevertheless, the concept of inertial attracting manifold has more wide 8eld of applications than the theory, based on the linear dominance assumption. The Newton method with incomplete linearization as well as the relaxation method allow us to 8nd an approximate slow invariant manifolds without the preliminary stage of Jacobian 8eld spectral decomposition. Moreover, a necessary slow invariant subspace of Jacobian in equilibrium point appears as a by-product of the Newton iterations (with incomplete linearization), or of the relaxation method. It is of importance to search for minimal (or subminimal) sets of natural parameters that uniquely determine the long-time behavior of a system. This problem was 8rst discussed by Foias and Prodi
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[73] and by Ladyzhenskaya [74] for the 2D Navier–Stokes equations. They have proved that the long-time behavior of solutions is completely determined by the dynamics of suRciently large amount of the 8rst Fourier modes. A general approach to the problem on the existence of a 8nite number of determining parameters has been discussed [75,76]. Past decade witnessed a rapid development of the so-called set oriented numerical methods [77]. The purpose of these methods is to compute attractors, invariant manifolds (often, computation of stable and unstable manifolds in hyperbolic systems [78–80]). Also, one of the central tasks of these methods is to gain statistical information, i.e. computations of physically observable invariant measures. The distinguished feature of the modern set-oriented methods of numerical dynamics is the use of ensembles of trajectories within a relatively short propagation time instead of a long time single trajectory. In this paper we systematically consider a discrete analogue of the slow (stable) positively invariant manifolds for dissipative systems, invariant grids. These invariant grids were introduced in [10]. Here we will describe the Newton method subject to incomplete linearization and the relaxation methods for the invariant grids [81]. It is worth to mention, that the problem of the grid correction is fully decomposed into the tasks of the grid’s nodes correction. The edges between the nodes appear only in the calculation of the tangent spaces at the nodes. This fact determines high computational eRciency of the invariant grids method. Let the (approximate) slow invariant manifold for a dissipative system be found. What for have we constructed it? One important part of the answer to this question is: We have constructed it to create models of open system dynamics in the neighborhood of this manifold. Di6erent approaches for this modeling are described. We apply these methods to the problem of reduced description in polymer dynamics and derive the universal limit in dynamics of dilute polymeric solutions. It is represented by the revised Oldroyd 8 constants constitutive equation for the polymeric stress tensor. CoeRcients of this constitutive equation are expressed in terms of the microscopic parameters. This limit of dynamics of dilute polymeric solutions is universal in the same sense, as Korteweg–De-Vries equation is universal in the description of the dispersive dissipative nonlinear waves: any physically consistent equation should contain the obtained equation as a limit. The phenomenon of invariant manifold explosion in driven open systems is demonstrated on the example of dumbbell models of dilute polymeric solutions [84]. This explosion gives us a possible mechanism of drag reduction in dilute polymeric solutions [85]. Suppose that for the kinetic system the approximate invariant manifold has been constructed and the slow motion equations have been derived. Suppose that we have solved the slow motion system and obtained xsl (t). We consider the following two questions: • How well this solution approximates the true solution x(t) given the same initial conditions? • How is it possible to use the solution xsl (t) for it’s re8nement without solving the slow motion system (or it’s modi8cations) again? These two questions are interconnected. The 8rst question states the problem of the accuracy estimation. The second one states the problem of postprocessing [244–246,277]. We propose various algorithms for postprocessing and accuracy estimation, and give an example of application.
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4
8 7
12
9 5
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11 6
10
Fig. 1. Logical connections between sections. All the sections depend on Section 3. There are many possible routes for reading, for example the following route gives the invariance equation and one of the main methods for its solution with applications to the Boltzmann equation: Sections 3–5 (with Example 1), Section 6 (with Example 3). Another possibility gives the shortest way to rheology applications: Sections 3, 5, 6 and 11 (with Example 12). The formalization of the classical Ehrenfests idea of coarse-graining and its application for derivation of the correct high-order hydrodynamic equations can be reached in such a way: Sections 3, 5, 10 (with Examples 10, 11). The shortest road to numerical representation of invariant manifolds and to the method of invariant grids is as follows: Sections 3–6, 8, 9 (with Examples 8, 9).
Our collection of methods and algorithms can be incorporated into recently developed technologies of computer-aided multiscale analysis which enable the “level jumping” between microscopic and macroscopic (system) levels. It is possible both for traditional technique based on transition from microscopic equations to macroscopic equations and for the “equation-free” approach [82]. This approach developed in recent series of work [83], when successful, can bypass the derivation of the macroscopic evolution equations when these equations conceptually exist but are not available in closed form. The mathematics-assisted development of a computational superstructure may enable alternative descriptions of the problem physics (e.g. Lattice Boltzmann (LB), kinetic Monte-Carlo (KMC) or Molecular Dynamics (MD) microscopic simulators, executed over relatively short time and space scales) to perform systems level tasks (integration over relatively large time and space scales, coarse bifurcation analysis, optimization, and control) directly. In e6ect, the procedure constitutes a system identi8cation based, closure-on-demand computational toolkit. It is possible to use macroscopic invariant manifolds in this environment without explicit equations. The present paper comprises sections of the two kinds. Numbered sections contain basic notions, methods and algorithmic realizations. Sections entitled “Examples” contain various case studies where the methods are applied to speci8c equations. Exposition in the “Examples” sections is not as consequent as in the numbered sections. Most of the examples can be read more or less independently. Logical connections between sections are presented in Figs. 1, 2. The list of cited literature is by no means complete although we spent e6ort in order to re=ect at least the main directions of studies related to computations of the invariant manifolds. We think that this list is more or less exhaustive in the second-order approximation.
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Sec. 5 Ex.1, Ex.2
Sec. 6 Ex.3, Ex.4, Ex.5
Sec. 9 Ex.8, Ex.9
Sec. 10 Ex.10, Ex.11
Sec. 7 Ex.6 Sec. 11 Ex.12, Ex.13
Sec. 8 Ex.7 Sec. 12 Ex.14
Fig. 2. Logical connections between sections and examples. Only one connection between examples is signi8cant: Example 3 depends on Example 1. All the examples depend on corresponding subsections of Section 2.
Mathematical notation and some terminology • The operator L from space W to space E: L : W → E. • The kernel of a linear operator L : W → E is a subspace ker L ⊂ W that transforms by L into 0: ker L = {x ∈ W | Lx = 0}. • The image of a linear operator L : W → E is a subspace im L = L(W ) ⊂ E. • Projector is a linear operator P : E → E with the property P 2 = P. Projector P is orthogonal one, if ker P ⊥ im P (the kernel of P is orthogonal to the image of P). • If F : U → V is a map of domains in normed spaces (U ⊂ W; V ⊂ E) then the diFerential of F at a point x is a linear operator Dx F : W → E with the property: F(x + x) − F(x) − (Dx F)(x) = o( x ). This operator (if it exists) is the best linear approximation of the map F(x + x) − F(x). • The di6erential of the function f(x) is the linear functional Dx f. The gradient of the function f(x) can be de8ned, if there is a given scalar product | , and if there exists a Riesz representation for functional Dx f: (Dx f)(a) = grad x f | a. The gradient grad x f is a vector. • The second diFerential of a map F : U → V is a bilinear operator Dx2 F : W × W → E which can be de8ned by Taylor formula: F(x + x) = F(x) + (Dx F)(x) + 12 (Dx2 F)(x; x) + o( x 2 ). • The di6erentiable map of domains in normed spaces F : U → V is an immersion, if for any x ∈ U the operator Dx F is injective: ker Dx F = {0}. In this case the image of F, i.e. (F(U )) is called the immersed manifold, and the image of Dx F is called the tangent space to the immersed manifold F(U ). We shall use the notation Tx for this tangent space: im Dx F = Tx . • The subset U of the vector space E is convex, if for every two points x1 ; x2 ∈ U it contains the segment between x1 and x2 : x1 + (1 − )x2 ∈ U for every ∈ [0; 1]. • The function f, de8ned on the convex set U ⊂ E, is convex, if its epigraph, i.e. the set of pairs Epi f = {(x; g) | x ∈ U; g ¿ f(x)}, is the convex set in E × R. The twice di6erentiable function f is convex if and only if the quadratic form (Dx2 f)(x; x) is nonnegative. • The convex function f is called strictly convex if in the domain of de8nition there is no line segment on which it is constant and 8nite (f(x) = const = ∞). The suRcient condition for the twice di6erentiable function f to be strictly convex is that the quadratic form (Dx2 f)(x; x) is positive de8ned (i.e. it is positive for all x = 0). • We use summation convention for vectors and tensors, c g = i i i ci gi , when it cannot cause a confusion, in more complicated cases we use the sign .
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2. The source of examples In this section we follow, partially, the paper [50], where nonlinear kinetic equations and methods of reduced description are reviewed for a wide audience of specialists and postgraduate students in physics, mathematical physics, material science, chemical engineering and interdisciplinary research. 2.1. The Boltzmann equation 2.1.1. The equation The Boltzmann equation is the 8rst and the most celebrated nonlinear kinetic equation introduced by the great Austrian scientist Ludwig Boltzmann in 1872 [86]. This equation describes the dynamics of a moderately rare8ed gas, taking into account the two processes: the free =ight of the particles, and their collisions. In its original version, the Boltzmann equation has been formulated for particles represented by hard spheres. The physical condition of rarefaction means that only pair collisions are taken into account, a mathematical speci8cation of which is given by the Grad-Boltzmann limit: If N is the number of particles, and is the diameter of the hard sphere, then the Boltzmann equation is expected to hold when N tends to in8nity, tends to zero, N 3 (the volume occupied by the particles) tends to zero, while N 2 (the total collision cross section) remains constant. The microscopic state of the gas at time t is described by the one-body distribution function P(x; C; t), where x is the position of the center of the particle, and C is the velocity of the particle. The distribution function is the probability density of 8nding the particle at time t within the in8nitesimal phase space volume centered at the phase point (x; C). The collision mechanism of two hard spheres is presented by a relation between the velocities of the particles before [C and w] and after [C and w ] their impact: C = C − n(n; C − w) ; w = w + n(n; C − w) ; where n is the unit vector along C−C . Transformation of the velocities conserves the total momentum of the pair of colliding particles (C + w = C + w), and the total kinetic energy (C 2 + w 2 = C2 + w2 ) The Boltzmann equation reads: 9P 9P + C; 9t 9x =N 2 (P(x; C ; t)P(x; w ; t) − P(x; C; t)P(x; w; t)) | (w − C; n) | dw dn ; (1) R
B−
where integration in n is carried over the unit sphere R3 , while integration in w goes over a hemisphere B− = {w | (w − C; n) ¡ 0}. This hemisphere corresponds to the particles entering the collision. The nonlinear integral operator on the right hand side of Eq. (1) is nonlocal in the velocity variable, and local in space. The Boltzmann equation for arbitrary hard-core interaction is a generalization of the Boltzmann equation for hard spheres under the proviso that the true in8nite-range interaction potential between the particles is cut-o6 at some distance. This generalization amounts to
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a replacement, 2 | (w − C; n) | dn → B( ; | w − C |) d d! ;
(2)
where function B is determined by the interaction potential, and vector n is identi8ed with two angles, and !. In particular, for potentials proportional to the nth inverse power of the distance, the function B reads, B( ; | C − w |) = "( ) | C − w |(n−5)=(n−1) :
(3)
In the special case n=5, function B is independent of the magnitude of the relative velocity (Maxwell molecules). Maxwell molecules occupy a distinct place in the theory of the Boltzmann equation, they provide exact results. Three most important 8ndings for the Maxwell molecules are mentioned here: (1) The exact spectrum of the linearized Boltzmann collision integral, found by Truesdell and Muncaster, (2) Exact transport coeRcients found by Maxwell even before the Boltzmann equation was formulated, (3) Exact solutions to the space-free model version of the nonlinear Boltzmann equation. Pivotal results in this domain belong to Galkin who has found the general solution to the system of moment equations in a form of a series expansion, to Bobylev, Krook and Wu who have found an exact solution of a particular elegant closed form, and to Bobylev who has demonstrated the complete integrability of this dynamic system. A broad review of the Boltzmann equation and analysis of analytical solutions to kinetic models is presented in the book of Cercignani [87]. A modern account of rigorous results on the Boltzmann equation is given in the book [88]. Proof of the existence theorem for the Boltzmann equation was done by DiPerna and Lions [92]. It is customary to write the Boltzmann equation using another normalization of the distribution function, f(x; C; t) dx dC, taken in such a way that the function f is compliant with the de8nition of the hydrodynamic 8elds: the mass density %, the momentum density %u, and the energy density !: f(x; C; t)m dC = %(x; t) ; f(x; C; t)mC dC = %u(x; t) ; f(x; C; t)m
v2 dC = !(x; t) : 2
Here m is the particle’s mass. The Boltzmann equation for the distribution function f reads, 9f 9 + C; f = Q(f; f) ; 9t 9x
(4)
(5)
where the nonlinear integral operator on the right hand side is the Boltzmann collision integral, Q= (f(C )f(w ) − f(C)f(w))B( ; C) dw d d! : (6) R3
B−
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Finally, we mention the following form of the Boltzmann collision integral (sometimes referred to as the scattering or the quasi-chemical representation), Q = W (C; w | C ; w )[(f(C )f(w ) − f(C)f(w))] dw dw dC ; (7) where W is a generalized function which is called the probability density of the elementary event, 2
2
W = w(C; w | C ; w )(C + w − C − w )(v2 + w2 − v − w ) :
(8)
2.1.2. The basic properties of the Boltzmann equation Generalized function W has the following symmetries: W (C ; w | C; w) ≡ W (w ; C | C; w) ≡ W (C ; w | w; C) ≡ W (C; w | C ; w ) :
(9)
The 8rst two identities re=ect the symmetry of the collision process with respect to labeling the particles, whereas the last identity is the celebrated detail balance condition which is underpinned by the time-reversal symmetry of the microscopic (Newton’s) equations of motion. The basic properties of the Boltzmann equation are: 1. Additive invariants of collision operator: Q(f; f){1; C; v2 } dC = 0 ; (10) for any function f, assuming integrals exist. Equality (10) re=ects the fact that the number of particles, the three components of particle’s momentum, and the particle’s energy are conserved by the collision. Conservation laws (10) imply that the local hydrodynamic 8elds (4) can change in time only due to redistribution in the space. 2. Zero point of the integral (Q = 0) satisfy the equation (which is also called the detail balance): for almost all velocities, f(C ; x; t)f(w ; x; t) = f(C; x; t)f(w; x; t) : 3. Boltzmann’s local entropy production inequality: (x; t) = −kB Q(f; f)ln f dC ¿ 0 ;
(11)
for any function f, assuming integrals exist. Dimensional Boltzmann’s constant (kB ≈ 6×10−23 J=K) in this expression serves for a recalculation of the energy units into the absolute temperature units. Moreover, equality sign takes place if lnf is a linear combination of the additive invariants of collision. Distribution functions f whose logarithm is a linear combination of additive collision invariants, with coeRcients dependent on x, are called local Maxwell distribution functions fLM , % 2,kB T −3=2 −m(C − u)2 : (12) fLM = exp m m 2kB T
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Local Maxwellians are parametrized by values of 8ve hydrodynamic variables, %; u and T . This parametrization is consistent with the de8nitions of the hydrodynamic 8elds (4), fLM (m; mC; mv2 =2) = (%; %u; !) provided the relation between the energy and the kinetic temperature T , holds, ! = 3%=2mkB T . 4. Boltzmann’s H theorem: The function (13) S[f] = −kB f ln f dC ; is called the entropy density. The local H theorem for distribution functions independent of space states that the rate of the entropy density increase is equal to the nonnegative entropy production, dS = ¿0 : dt
(14)
Thus, if no space dependence is concerned, the Boltzmann equation describes relaxation to the unique global Maxwellian (whose parameters are 8xed by initial conditions), and the entropy density grows monotonically along the solutions. Mathematical speci8cations of this property has been initialized by Carleman, and many estimations of the entropy growth were obtained over the past two decades. In the case of space-dependent distribution functions, the local entropy density obeys the entropy balance equation: 9S(x; t) 9 ; Js (x; t) = (x; t) ¿ 0 ; (15) + 9t 9x where Js is the entropy =ux, Js (x; t) = −kB ln f(x; t)Cf(x; t) dC. For suitable boundary conditions, such as, specularly re=ecting or at the in8nity, the entropy =ux gives no contribution to the equation for the total entropy, Stot = S(x; t) dx and its rate of changes is then equal to the nonnegative total entropy production tot = (x; t) dx (the global H theorem). For more general boundary conditions which maintain the entropy in=ux the global H theorem needs to be modi8ed. A detailed discussion of this question is given by Cercignani. The local Maxwellian is also speci8ed as the maximizer of the Boltzmann entropy function (13), subject to 8xed hydrodynamic constraints (4). For this reason, the local Maxwellian is also termed as the local equilibrium distribution function. 2.1.3. Linearized collision integral Linearization of the Boltzmann integral around the local equilibrium results in the linear integral operator, Lh(C) =
h(w ) h(C) h(w) h(C ) W (C; w | C ; w )fLM (C)fLM (w) + − − dw dC dw : fLM (C ) fLM (w ) fLM (C) fLM (w)
Linearized collision integral is symmetric with respect to scalar product de8ned by the second derivative of the entropy functional, −1 −1 fLM (C)g(C)Lh(C) dC = fLM (C)h(C)Lg(C) dC ;
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it is nonpositively de8nite, −1 fLM (C)h(C)Lh(C) dC 6 0 ; −1 is a linear combination of collision invariants, where equality sign takes place if the function hfLM which characterize the null-space of the operator L. Spectrum of the linearized collision integral is well studied in the case of the small angle cut-o6.
2.2. Phenomenology and Quasi-chemical representation of the Boltzmann equation Boltzmann’s original derivation of his collision integral was based on a phenomenological “bookkeeping” of the gain and of the loss of probability density in the collision process. This derivation postulates that the rate of gain G equals G = W + (C; w | C ; w )f(C )f(w ) dC dw dw ; while the rate of loss is L = W − (C; w | C ; w )f(C)f(w) dC dw dw : The form of the gain and of the loss, containing products of one-body distribution functions in place of the two-body distribution, constitutes the famous Stosszahlansatz. The Boltzmann collision integral follows now as (G − L), subject to the detail balance for the rates of individual collisions, W + (C; w | C ; w ) = W − (C; w | C ; w ) : This representation for interactions di6erent from hard spheres requires also the cut-o6 of functions " (3) at small angles. The gain-loss form of the collision integral makes it evident that the detail balance for the rates of individual collisions is suRcient to prove the local H theorem. A weaker condition which is also suRcient to establish the H theorem was 8rst derived by Stueckelberg (so-called semi-detailed balance), and later generalized to inequalities of concordance: dC dw (W + (C; w | C ; w ) − W − (C; w | C ; w )) ¿ 0 ;
dC
dw(W + (C; w | C ; w ) − W − (C; w | C ; w )) 6 0 :
The semi-detailed balance follows from these expressions if the inequality signs are replaced by equalities.
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The pattern of Boltzmann’s phenomenological approach is often used in order to construct nonlinear kinetic models. In particular, nonlinear equations of chemical kinetics are based on this idea: If n chemical species Ai participate in a complex chemical reaction, 1si Ai ↔ "si Ai ; i
i
where 1si and "si are nonnegative integers (stoichiometric coeRcients) then equations of chemical kinetics for the concentrations of species cj are written n n n 9G 9G dci ("si − 1si ) ’+ 1sj − ’− "sj : = s exp s exp dt 9c 9cj j s=1 j=1 j=1 − Functions ’+ s and ’s are interpreted as constants of the direct and of the inverse reactions, while the function G is an analog of the Boltzmann’s H -function. Modern derivation of the Boltzmann equation, initialized by the seminal work of N.N. Bogoliubov, seeks a replacement condition, and which would be more closely related to many-particle dynamics. Such conditions are applied to the N -particle Liouville equation should factorize in the remote enough past, as well as in the remote in8nity (the hypothesis of weakening of correlations). Di6erent conditions has been formulated by D.N. Zubarev, J. Lewis and others. The advantage of these formulations is the possibility to systematically 8nd corrections not included in the Stosszahlansatz.
2.3. Kinetic models Mathematical complications caused by the nonlinearly Boltzmann collision integral are traced back to the Stosszahlansatz. Several approaches were developed in order to simplify the Boltzmann equation. Such simpli8cations are termed kinetic models. Various kinetic models preserve certain features of the Boltzmann equation, while sacri8cing the rest of them. The most well known kinetic model which preserves the H theorem is the nonlinear Bhatnagar–Gross–Krook model (BGK) [89]. The BGK collision integral reads: 1 QBGK = − (f − fLM (f)) : 4 The time parameter 4 ¿ 0 is interpreted as a characteristic relaxation time to the local Maxwellian. The BGK is a nonlinear operator: parameters of the local Maxwellian are identi8ed with the values of the corresponding moments of the distribution function f. This nonlinearly is of “lower dimension” than in the Boltzmann collision integral because fLM (f) is a nonlinear function of only the moments of f whereas the Boltzmann collision integral is nonlinear in the distribution function f itself. This type of simpli8cation introduced by the BGK approach is closely related to the family of so-called mean-8eld approximations in statistical mechanics. By its construction, the BGK collision integral preserves the following three properties of the Boltzmann equation: additive invariants of collision, uniqueness of the equilibrium, and the H theorem. A class of kinetic models which generalized the BGK model to quasi-equilibrium approximations of a general form is described as follows: The quasi-equilibrium f∗ for the set of linear functionals M (f) is a distribution function f∗ (M )(x; C)
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which maximizes the entropy under 8xed values of functions M . The quasi-equilibrium (QE) models are characterized by the collision integral [90], 1 QQE (f) = − [f − f∗ (M (f))] + QB (f∗ (M (f)); f∗ (M (f))) : 4 Same as in the case of the BGK collision integral, operator QQE is nonlinear in the moments M only. The QE models preserve the following properties of the Boltzmann collision operator: additive invariants, uniqueness of the equilibrium, and the H theorem, provided the relaxation time 4 to the quasi-equilibrium is suRciently small. A di6erent nonlinear model was proposed by Lebowitz et al. [91]: 9 9 m 9 QD = D f+ (C − u(f))f : 9C 9C kB T 9C The collision integral has the form of the self-consistent Fokker–Planck operator, describing di6usion (in the velocity space) in the self-consistent potential. Di6usion coeRcient D ¿ 0 may depend on the distribution function f. Operator QD preserves the same properties of the Boltzmann collision operator as the BGK model. The kinetic BGK model has been used for obtaining exact solutions of gasdynamic problems, especially its linearized form for stationary problems. Linearized BGK collision model has been extended to model more precisely the linearized Boltzmann collision integral. 2.4. Methods of reduced description One of the major issues raised by the Boltzmann equation is the problem of the reduced description. Equations of hydrodynamics constitute a closed set of equations for the hydrodynamic 8eld (local density, local momentum, and local temperature). From the standpoint of the Boltzmann equation, these quantities are low-order moments of the one-body distribution function, or, in other words, the macroscopic variables. The problem of the reduced description consists in giving an answer to the following two questions: 1. What are the conditions under which the macroscopic description sets in? 2. How to derive equations for the macroscopic variables from kinetic equations? The classical methods of reduced description for the Boltzmann equation are: the Hilbert method, the Chapman–Enskog method, and the Grad moment method. 2.4.1. The Hilbert method In 1911, David Hilbert introduced the notion of normal solutions, fH (C; n(x; t); u(x; t); T (x; t)) ; that is, solution to the Boltzmann equation which depends on space and time only through 8ve hydrodynamic 8elds [52]. The normal solutions are found from a singularly perturbed Boltzmann equation, Dt f =
1 Q(f; f) ; !
(16)
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where ! is a small parameter, and 9 9 f : Dt f ≡ f + C; 9t 9x Physically, parameter ! corresponds to the Knudsen number, the ratio between the mean free path of the molecules between collisions, and the characteristic scale of variation of the hydrodynamic 8elds. In the Hilbert method, one seeks functions n(x; t); u(x; t); T (x; t), such that the normal solution in the form of the Hilbert expansion, fH =
∞
!i fH(i)
(17)
i=0
satis8es the Eq. (16) order by order. Hilbert was able to demonstrate that this is formally possible. Substituting (17) into (16), and matching various order in !, we have the sequence of integral equations Q(fH(0) ; fH(0) ) = 0 ;
(18)
LfH(1) = Dt fH(0) ;
(19)
LfH(2) = Dt fH(1) − Q(fH(0) ; fH(1) ) ;
(20)
and so on for higher orders. Here L is the linearized collision integral. From Eq. (18), it follows that fH(0) is the local Maxwellian with parameters not yet determined. The Fredholm alternative, as applied to the second Eq. (19) results in (a) Solvability condition, Dt fH(0) {1; C; v2 } dC = 0 ; which is the set of compressible Euler equations of the nonviscous hydrodynamics. Solution to the Euler equation determine the parameters of the Maxwellian fH0 . (b) General solution fH(1) = fH(1)1 + fH(1)2 , where fH(1)1 is the special solution to the linear integral equation (19), and fH(1)2 is yet undetermined linear combination of the additive invariants of collision. (c) Solvability condition to the next equation (19) determines coeRcients of the function fH(1)2 in terms of solutions to the linear hyperbolic di6erential equations, Dt (fH(1)1 + fH(1)2 ){1; C; v2 } dC = 0 : Hilbert was able to demonstrate that this procedure of constructing the normal solution can be carried out to arbitrary order n, where the function fH(n) is determined from the solvability condition at the next, (n+1)th order. In order to summarize, implementation of the Hilbert method requires solutions for the function n(x; t); u(x; t), and T (x; t) obtained from a sequence of partial di6erential equations.
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2.4.2. The Chapman–Enskog method A completely di6erent approach to the reduced description was invented in 1917 by David Enskog [93], and independently by Sidney Chapman [51]. The key innovation was to seek an expansion of the time derivatives of the hydrodynamic variables rather than seeking the time-space dependencies of these functions as in the Hilbert method. The Chapman–Enskog method starts also with the singularly perturbed Boltzmann equation, and with the expansion ∞ (n) fCE = !n fCE : n=0 (n) di6ers from the Hilbert method: However, the procedure of evaluation of the functions fCE (0) (0) Q(fCE ; fCE )=0 ; (1) (0) (0) = −Q(fCE ; fCE )+ LfCE
9(0) (0) 9 fCE + C; 9t 9x
(21) (0) fCE :
(22)
Operator 9(0) =9t is de8ned from the expansion of the right hand side of hydrodynamic equation, 9 9(0) mv2 (0) C; fCE dC : (23) {%; %u; e} ≡ − m; mC; 9t 2 9x (0) From Eq. (21), function fCE is again the local Maxwellian, whereas (23) is the Euler equations, (0) and 9 =9t acts on various functions g(%; %u; e) according to the chain rule, 9g 9(0) 9g 9(0) 9g 9(0) e 9(0) g= %+ %u + ; 9t 9% 9t 9(%u) 9t 9e 9t
while the time derivatives 9(0) =9t of the hydrodynamic 8elds are expressed using the right hand side of Eq. (23). The result of the Chapman–Enskog de8nition of the time derivative 9(0) =9t, is that the Fredholm alternative is satis8ed by the right hand side of Eq. (22). Finally, the solution to the homogeneous equation is set to be zero by the requirement that the hydrodynamic variables as de8ned by the function f(0) + !f(1) coincide with the parameters of the local Maxwellian f(0) : (1) {1; C; v2 }fCE dC = 0 : (1) of the Chapman–Enskog method adds the terms The 8rst correction fCE (1) 9 9 mv2 (1) C; fCE dC {%; %u; e} = − m; mC; 9t 2 9x
to the time derivatives of the hydrodynamic 8elds. These terms correspond to the dissipative hydrodynamics where viscous momentum transfer and heat transfer are in the Navier–Stokes and Fourier form. The Chapman–Enskog method was the 8rst true success of the Boltzmann equation since it had made it possible to derive macroscopic equation without a priori guessing (the generalization of the Boltzmann equation onto mixtures predicted existence of the thermodi6usion before it has been found experimentally), and to express the kinetic coeRcient in terms of microscopic particle’s interaction.
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However, higher-order corrections of the Chapman–Enskog method, resulting in hydrodynamic equations with derivatives (Burnett hydrodynamic equations) face serve diRculties both from the theoretical, as well as from the practical sides. In particular, they result in unphysical instabilities of the equilibrium. 2.4.3. The Grad moment method In 1949, Harold Grad extended the basic assumption behind the Hilbert and the Chapman–Enskog methods (the space and time dependence of the normal solutions is mediated by the 8ve hydrodynamic moments) [158]. A physical rationale behind the Grad moment method is an assumption of the decomposition of motions: (i) During the time of order 4, a set of distinguished moments M (which include the hydrodynamic moments and a subset of higher-order moments) does not change signi8cantly as compared to the rest of the moments M (the fast evolution). (ii) Towards the end of the fast evolution, the values of the moments M become unambiguously determined by the values of the distinguished moments M . (iii) On the time of order 4, dynamics of the distribution function is determined by the dynamics of the distinguished moments while the rest of the moments remain to be determined by the distinguished moments (the slow evolution period). Implementation of this picture requires an ansatz for the distribution function in order to represent the set of states visited in the course of the slow evolution. In Grad’s method, these representative sets are 8nite-order truncations of an expansion of the distribution functions in terms of Hermite velocity tensors: N a(1) (M )H(1) (C − u) ; (24) fG (M ; C) = fLM (%; u; E; C) 1 + (1)
where H(1) (C − u) are various Hermite tensor polynomials, orthogonal with the weight fLM , while coeRcient a(1) (M ) are known functions of the distinguished moments M , and N is the highest order of M . Other moments are functions of M : M = M (fG (M )). Slow evolution of distinguished moments is found upon substitution of Eq. (24) into the Boltzmann equation and 8nding the moments of the resulting expression (Grad’s moment equations). Following Grad, this extremely simple approximation can be improved by extending the list of distinguished moments. The most well known is Grad’s 13-moment approximation where the set of distinguished moments consists of 8ve hydrodynamic moments, 8ve components of the traceless stress tensor ij = m[(vi − ui )(vj − uj ) − ij (C − u)2 =3]f dC, and of the three components of the heat =ux vector qi = (vi − ui )m(C − u)2 =2f dC. The time evolution hypothesis cannot be evaluated for its validity within the framework of Grad’s approach. It is not surprising therefore that Grad’s methods failed to work in situations where it was (unmotivatedly) supposed to, primarily, in the phenomena with sharp time-space dependence such as the strong shock wave. On the other hand, Grad’s method was quite successful for describing transition between parabolic and hyperbolic propagation, in particular, the second sound e6ect in massive solids at low temperatures, and, in general, situations slightly deviating from the classical Navier–Stokes–Fourier domain. Finally, the Grad method has been important background for
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development of phenomenological nonequilibrium thermodynamics based on hyperbolic 8rst-order equation, the so-called EIT (extended irreversible thermodynamics). 2.4.4. Special approximations Special approximation of the solutions to the Boltzmann equation has been found for several problems, and which perform better than results of “regular” procedures. The most well known is the ansatz introduced independently by Mott–Smith and Tamm for the strong shock wave problem: The (stationary) distribution function is thought as fTMS (a(x)) = (1 − a(x))f+ + a(x)f− ;
(25)
where f± are upstream and downstream Maxwell distribution functions, whereas a(x) is an undetermined scalar function of the coordinate along the shock tube. Equation for function a(x) has to be found upon substitution of Eq. (25) into the Bolltzmann equation, and upon integration with some velocity-dependent function ’(C). Two general problems arise with the special approximation thus constructed: Which function ’(C) should be taken, and how to 8nd correction to the ansatz like Eq. (25). 2.4.5. The method of invariant manifold The general approach to the problem of reduced description for dissipative system was recognized as the problem of 8nding stable invariant manifolds in the space of distribution functions [4–6]. The notion of invariant manifold generalizes the normal solution in the Hilbert and in the Chapman –Enskog method, and the 8nite-moment sets of distribution function in the Grad method: If 8 is a smooth manifold in the space of distribution function, and if f8 is an element of 8, then 8 is invariant with respect to the dynamic system, 9f = J (f) ; 9t if J (f8 ) ∈ T8
(26) for all f8 ∈ 8 ;
(27)
where T8 is the tangent bundle of the manifold 8. Application of the invariant manifold idea to dissipative systems is based on iterations, progressively improving the initial approximation, involves the following steps: Thermodynamic projector. Given a manifold 8 (not obligatory invariant), the macroscopic dynamics on this manifold is de8ned by the macroscopic vector 8eld, which is the result of a projection of vectors J (f8 ) onto the tangent bundle T8. The thermodynamic projector Pf∗8 takes advantage of dissipativity: ker Pf∗8 ⊆ ker Df S|f8 ;
(28)
where Df S|f8 is the di6erential of the entropy evaluated in f8 . This condition of thermodynamicity means that each state of the manifold 8 is regarded as the result of decomposition of motions occurring near 8: The state f8 is the maximum entropy state on the set of states f8 + ker Pf∗8 . Condition of thermodynamicity does not de8ne projector completely; rather, it is the condition that should be satis8ed by any projector used to de8ne the macroscopic vector 8eld, J8 = Pf∗8 J (f8 ). For, once the condition (28) is met, the macroscopic vector 8eld
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preserves dissipativity of the original microscopic vector 8eld J (f): Df S|f8 · Pf∗8 (J (f8 )) ¿ 0
for all f8 ∈ 8 :
The thermodynamic projector is the formalization of the assumption that 8 is the manifold of slow motion: If a fast relaxation takes place at least in a neighborhood of 8, then the states visited in this process before arriving at f8 belong to ker Pf∗8 . In general, Pf∗8 depends in a nontrivial way on f8 . Iterations for the invariance condition. The invariance condition for the manifold 8 reads P8 (J (f8 )) − J (f8 ) = 0 ; here P8 is arbitrary (not obligatory thermodynamic) projector onto the tangent bundle of 8. The invariance condition is considered as an equation which is solved iteratively, starting with initial approximation 80 . On the (n + 1)th iteration, the correction f(n+1) = f(n) + f(n+1) is found from linear equations, Df Jn∗ f(n+1) = Pn∗ J (f(n) ) − J (f(n) ) ; Pn∗ f(n+1) = 0 ;
(29)
here Df Jn∗ is the linear selfajoint operator with respect to the scalar product by the second di6erential of the entropy Df2 S|f(n) . Together with the above-mentioned principle of thermodynamic projecting, the selfadjoint linearization implements the assumption about the decomposition of motions around the nth approximation. The selfadjoint linearization of the Boltzmann collision integral Q (7) around a distribution function f is given by the formula, f(C)f(w) + f(C )f(w ) Df Q∗ f = W (C; w; | C ; w ) 2 f(C ) f(w ) f(C) f(w) dw dC dw : (30) × + − − f(C ) f(w ) f(C) f(w) If f = fLM , the self-adjoint operator (30) becomes the linearized collision integral. The method of invariant manifold is the iterative process: ∗ (f(n) ; Pn∗ ) → (f(n+1) ; Pn∗ ) → (f(n+1) ; Pn+1 ) :
On the each 1st part of the iteration, the linear equation (29) is solved with the projector known from the previous iteration. On the each 2nd part, the projector is updated, following the thermodynamic construction. The method of invariant manifold can be further simpli8ed if smallness parameters are known. The proliferation of the procedure in comparison to the Chapman–Enskog method is essentially twofold: First, the projector is made dependent on the manifold. This enlarges the set of admissible approximations. Second, the method is based on iteration rather than a series expansion in a smallness parameter. Importance of iteration procedures is well understood in physics, in particular, in the renormalization group approach to reducing the description in equilibrium statistical mechanics, and in the Kolmogorov–Arnold–Moser theory of 8nite-dimensional Hamiltonian systems.
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2.4.6. Quasi-equilibrium approximations Important generalization of the Grad moment method is the concept of the quasi-equilibrium approximations already mentioned above (we will discuss this approximation in detail in a separate section). The quasi-equilibrium distribution function for a set of distinguished moment M = m(f) maximizes the entropy density S for 8xed M . The quasi-equilibrium manifold 8∗ (M ) is the collection of the quasi-equilibrium distribution functions for all admissible values of M . The quasi-equilibrium approximation is the simplest and extremely useful (not only in the kinetic theory itself) implementation of the hypothesis about a decomposition of motions: if M are considered as slow variables, then states which could be visited in the course of rapid motion in the vicinity of 8∗ (M ) belong to the planes 9M = {f | m(f − f∗ (M )) = 0}. In this respect, the thermodynamic construction in the method of invariant manifold is a generalization of the quasi-equilibrium approximation where the given manifold is equipped with a quasi-equilibrium structure by choosing appropriately the macroscopic variables of the slow motion. In contrast to the quasi-equilibrium, the macroscopic variables thus constructed are not obligatory moments. A text book example of the quasi-equilibrium approximation is the generalized Gaussian function for M = {%; %u; P} where Pij = vi vj f dC is the pressure tensor. The thermodynamic projector P ∗ for a quasi-equilibrium approximation was 8rst introduced by Robertson [95] (in a di6erent context of conservative dynamics and for a special case of the Gibbs–Shannon entropy). It acts on a function : as follows: 9f∗ ∗ mi : dC ; PM := 9Mi i where M = mi f dC. The quasi-equilibrium approximation does not exist if the highest order moment is an odd polynomial of velocity (therefore, there exists no quasi-equilibrium for thirteen Grad’s moments). Otherwise, the Grad moment approximation is the 8rst-order expansion of the quasi-equilibrium around the local Maxwellian. 2.5. Discrete velocity models If the number of microscopic velocities is reduced drastically to only a 8nite set, the resulting discrete velocity, continuous time and continuous space models can still mimic the gas-dynamic =ows. This idea was introduced in Broadwell’s paper in 1963 to mimic the strong shock wave [94]. Further important development of this idea was due to Cabannes and Gatignol in the seventies who introduced a systematic class of discrete velocity models [98]. The structure of the collision operators in the discrete velocity models mimics the polynomial character of the Boltzmann collision integral. Discrete velocity models are implemented numerically by using the natural operator splitting in which each update due to free =ight is followed by the collision update, the idea which dates back to Grad. One of the most important recent results is the proof of convergence of the discrete velocity models with pair collisions to the Boltzmann collision integral. 2.6. Direct simulation Besides the analytical approach, direct numerical simulation of Boltzmann-type nonlinear kinetic equations have been developed since mid of 1960s [96,97]. The basis of the approach is a representation of the Boltzmann gas by a set of particles whose dynamics is modeled as a sequence of free
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propagation and collisions. The modeling of collisions uses a random choice of pairs of particles inside the cells of the space, and changing the velocities of these pairs in such a way as to comply with the conservation laws, and in accordance with the kernel of the Boltzmann collision integral. At present, there exists a variety of this scheme known under the common title of the direct simulation Monte-Carlo method [96,97]. The DSMC, in particular, provides data to test various analytical theories. 2.7. Lattice gas and Lattice Boltzmann models Since mid of 1980s, the kinetic theory based approach to simulation of complex macroscopic phenomena such as hydrodynamics has been developed. The main idea of the approach is construction of minimal kinetic system in such a way that their long-time and large-scale limit matches the desired macroscopic equations. For this purpose, the fully discrete (in time–space–velocity) nonlinear kinetic equations are considered on suRciently isotropic lattices, where the links represent the discrete velocities of 8ctitious particles. In the earlier version of the lattice methods, the particle-based picture has been exploited, subject to the exclusion rule (one or zero particle per lattice link) [the lattice gas model [99]]. Most of the present versions use the distribution function picture, where populations of the links are noninteger [the Lattice Boltzmann model [100–104]]. Discrete-time dynamics consists of a propagation step where populations are transmitted to adjacent links and collision step where populations of the links at each node of the lattice are equilibrated by a certain rule. Most of the present versions use the BGK-type equilibration, where the local equilibrium is constructed in such a way as to match desired macroscopic equations. The Lattice Boltzmann method is a useful approach for computational =uid dynamics, e6ectively compliant with parallel architectures. The proof of the H theorem for the Lattice gas models is based on the semi-detailed (or Stueckelberg’s) balance principle. The proof of the H theorem in the framework of the Lattice Boltzmann method has been only very recently achieved [105–110]. 2.8. Other kinetic equations 2.8.1. The Enskog equation for hard spheres The Enskog equation for hard spheres is an extension of the Boltzmann equation to moderately dense gases. The Enskog equation explicitly takes into account the nonlocality of collisions through a two-fold modi8cation of the Boltzmann collision integral: First, the one-particle distribution functions are evaluated at the locations of the centers of spheres, separated by the nonzero distance at the impact. This makes the collision integral nonlocal in space. Second, the equilibrium pair distribution function at the contact of the spheres enhances the scattering probability. The proof of the H theorem for the Enskog equation has posed certain diRculties, and has led to a modi8cation of the collision integral. Methods of solution of the Enskog equation are immediate generalizations of those developed for the Boltzmann equation, but there is one additional diRculty. The Enskog collision integral is nonlocal in space. The Chapman–Enskog method, when applied to the Enskog equation, is supplemented with a gradient expansion around the homogeneous equilibrium state.
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2.8.2. The Vlasov equation The Vlasov equation (or kinetic equation for a self-consistent force) is the nonlinear equation for the one-body distribution function, which takes into account a long-range interaction between particles: 9 9 9 f + C; f + F; f = 0 ; 9t 9x 9C where F = ;(|x − x |)(x − x )=(|x − x |)n(x ) dx is the self-consistent force. In this expression ;(|x − x |)(x − x )=(|x − x |) is the microscopic force the two particles, and n(x ) is the between density of particles, de8ned self-consistently, n(x ) = f(x ; C) dC. The Vlasov equation is used for a description of collisionless plasmas in which case it is completed by a set of Maxwell equation for the electromagnetic 8eld [135]. It is also used for a description of the gravitating gas. The Vlasov equation is an in8nite-dimensional Hamiltonian system. Many special and approximate (wave-like) solutions to the Vlasov equation are known and they describe important physical e6ects. One of the most well known e6ects is the Landau damping: The energy of a volume element dissipates with the rate 2 !(k) df0 Q ≈ −|E| ; k 2 dv v=!=k where f0 is the Maxwell distribution function, |E| is the amplitude of the applied monochromatic electric 8eld with the frequency !(k), k is the wave vector. The Landau damping is thermodynamically reversible e6ect, and it is not accompanied with an entropy increase. Thermodynamically reversed to the Landau damping is the plasma echo e6ect. 2.8.3. The Fokker–Planck equation The Fokker–Planck equation (FPE) is a familiar model in various problems of nonequilibrium statistical physics [111,112]. We consider the FPE of the form 9W (x; t) 9 9 9 = D W U+ W : (31) 9t 9x 9x 9x Here W (x; t) is the probability density over the con8guration space x, at the time t, while U (x) and D(x) are the potential and the positively semi-de8nite ((y; Dy) ¿ 0) di6usion matrix. The FPE (31) is particularly important in studies of polymer solutions [113–115]. Let us recall the two properties of the FPE (31), important to what will follow: (i). Conservation of the total probability: W (x; t) d x = 1. (ii). Dissipation: The equilibrium distribution, Weq ˙ exp(−U ), is the unique stationary solution to the FPE (31). The entropy, W (x; t) dx ; (32) S[W ] = − W (x; t)ln Weq (x)
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is a monotonically growing function due to the FPE (31), and it arrives at the global maximum in the equilibrium. These properties are most apparent when the FPE (31) is rewritten as follows: S[W ] ; (33) 9t W (x; t) = Mˆ W W (x; t) where
9 9 ˆ MW = − W (x; t)D(x) 9x 9x
is a positive semi-de8nite symmetric operator with kernel 1. Form (33) is the dissipative part of a structure termed GENERIC (the dissipative vector 8eld is a metric transform of the entropy gradient) [116,117]. The entropy does not depend on kinetic constants. It is the same for di6erent details of kinetics and depends only on equilibrium data. Let us call this property “universality”. It is known that for the Boltzmann equation there exists only one universal Lyapunov functional. It is the entropy (we do not distinguish functionals which are connected by multiplication on a constant or adding a constant). But for the FPE there exists a big family of universal Lyapunov functionals. Let h(a) be a convex function of one variable a ¿ 0, h (a) ¿ 0, W (x; t) dx : (34) Sh [W ] = − Weq (x)h Weq (x) The production of the generalized entropy Sh , h is nonnegative: 9 W (x; t) 9 W (x; t) W (x; t) ;D ¿0 : h (x) = Weq (x)h Weq (x) 9x Weq (x) 9x Weq (x)
(35)
The most important variants for choice of h: h(a) = a ln a, Sh is the Boltzmann–Gibbs–Shannon entropy (in the Kullback form [118,119]), h(a) = a ln a − j ln a, Shj is the maximal family of additive entropies [120–122] (these entropies are additive for composition of independent subsystems). h(a) = (1 − aq )=(1 − q), Shq is the family of Tsallis entropies [123,124]. These entropies are not additive, but become additive after nonlinear monotonous transformation. This property can serve as de8nition of the Tsallis entropies in the class of generalized entropies (34) [122]. 2.9. Equations of chemical kinetics and their reduction 2.9.1. Outline of the dissipative reaction kinetics We begin with an outline of the reaction kinetics (for details see e.g. the book [59]). Let us consider a closed system with n chemical species A1 ; : : : ; An , participating in a complex reaction. The complex reaction is represented by the following stoichiometric mechanism: 1s1 A1 + · · · + 1sn An "s1 A1 + · · · + "sn An ;
(36)
where the index s = 1; : : : ; r enumerates the reaction steps, and where integers, 1si and "si , are stoichiometric coeRcients. For each reaction step s, we introduce n-component vectors s and s with components 1si and "si . Notation s stands for the vector with integer components >si = "si − 1si (the stoichiometric vector).
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For every Ai an extensive variable Ni , “the number of particles of that species”, is de8ned. The concentration of Ai is ci = Ni =V , where V is the volume. Given the stoichiometric mechanism (36), the reaction kinetic equations read: r N˙ = V J (c); J (c) = s Ws (c) ; (37) s=1
where dot denotes the time derivative, and Ws is the reaction rate function of the step s. In particular, the mass action law suggests the polynomial form of the reaction rates: n n Ws (c) = Ws+ (c) − Ws− (c) = ks+ (T ) ci1i − ks− (T ) ci"i ; (38) i=1
i=1
where ks+ (T ) and ks− (T ) are the constants of the direct and of the inverse reactions rates of the sth reaction step, T is the temperature. The (generalized) Arrhenius equation gives the most popular form of dependence ks+ (T ): bs ks± (T ) = a± exp(Ss± =kB ) exp(−Hs± =kB T ) ; s T ±
a± ,
b±
H±
(39) S±
where s s are constants, s are activation enthalpies, s are activation entropies. The rate constants are not independent. The principle of detail balance gives the following connection between these constants: There exists such a positive vector c eq (T ) that Ws+ (c eq ) = Ws− (c eq )
for all s = 1; : : : ; r :
(40) eq
The necessary and suRcient conditions for existence of such c can be formulate as the system of polynomial equalities for {ks± }, if the stoichiometric vectors {s } are linearly dependent (see, for example, [59]). The reaction kinetic equations (37) do not give us a closed system of equations, because dynamics of the volume V is not de8ned still. Four classical conditions for closure of this system are well studied: U , V = const (isolated system, U is the internal energy); H , P = const (thermal isolated isobaric system, P is the pressure, H = U + PV is the enthalpy), V , T = const (isochoric isothermal conditions); P, T =const (isobaric isothermal conditions). For V , T =const we do not need additional equations and data. It is possible just to divide Eq. (37) on the constant volume and write r c˙ = s Ws (c) : (41) s=1
For nonisothermal and nonisochoric conditions we do need addition formulae to derive T and V . For all four classical conditions the thermodynamic Lyapunov functions G for kinetic equations are known: U; V = const; GU; V = −S=kB ;
V; T = const; GV; T = F=kB T = U=kB T − S=kB ;
H; P = const; GH; P = −S=kB ;
P; T = const; GP; T = G=T = H=kB T − S=kB ;
(42)
where F = U − TS is the free energy (Helmholtz free energy), G = H − TS is the free enthalpy (Gibbs free energy). All the thermodynamic Lyapunov functions are normalized to dimensionless scale (if one measures the number of particles in moles, then it is necessary to change kB to R). All these functions decrease in time. For classical conditions the correspondent thermodynamic Lyapunov
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functions can be written in the form: G• (const; N ). The derivatives 9G• (const; N )=9Ni are the same functions of c and T for all classical conditions: 9G• (const; N ) @chem i (c; T ) = ; 9Ni kB T
@i (c; T ) =
(43)
where @chemi (c; T ) is the chemical potential of Ai . Usual G• (const; N ) are strictly convex functions of N , and the matrix 9@i =9cj is positively de8ned. The dissipation inequality (44) holds dG• = V ( ; J ) 6 0 : dt
(44)
This inequality is the restriction on possible kinetic low and on possible values of kinetic constants. The most important generalization of the mass action law (38) is the Marcelin–De Donder kinetic function. This generalization [187,188] is based on ideas of the thermodynamic theory of aRnity [189]. We use the kinetic function suggested in its 8nal form in [188]. Within this approach, the functions Ws are constructed as follows: For a given (c; T ) (43), and for a given stoichiometric mechanism (36), we de8ne the gain (+) and the loss (−) rates of the sth step, Ws− = ’− s exp( ; s ) ;
Ws+ = ’+ s exp( ; s );
(45)
+ − where ’± s ¿ 0 are kinetic factors. The Marcelin–De Donder kinetic function reads: Ws = Ws − Ws , and the right hand side of the kinetic equation (37) becomes,
J=
r
− s {’+ s exp( ; s ) − ’s exp( ; s )} :
(46)
s=1
For the Marcelin–De Donder reaction rate (45), the dissipation inequality (44) reads: G˙ =
r
( ; s ) ( ; s ) [( ; s ) − ( ; s )]{’+ − ’− }60 : s e s e
(47)
s=1
The kinetic factors ’± s should satisfy certain conditions in order to make valid the dissipation inequality (47). A well known suRcient condition is the detail balance: − ’+ s = ’s ;
(48)
other suRcient conditions are discussed in detail elsewhere [125,59,126]. For ideal systems, function G• is constructed from the thermodynamic data of individual species. It is convenient to start from the isochoric isothermal conditions. The Helmholtz free energy for ideal system is F = kB T Ni [ln ci − 1 + @0i ] + const T; V ; (49) i
where the internal energy is assumed to be a linear function: U= Ni ui (T ) = Ni (u0i + CVi T ) i
i
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in given interval of c, T , ui (T ) is the internal energy of Ai per particle. It is well known that S = −(9F=9T )V; N =const , U = F + TS = F − T (9F=9T )V; N =const , hence, ui (T ) = −kB T 2 d@0i =dT and @0i = i + u0i =kB T − (CVi =kB )ln T ;
(50)
where i = const, CVi is the Ai heat capacity at constant volume (per particle). In concordance with the form of ideal free energy (49) the expression for is: @i = ln ci + i + u0i =kB T − (CVi =kB )ln T :
(51)
For the function of form (51), the Marcelin–De Donder equation casts into the more familiar mass action law form (38). Taking into account the principle of detail balance (48) we get the ideal rate functions: Ws (c) = Ws+ (c) − Ws− (c) ; Ws+ (c)
= ’(c; T )T
−
i
1si CVi =kB
e
i
1si (i +u0i =kB T )
n
ci1i ;
i=1
Ws− (c) = ’(c; T )T −
i
"si CVi =kB
e
i
"si (i +u0i =kB T )
n
ci"i :
(52)
i=1
where ’(c; T ) is an arbitrary positive function (from thermodynamic point of view). Let us discuss further the vector 8eld J (c) in the concentration space (41). Conservation laws (balances) impose linear constrains on admissible vectors dc=dt: (bi ; c) = Bi = const;
i = 1; : : : ; l ;
(53)
where bi are 8xed and linearly independent vectors. Let us denote as B the set of vectors which satisfy the conservation laws (53) with given Bi : B = {c | (b1 ; c) = B1 ; : : : ; (bl ; c) = Bl } : The natural phase space X of system (41) is the intersection of the cone of n-dimensional vectors with nonnegative components, with the set B, and dim X = d = n − l. In the sequel, we term a vector c ∈ X the state of the system. In addition, we assume that each of the conservation laws is supported by each elementary reaction step, that is (s ; bi ) = 0 ;
(54)
for each pair of vectors s and bi . Reaction kinetic equations describe variations of the states in time. The phase space X is positiveinvariant of system (41): If c(0) ∈ X , then c(t) ∈ X for all the times t ¿ 0. In the sequel, we assume that the kinetic equation (41) describes evolution towards the unique equilibrium state, c eq , in the interior of the phase space X . Furthermore, we assume that there exists a strictly convex function G(c) which decreases monotonically in time due to Eq. (41): Here ∇G is the vector of partial derivatives 9G=9ci , and the convexity assumes that the n × n matrices Hc = 92 G(c)=9ci 9cj ;
(55)
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are positive de8nite for all c ∈ X . In addition, we assume that the matrices (55) are invertible if c is taken in the interior of the phase space. The function G is the Lyapunov function of system (37), and c eq is the point of global minimum of the function G in the phase space X . Otherwise stated, the manifold of equilibrium states c eq (B1 ; : : : ; Bl ) is the solution to the variational problem, G → min for (bi ; c) = Bi ;
i = 1; : : : ; l :
(56)
For each 8xed value of the conserved quantities Bi , the solution is unique. In many cases, however, it is convenient to consider the whole equilibrium manifold, keeping the conserved quantities as parameters. For example, for perfect systems in a constant volume under a constant temperature, the Lyapunov function G reads G=
n
ci [ln(ci =cieq ) − 1] :
(57)
i=1
It is important to stress that c eq in Eq. (57) is an arbitrary equilibrium of the system, under arbitrary values of the balances. In order to compute G(c), it is unnecessary to calculate the speci8c equilibrium c eq which corresponds to the initial state c. Let us compare the Lyapunov function G (57) with the classical formula for the free energy (49). This comparison gives a possible choice for c eq : ln cieq = −i − u0i =kB T + (CVi =kB )ln T :
(58)
2.9.2. The problem of reduced description in chemical kinetics What does it mean, “to reduce the description of a chemical system”? This means the following: 1. To (i) (ii) 2. To (i)
shorten the list of species. This, in turn, can be achieved in two ways: To eliminate inessential components from the list; To lump some of the species into integrated components. shorten the list of reactions. This also can be done in several ways: To eliminate inessential reactions, those which do not signi8cantly in=uence the reaction process; (ii) To assume that some of the reactions “have been already completed”, and that the equilibrium has been reached along their paths (this leads to dimensional reduction because the rate constants of the “completed” reactions are not used thereafter, what one needs are equilibrium constants only). 3. To decompose the motions into fast and slow, into independent (almost-independent) and slaved, etc. As the result of such a decomposition, the system admits a study “in parts”. After that, results of this study are combined into a joint picture. There are several approaches which fall into this category. The famous method of the quasi-steady state (QSS), pioneered by Bodenstein and Semenov, follows the Chapman–Enskog method. The partial equilibrium approximations are predecessors of the Grad method and quasi-equilibrium approximations in physical kinetics. These two family of methods have di6erent physical backgrounds and mathematical forms.
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2.9.3. Partial equilibrium approximations Quasi-equilibrium with respect to reactions is constructed as follows: from the list of reactions (36), one selects those which are assumed to equilibrate 8rst. Let they be indexed with the numbers s1 ; : : : ; sk . The quasi-equilibrium manifold is de8ned by the system of equations, Ws+i = Ws−i ;
i = 1; : : : ; k :
(59)
This system of equations looks particularly elegant when written in terms of conjugated (dual) variables, = ∇G: (si ; ) = 0;
i = 1; : : : ; k :
(60)
In terms of conjugated variables, the quasi-equilibrium manifold forms a linear subspace. This subspace, L⊥ , is the orthogonal completement to the linear envelope of vectors, L = lin{s1 ; : : : ; sk }. Quasi-equilibrium with respect to species is constructed practically in the same way but without selecting the subset of reactions. For a given set of species, Ai1 ; : : : ; Aik , one assumes that they evolve fast to equilibrium, and remain there. Formally, this means that in the k-dimensional subspace of the space of concentrations with the coordinates ci1 ; : : : ; cik , one constructs the subspace L which is de8ned by the balance equations, (bi ; c) = 0. In terms of the conjugated variables, the quasi-equilibrium manifold, L⊥ , is de8ned by equations, ∈ L⊥ ;
( = (@1 ; : : : ; @n )) :
(61)
The same quasi-equilibrium manifold can be also de8ned with the help of 8ctitious reactions: Let g1 ; : : : ; gq be a basis in L. Then Eq. (61) may be rewritten as follows: (gi ; ) = 0;
i = 1; : : : ; q :
(62)
Illustration: Quasi-equilibrium with respect to reactions in hydrogen oxidation: Let us assume equilibrium with respect to dissociation reactions, H2 2H, and, O2 2O, in some subdomain of reaction conditions. This gives k1+ cH2 = k1− cH2 ;
k2+ cO2 = k2− cO2 :
Quasi-equilibrium with respect to species: for the same reaction, let us assume equilibrium over H, O, OH, and H2 O2 , in a subdomain of reaction conditions. Subspace L is de8ned by balance constraints: cH + cOH + 2cH2 O2 = 0;
cO + cOH + 2cH2 O2 = 0 :
Subspace L is two-dimensional. Its basis, {g1 ; g2 } in the coordinates cH , cO , cOH , and cH2 O2 reads: g1 = (1; 1; −1; 0);
g2 = (2; 2; 0; −1) :
Corresponding Eq. (62) is @H + @O = @OH ;
2@H + 2@O = @H2 O2 :
General construction of the quasi-equilibrium manifold: In the space of concentration, one de8nes a subspace L which satis8es the balance constraints: (bi ; L) ≡ 0 :
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The orthogonal complement of L in the space with coordinates = ∇G de8nes then the quasiequilibrium manifold L . For the actual computations, one requires the inversion from to c. Duality structure ↔ c is well studied by many authors [127,126]. Quasi-equilibrium projector. It is not suRcient to just derive the manifold, it is also required to de8ne a projector which would transform the vector 8eld de8ned on the space of concentrations to a vector 8eld on the manifold. Quasi-equilibrium manifold consists of points which minimize G on the aRne spaces of the form c + L. These aRne planes are hypothetic planes of fast motions (G is decreasing in the course of the fast motions). Therefore, the quasi-equilibrium projector maps the whole space of concentrations on L parallel to L. The vector 8eld is also projected onto the tangent space of L parallel to L. Thus, the quasi-equilibrium approximation implies the decomposition of motions into the fast—parallel to L, and the slow—along the quasi-equilibrium manifold. In order to construct the quasi-equilibrium approximation, knowledge of reaction rate constants of “fast” reactions is not required (stoichiometric vectors of all these fast reaction are in L, fast ∈ L, thus, knowledge of L suRces), one only needs some con8dence in that they all are suRciently fast [128]. The quasiequilibrium manifold itself is constructed based on the knowledge of L and of G. Dynamics on the quasi-equilibrium manifold is de8ned as the quasi-equilibrium projection of the “slow component” of kinetic equations (37). 2.9.4. Model equations The assumption behind the quasi-equilibrium is the hypothesis of the decomposition of motions into fast and slow. The quasi-equilibrium approximation itself describes slow motions. However, sometimes it becomes necessary to restore to the whole system, and to take into account the fast motions as well. With this, it is desirable to keep intact one of the important advantages of the quasi-equilibrium approximation—its independence of the rate constants of fast reactions. For this purpose, the detailed fast kinetics is replaced by a model equation (single relaxation time approximation). Quasi-equilibrium models (QEM) are constructed as follows: For each concentration vector c, consider the aRne manifold, c + L. Its intersection with the quasi-equilibrium manifold L consists of one point. This point delivers the minimum to G on c + L. Let us denote this point as cL∗ (c). The equation of the quasi-equilibrium model reads: 1 c˙ = − [c − cL∗ (c)] + s Ws (cL∗ (c)) ; (63) 4 slow
where 4 ¿ 0 is the relaxation time of the fast subsystem. Rates of slow reactions are computed at the points cL∗ (c) (the second term on the right hand side of Eq. (63)), whereas the rapid motion is taken into account by a simple relaxational term (the 8rst term on the right hand side of Eq. (63)). The most famous model kinetic equation is the BGK equation in the theory of the Boltzmann equation [89]. The general theory of the quasi-equilibrium models, including proofs of their thermodynamic consistency, was constructed in the paper [90]. Single relaxation time gradient models (SRTGM) were considered in the context of the lattice Boltzmann method for hydrodynamics [109,129]. These models are aimed at improving the obvious drawback of quasi-equilibrium models (63): In order to construct the QEM, one needs to compute
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the function, cL∗ (c) = arg
min
x∈c+L; x¿0
G(x) :
(64)
This is a convex programming problem. It does not always has a closed-form solution. Let g1 ; : : : ; gk is the orthonormal basis of L. We denote as D(c) the k × k matrix with the elements (gi ; Hc gj ), where Hc is the matrix of second derivatives of G (55). Let C (c) be the inverse of D(c). The single relaxation time gradient model has the form: 1 gi C (c)ij (gj ; ∇G) + s Ws (c) : (65) c˙ = − 4 i; j slow
The 8rst term drives the system to the minimum of G on c + L, it does not require solving the problem (64), and its spectrum in the quasi-equilibrium is the same as in the quasi-equilibrium model (63). Note that the slow component is evaluated in the “current” state c. The 8rst term in Eq. (65) has a simple form 1 (66) c˙ = − grad G ; 4 if one calculates grad G with the entropic scalar product 3 x; y = (x; Hc y). Models (63) and (65) lift the quasi-equilibrium approximation to a kinetic equation by approximating the fast dynamics with a single “reaction rate constant”—relaxation time 4. 2.9.5. Quasi-steady state approximation The quasi-steady state approximation (QSS) is a tool used in a major number of works. Let us split the list of species in two groups: The basic and the intermediate (radicals etc). Concentration vectors are denoted accordingly, c s (slow, basic species), and c f (fast, intermediate species). The concentration vector c is the direct sum, c =c s ⊕c f . The fast subsystem is Eq. (37) for the component c f at 8xed values of c s . If it happens that this way de8ned fast subsystem relaxes to a stationary f f state, c f → cqss (c s ), then the assumption that c f = cqss (c) is precisely the QSS assumption. The slow s subsystem is the part of system (37) for c , on the right hand side of which the component c f is f replaced with cqss (c). Thus, J = Js ⊕ Jf , where c˙f = Jf (c s ⊕ c f );
c s = const;
f (c s )) : c˙s = Js (c s ⊕ cqss
f c f → cqss (c s ) ;
(67) (68)
Bifurcations in system (67) under variation of c s as a parameter are confronted to kinetic critical phenomena. Studies of more complicated dynamic phenomena in the fast subsystem (67) require various techniques of averaging, stability analysis of the averaged quantities, etc. Various versions of the QSS method are well possible, and are actually used widely, for example, the hierarchical QSS method. There, one de8nes not a single fast subsystem but a hierarchy of them, c f1 ; : : : ; c fk . Each subsystem c fi is regarded as a slow system for all the foregoing subsystems, and it 3 Let us remind that grad G is the Riesz representation of the di6erential of G in the phase space X : G(c + Yc) = G(c) + grad G(c); Yc + o(Yc). It depends on the scalar product, and from thermodynamic point of view there is only one distinguished scalar product in concentration space. Usual de8nition of grad G as the vector of partial derivatives corresponds to the standard scalar product (•; •).
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is regarded as a fast subsystem for the following members of the hierarchy. Instead of one system of Eqs. (67), a hierarchy of systems of lower-dimensional equations is considered, each of these subsystem is easier to study analytically. Theory of singularly perturbed systems of ordinary di6erential equations is used to provide a mathematical background and further development of the QSS approximation. In spite of a broad literature on this subject, it remains, in general, unclear, what is the smallness parameter that separates the intermediate (fast) species from the basic (slow). Reaction rate constants cannot be such a parameter (unlike in the case of the quasi-equilibrium). Indeed, intermediate species participate in the same reactions, as the basic species (for example, H2 2H, H + O2 OH + O). It is therefore incorrect to state that c f evolve faster than c s . In the sense of reaction rate constants, c f is not faster. For catalytic reactions, it is not diRcult to 8gure out what is the smallness parameter that separates the intermediate species from the basic, and which allows to upgrade the QSS assumption to a singular perturbation theory rigorously [59]. This smallness parameter is the ratio of balances: Intermediate species include the catalyst, and their total amount is simply signi8cantly less than the amount of all the ci ’s. After renormalizing to the variables of one order of magnitude, the small parameter appears explicitly. The simplest example gives the catalytic reaction A + Z AZ P + Z (here Z is a catalyst, A and P are an initial substrate and a product). The kinetic equations are (in obvious notations): c˙A = −k1+ cA cZ + k1− cAZ ; c˙Z = −k1+ cA cZ + k1− cAZ + k2+ cAZ − k2− cZ cP ; c˙AZ = k1+ cA cZ − k1− cAZ − k2+ cAZ + k2− cZ cP ; c˙P = k2+ cAZ − k2− cZ cP :
(69)
The constants and the reactions rates are the same for concentrations cA ; cP , and for cZ ; cAZ , and cannot give a reason for relative slowness of cA ; cP in comparison with cZ ; cAZ , but there may be another source of slowness. There are two balances for this kinetics: cA +cP +cAZ =BA ; cZ +cAZ =BZ . Let us go to dimensionless variables: &A = cA =BA ; &P = cP =BA ; &Z = cZ =BZ ; &AZ = cAZ =BZ ; k1− + &AZ ; &˙A = BZ −k1 &A &Z + BA k1− k2+ − + &AZ + &AZ − k2 &Z &P ; &˙Z = BA −k1 &A &Z + BA BA & A + &P +
BZ &AZ = 1; BA
&Z + &AZ = 1;
&• ¿ 0 :
(70)
For BZ BA the slowness of &A , &P is evident from these Eqs. (70). For usual radicals, the origin of the smallness parameter is quite similar. There are much less radicals than the basic species (otherwise, the QSS assumption is inapplicable). In the case of radicals, however, the smallness parameter cannot be extracted directly from balances Bi (53). Instead, one can come up with a thermodynamic estimate: Function G decreases in the course of reactions,
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whereupon we obtain the limiting estimate of concentrations of any species: ci 6
max
G(c)6G(c(0))
ci ;
(71)
where c(0) is the initial composition. If the concentration cR of the radical R is small both initially and in the equilibrium, then it should remain small also along the path to the equilibrium. For example, in the case of ideal G (57) under relevant conditions, for any t ¿ 0, the following inequality is valid: cR [ln(cR (t)=cReq ) − 1] 6 G(c(0)) :
(72)
Inequality (72) provides the simplest (but rather coarse) thermodynamic estimate of cR (t) in terms of G(c(0)) and cReq uniformly for t ¿ 0. Complete theory of thermodynamic estimates of dynamics has been developed in the book [125]. One can also do computations without a priori estimations, if one accepts the QSS assumption until the values c f stay suRciently small. It is the simplest way to operate with QSS: Just use it until c f are small. Let us assume that an a priori estimate has been found, ci (t) 6 ci max , for each ci . These estimate may depend on the initial conditions, thermodynamic data etc. With these estimates, we are able to renormalize the variables in the kinetic equations (37) in such a way that the renormalized variables take their values from the unit segment [0; 1]: c˜i = ci =ci max . Then system (37) can be written as follows: d c˜i 1 = Ji (c) : dt ci max
(73)
The system of dimensionless parameters, ji = ci max =maxi ci max de8nes a hierarchy of relaxation times, and with its help one can establish various realizations of the QSS approximation. The simplest version is the standard QSS assumption: Parameters ji are separated in two groups, the smaller ones, and of the order 1. Accordingly, the concentration vector is split into c s ⊕ c f . Various hierarchical QSS are possible, with this, the problem becomes more tractable analytically. There exist a variety of ways to introduce the smallness parameter into kinetic equations, and one can 8nd applications to each of the realizations. However, the two particular realizations remain basic for chemical kinetics: (i) fast reactions (under a given thermodynamic data); (ii) small concentrations. In the 8rst case, one is led to the quasi-equilibrium approximation, in the second case—to the classical QSS assumption. Both of these approximations allow for hierarchical realizations, those which include not just two but many relaxation time scales. Such a multi-scale approach essentially simpli8es analytical studies of the problem. 2.9.6. Thermodynamic criteria for selection of important reactions One of the problems addressed by the sensitivity analysis is the selection of the important and discarding the unimportant reactions. A simple principle was suggested in the paper [130] to compare importance of di6erent reactions according to their contribution to the entropy production (or, ˙ Based on this principle, Dimitrov [133] which is the same, according to their contribution to G). described domains of parameters in which the reaction of hydrogen oxidation, H2 + O2 + M, proceeds due to di6erent mechanisms. For each elementary reaction, he has derived the domain inside
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which the contribution of this reaction is essential (nonnegligible). Due to its simplicity, this entropy production principle is especially well suited for analysis of complex problems. In particular, recently, a version of the entropy production principle was used in the problem of selection of boundary conditions for Grad’s moment equations [131,132]. For ideal systems (57), as well, as for the Marcelin–De Donder kinetics (47) the contribution of the sth reaction to G˙ has a particularly simple form: + r Ws ˙= G˙ s = −Ws ln ; G (74) G˙ s : Ws− s=1 2.9.7. Opening One of the problems to be focused on when studying closed systems is to prepare extensions of the result for open or driven by =ows systems. External =ows are usually taken into account by additional terms in the kinetic equations (37): N˙ = V J (c) + (c; t) :
(75)
It is important to stress here that the vector 8eld J (c) in Eqs. (75) is the same, as for the closed system, with thermodynamic restrictions, Lyapunov functions, etc. The thermodynamic structures are important for analysis of open systems (75), if the external =ow is small in some sense, is linear function of c, has small time derivatives, etc. There are some general results for such “weakly open” systems, for example the Prigogine minimum entropy production theorem [134] and the estimations of possible of steady states and limit sets for open systems, based on thermodynamic functions and stoichiometric equations [125]. There are general results for another limit case: for very intensive =ow the dynamics is very simple again [59]. Let the =ow have a natural structure: (c; t) = vin (t)cin (t) − vout (t)c(t), where vin and vout are the rates of in=ow and out=ow, cin (t) is the concentration vector for in=ow. If vout is suRciently big, vout (t) ¿ v0 for some critical value v0 and all t ¿ 0, then for the open system (75) the Lyapunov norm exists: for any two solutions c 1 (t) and c 2 (t) the function c 1 (t) − c 2 (t) monotonically decreases in time. Such a critical value v0 exists for any norm, for example, for usual Euclidian norm • 2 = (•; •). For arbitrary form of , system (75) can lose all signs of being thermodynamic one. Nevertheless, thermodynamic structures often can help in the study of open systems. The seminal questions are: What happens with slow/fast motion separation after opening? Which slow invariant manifold for the closed system can be deformed to the slow invariant manifold for the open system? Which slow invariant manifold for the closed system can be used as approximate slow invariant manifold for the open system? There is more or less useful technique to seek the answers for speci8c systems under consideration. The way to study an open system as the result of opening a closed system may be fruitful. In any case, out of this way we have just a general dynamical system (75) and no hints what to do with. Basic introductory textbook on physical kinetics of the Landau L.D. and Lifshitz E.M. Course of Theoretical Physics [135] contains many further examples and their applications. Modern development of kinetics follows the route of speci8c numerical methods, such as direct simulations. An opposite tendency is also clearly observed, and the kinetic theory based schemes are
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increasingly used for the development of numerical methods and models in mechanics of continuous media.
3. Invariance equation in the di"erential form The notions and notations of this section will apply elsewhere below. De8nition of the invariance in terms of motions and trajectories assumes, at least, existence and uniqueness theorems for solutions of the original dynamic system. This prerequisite causes diRculties when one studies equations relevant to physical and chemical kinetics, such as, for example, equations of hydrodynamics. Nevertheless, there exists a necessary diFerential condition of invariance: The vector 8eld of the original dynamic system touches the manifold in every point. Let us write down this condition in order to set up notation. Let E be a linear space, let U (the phase space) be a domain in E, and let a vector 8eld J : U → E be de8ned in U . This vector 8eld de8nes the original dynamic system, dx = J (x); dt
x∈U :
(76)
In the sequel, we consider submanifolds in U which are parameterized with a given set of parameters. Let a linear space of parameters L be de8ned, and let W be a domain in L. We consider di6erentiable maps, F : W → U , such that, for every y ∈ W , the di6erential of F, Dy F : L → E, is an isomorphism of L on a subspace of E. That is, F are the manifolds, immersed in the phase space of the dynamic system (76), and parametrized by parameter set W . Remark. One never discusses the choice of norms and topologies are such a general setting. It is assumed that the corresponding choice is made appropriately in each speci8c case. We denote Ty the tangent space at the point y, Ty = (Dy F)(L). The diFerential condition of invariance has the following form: For every y ∈ W , J (F(y)) ∈ Ty :
(77)
Let us rewrite the di6erential condition of invariance (77) in a form of a di6erential equation. In order to achieve this, one needs to de8ne a projector Py : E → Ty for every y ∈ W . Once a projector Py is de8ned, then condition (77) takes the form: Gy = (1 − Py )J (F(y)) = 0 :
(78)
Obviously, by Py2 = Py we have, Py Gy = 0. We refer to the function Gy as the defect of invariance at the point y. The defect of invariance will be encountered oft in what will follow. Eq. (78) is the 8rst-order di6erential equation for the function F(y). Projectors Py should be tailored to the speci8c physical features of the problem at hand. A separate section below is devoted to the construction of projectors. There we shall demonstrate how to construct a projector,
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P(x; T ) : E → T , given a point x ∈ U and a speci8ed subspace T . We then set Py = P(F(y); Ty ) in Eq. (78). 4 There are two possible senses of the notion “approximate solution of invariance equations” (78): 1. The approximation of the solution; 2. The map F with small defect of invariance (the right hand side approximation). If someone is looking for the approximation of the 8rst kind, then he needs theorems about existence of solutions, he should 8nd the estimations of deviations from the exact solution, because the right hand side not always gives the good estimation, etc. The second kind of approximations does not require hypothesis of existence of solutions. Moreover, the manifold with suRciently small defect of invariance can serve as a slow manifold by itself. So, we shall accept the concept of approximate invariant manifold (the manifold with small defect of invariance) instead of the approximation of the invariant manifold (see also [13,245] and other works about approximate inertial manifolds). Sometimes these approximate invariant manifolds will give approximations of the invariant manifolds, sometimes not, but it is additional and often diRcult problem to make a distinction between these situations. In addition to defect of invariance, the key role in analysis of motion separation into the fast and the slow components play Jacobians, the di6erentials of J (x). Some estimations of errors of this separation will be presented below in the subsection devoted to post-processing. Our paper is focused on nonperturbative methods for computing invariant manifolds, but it should be mentioned that in the huge amount of applications the Taylor expansion is in use, and sometimes it works rather well. The main idea is the continuation of slow manifold with respect to a small parameter: Let our system depends on the parameter !, and let a manifold of steady states exist for ! = 0, as well, as 8bers of motions towards these steady states, for example x˙ = !f(x; y);
y˙ = g(x; y) :
(79)
For ! = 0 a value of (vector) variable x is a vector of conserved quantities. Let for every x the equation of fast motion, y˙ = g(x; y), be globally stable: Its solution y(t) tends to the unique (for given x) stable 8xed point yx . If the function g(x; y) meets the conditions of the implicit function theorem, then the graph of the map x → yx forms a manifold 80 = {(x; yx )} of steady states. For small ! ¿ 0 we can look for the slow manifold in a form of a series in powers of !: 8! = {x; y(x; !)}, y(x; !) = yx + !y1 (x) + !2 y2 (x) + · · · : The 8bers of fast motions can be constructed in a form of a power series too (the zero term is the fast motion y˙ = g(x; y) in the aRne planes x = const). This analytic continuation with respect to the parameter ! for small ! ¿ 0 is studied in the “Geometric singular perturbation theory” [247,248]. As it was mentioned above, the 8rst successful application 4
One of the main routes to de8ne the 8eld of projectors P(x; T ) would be to make use of a Riemannian structure. To this end, one de8nes a scalar product in E for every point x ∈ U , that is, a bilinear form p|qx with a positive de8nite quadratic form, p|px ¿ 0, if p = 0. A good candidate for such a scalar product is the bilinear form de8ned by the negative second di6erential of the entropy at the point x, −D2 S(x). As we demonstrate it later in this review, this choice is essentially the only correct one close to the equilibrium. However, far from the equilibrium, an improvement is required in order to guarantee the thermodymamicity condition, ker Py ⊂ ker(Dx S)x=F(y) , for the 8eld of projectors, P(x; T ), de8ned for any x and T , if T ⊂ ker Dx S: The thermodymamicity condition provides the preservation of the type of dynamics: if dS=dt ¿ 0 for initial vector 8eld (76) in point x = F(y), then dS=dt ¿ 0 in this point x for projected vector 8eld Py (J (F(y))) too.
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of such an approach to construction of a slow invariant manifold in a form of Taylor expansion in powers of small parameter of singular perturbation ! was the Chapman–Enskog expansion [51]. It is well-known in various applications that there are many di6erent ways to introduce a small parameter into a system, there are many ways to present a given system as a member of a one-parametric family with a manifold of 8xed points for zero value of a parameter. And di6erent ways of speci8cation of such a parameter result in di6erent de8nitions of slowness of positively invariant manifold. Therefore it is desirable to study the notion of separation of motions without such an arti8cial speci8cation. The notion of slow positively invariant manifold should be intrinsic. At least we should try to invent such a notion. 4. Film extension of the dynamics: slowness as stability 4.1. Equation for the 1lm motion One of the diRculties in the problem of reducing the description is caused by the fact that there exists no commonly accepted formal de8nition of slow (and stable) positively invariant manifolds. Classical de8nitions of stability and of the asymptotic stability of the invariant sets sound as follows: Let a dynamic system be de8ned in some metric space, (so that we can measure distances between points), and let x(t; x0 ) be a motion of this system at time t with the initial condition x(0) = x0 at time t = 0. The subset S of the phase space is called invariant if it is made of whole trajectories, that is, if x0 ∈ S then x(t; x0 ) ∈ S for all t ∈ (−∞; ∞). Let us denote as %(x; y) the distance between the points x and y. The distance from x to a closed set S is de8ned as usual: %(x; S) = inf {%(x; y) | y ∈ S}. The closed invariant subset S is called stable, if for every j ¿ 0 there exists ¿ 0 such that if %(x0 ; S) ¡ , then for every t ¿ 0 it holds %(x(t; x0 ); S) ¡ j. A closed invariant subset S is called asymptotically stable if it is stable and attractive, that is, there exists j ¿ 0 such that if %(x0 ; S) ¡ j, then %(x(t; x0 ); S) → 0 as t → ∞. Formally, one can reiterate the de8nitions of stability and of the asymptotic stability for positively invariant subsets. Moreover, since in the de8nitions mentioned above it goes only about t ¿ 0 or t → ∞, it might seem that positively invariant subsets can be a natural object of study for stability issues. Such conclusion is misleading, however. The study of the classical stability of the positively invariant subsets reduces essentially to the notion of stability of invariant sets—maximal attractors. Let Y be a closed positively invariant subset of the phase space. The maximal attractor for Y is the set MY , MY = Tt (Y ) ; (80) t ¿0
where Tt is the shift operator for the time t: Tt (x0 ) = x(t; x0 ) : The maximal attractor MY is invariant, and the stability of Y de8ned classically is equivalent to the stability of MY under any sensible assumption about homogeneous continuity (for example, it is so for a compact phase space). For systems which relax to a stable equilibrium, the maximal attractor is simply one and the same for any bounded positively invariant subset, and it consists of a single stable point.
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It is important to note that in de8nition (80) one considers motions of a positively invariant subset to equilibrium along itself: Tt Y ⊂ Y for t ¿ 0. It is precisely this motion which is uninteresting from the perspective of the comparison of stability of positively invariant subsets. If one subtracts this motion along itself out of the vector 8eld J (x) (76), one obtains a less trivial picture. We again assume submanifolds in U parameterized with a single parameter set F : W → U . Note that there exists a wide class of transformations which do not alter the geometric picture of motion: For a smooth di6eomorphism ’ : W → W (a smooth coordinate transform), maps F and F ◦ ’ de8ne the same geometric pattern in the phase space. Let us consider motions of the manifold F(W ) along solutions of Eq. (76). Denote as Ft the time-dependent map, and write equation of motion for this map: dFt (y) (81) = J (Ft (y)) : dt Let us now subtract the component of the vector 8eld responsible for the motion of the map Ft (y) along itself from the right hand side of Eq. (81). In order to do this, we decompose the vector 8eld J (x) in each point x = Ft (y) as J (x) = J (x) + J⊥ (x) ;
(82)
where J (x) ∈ Tt; y Tt; y = (Dy Ft (y)(L)). If projectors are well de8ned, Pt; y = P(Ft (y); Tt; y ), then decomposition (82) has the form J (x) = Pt; y J (x) + (1 − Pt; y )J (x) :
(83)
Subtracting the component J from the right hand side of Eq. (81), we obtain dFt (y) = (1 − Pt; y )J (Ft (y)) : (84) dt Note that the geometric pictures of motion corresponding to Eqs. (81) and (84) are identical locally in y and t. Indeed, the in8nitesimal shift of the manifold W along the vector 8eld is easily computed: (Dy Ft (y))−1 J (Ft (y)) = (Dy Ft (y))−1 (Pt; y J (Ft (y))) :
(85)
This de8nes a smooth change of the coordinate system (assuming all solutions exist). In other words, the component J⊥ de8nes the motion of the manifold in U , while we can consider (locally) the component J as a component which locally de8nes motions in W (a coordinate transform). The positive semi-trajectory of motion (for t ¿ 0) of any submanifold in the phase space along the solutions of initial di6erential equation (76) (without subtraction of J (x)) is the positively invariant manifold. The closure of such semi-trajectory is an invariant subset. The construction of the invariant manifold as a trajectory of an appropriate initial edge may be useful for producing invariant exponentially attracting set [136,137]. Very recently, the notion of exponential stability of invariants manifold for ODEs was revised by splitting motions into tangent and transversal (orthogonal) components in the work [138]. We further refer to Eq. (84) as the 1lm extension of the dynamic system (76). The phase space of the dynamic system (84) is the set of maps F (8lms). Fixed points of Eq. (84) are solutions to the invariance equation in the di6erential form (78). These include, in particular, all positively invariant manifolds. Stable or asymptotically stable 8xed points of Eq. (84) are slow manifolds we
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are interested in. It is the notion of stability associated with the 8lm extension of the dynamics which is relevant to our study. Below in Section 8, we consider relaxation methods for constructing slow positively invariant manifolds on the basis of the 8lm extension (84). 4.2. Stability of analytical solutions When studying the Cauchy problem for Eq. (84), one should ask a question of how to choose the boundary conditions: which conditions the function F must satisfy at the boundary of W ? Without 8xing the boundary conditions, the general solution of the Cauchy problem for the 8lm extension equations (84) in the class of smooth functions on W is essentially ambiguous. The boundary of W , 9W , splits in two pieces: 9W = 9W+ ∪ 9W− . For a smooth boundary these parts can be de8ned as 9W+ = {y ∈ 9W | (J(y); (DF(y))−1 (Py J (F(y)))) ¡ 0} ; 9W− = {y ∈ 9W | (J(y); (DF(y))−1 (Py J (F(y)))) ¿ 0} :
(86)
where J(y) denotes the unit outer normal vector in the boundary point y, (DF(y))−1 is the isomorphism of the tangent space Ty on the linear space of parameters L. One can understand the boundary splitting (86) in such a way: The projected vector 8eld Py J (F(y)) de8nes dynamics on the manifold F(W ), this dynamics is the image of some dynamics on W . The corresponding vector 8eld on W is v(y) = (DF(y))−1 (Py J (F(y))). The boundary part 9W+ consists of points y, where the velocity vector v(y) is pointed inside W , and for y ∈ 9W− this vector v(y) is directed outside of W (or is tangent to 9W ). The splitting 9W = 9W+ ∪ 9W− depends on t with the vector 8eld v(y): vt (y) = (DFt (y))−1 (Py J (Ft (y))) ; and dynamics of Ft (y) is determined by Eq. (84). If we would like to derive a solution of the 8lm extension, (84) F(y; t) for (y; t) ∈ W × [0; 4], for some time 4 ¿ 0, then it is necessary to 8x some boundary conditions on 9W+ (for the “incoming from abroad” part of the function F(y)). Nevertheless, there is a way to study Eq. (84) without introducing any boundary conditions. It is in the spirit of the classical Cauchy–Kovalevskaya theorem [139–141] about analytical Cauchy problem solutions with analytical data, as well as in the spirit of the classical Lyapunov auxiliary theorem about analytical invariant manifolds in the neighborhood of a 8xed point [144,33] and Poincar[e [145] theorem about analytical linearization of analytical nonresonant contractions (see [146]). We note in passing that recently, the interest to the classical analytical Cauchy problem revived in the mathematical physics literature [142,143]. In particular, analogs of the Cauchy–Kovalevskaya theorem were obtained for generalized Euler equations [142]. A technique to estimate the convergence radii of the series emerging therein was also developed. Analytical solutions to Eq. (84) do not require boundary conditions on the boundary of W . The analycity condition itself allows 8nding unique analytical solutions of the Eq. (84) with the analytical right hand side (1 − P)J for analytical initial conditions F0 in W (assuming that such solutions exist). Of course, the analytical continuation without additional regularity conditions is an
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ill-posed problem. However, it may be useful to go from functions to germs: 5 we can solve chains of ordinary di6erential equations for Taylor coeRcients instead of partial di6erential equations for functions (84), and after that it may be possible to prove the convergence of the Taylor series thus obtained. This is the way to prove the Lyapunov auxiliary theorem [144], and one of the known ways to prove the Cauchy–Kovalevskaya theorem. Let us consider system (1) with stable equilibrium point x∗ , real analytical right hand side J , and real analytical projector 8eld P(x; T ) : E → T . We shall study real analytical sub-manifolds, which include the equilibrium point x∗ (0 ∈ W; F(0) = x∗ ). Let us expand F in a Taylor series in the neighborhood of zero: F(y) = x∗ + A1 (y) + A2 (y; y) + · · · + Ak (y; y; : : : ; y) + · · · ;
(87)
where Ak (y; y; : : : ; y) is a symmetric k-linear operator (k = 1; 2; : : :). Let us expand also the right hand side of the 8lm equation (84). Matching operators of the same order, we obtain a chain of equations for A1 ; : : : ; Ak ; : : :: dAk = :k (A1 ; : : : ; Ak ) : (88) dt It is crucially important, that the dynamics of Ak does not depend on Ak+1 ; : : :, and Eqs. (88) can be studied in the following order: we 8rst study the dynamics of A1 , then the dynamics of A2 with the A1 motion already given, then A3 and so on. Let the projector Py in Eq. (84) be analytical function of the derivative Dy F(y) and of the deviation x − x∗ . Let the correspondent Taylor expansion at the point (A01 (•); x∗ ) have the form ∞ Dy F(y)(•) = A1 (•) + kAk (y; : : : ; •) ; k=2
Py =
∞ k; m=0
Pk; m (Dy F(y)(•) − A01 (•); : : : ; Dy F(y)(•) − A01 (•); F(y) − x∗ ; : : : ; F(y) − x∗ ) ; (89) k
m
A01 (•),
where A1 (•), Ak (y; : : : ; •) are linear operators. Pk; m is a k +m-linear operator (k; m=0; 1; 2; : : :) with values in the space of linear operators E → E. The operators Pk; m depend on the operator A01 (•) as on a parameter. Let the point of expansion A01 (•) be the linear part of F: A01 (•) = A1 (•). Let us represent the analytical vector 8eld J (x) as a power series: ∞ J (x) = Jk (x − x∗ ; : : : ; x − x∗ ) ; (90) k=1
where Jk is a symmetric k-linear operator (k = 1; 2; : : :). Let us write, for example, the 8rst two equations of the equation chain (88): dA1 (y) = (1 − P0; 0 )J1 (A1 (y)) ; dt dA2 (y; y) = (1 − P0; 0 )[J1 (A2 (y; y)) + J2 (A1 (y); A1 (y))] dt − [2P1; 0 (A2 (y; •)) + P0; 1 (A1 (y))]J1 (A1 (y)) : 5
The germ is the sequences of Taylor coeRcients that represent an analytical function near a given point.
(91)
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Here operators P0; 0 , P1; 0 (A2 (y; •)), P0; 1 (A1 (y)) parametrically depend on the operator A1 (•), hence, the 8rst equation is nonlinear, and the second is linear with respect to A2 (y; y). The leading term on the right hand side has the same form for all equations of the sequence (88): dAn (y; : : : ; y) = (1 − P0; 0 )J1 (An (y; : : : ; y)) − nP1; 0 (An (y; : : : ; y; •))J1 (A1 (y)) + · · · : (92) dt n− 1
Py2
= Py , because Py is a projector, and There are two important conditions on Py and Dy F(y): im Py = im Dy F(y); because Py projects on the image of Dy F(y). If we expand these conditions in the power series, then we get the conditions on the coeRcients. For example, from the 8rst condition we get P0;2 0 = P0; 0 ; P0; 0 [2P1; 0 (A2 (y; •)) + P0; 1 (A1 (y))] + [2P1; 0 (A2 (y; •)) + P0; 1 (A1 (y))]P0; 0 = 2P1; 0 (A2 (y; •)) + P0; 1 (A1 (y)); : : : :
(93)
After multiplication the second equation in (93) with P0; 0 we get P0; 0 [2P1; 0 (A2 (y; •)) + P0; 1 (A1 (y))]P0; 0 = 0 :
(94)
Similar identities can be obtained for any oder of the expansion. These equalities allow us to simplify the stationary equation for sequence (88). For example, for the 8rst two equations of this sequence (91) we obtain the following stationary equations: (1 − P0; 0 )J1 (A1 (y)) = 0; (1 − P0; 0 )[J1 (A2 (y; y)) + J2 (A1 (y); A1 (y))] − [2P1; 0 (A2 (y; •)) + P0; 1 (A1 (y))]J1 (A1 (y)) = 0 :
(95)
The operator P0; 0 is the projector on the space im A1 (the image of A1 ), hence, from the 8rst equation in (95) it follows: J1 (im A1 ) ⊆ im A1 . So, im A1 is a J1 -invariant subspace in E (J1 = Dx J (x)|x∗ ) and P0; 0 (J1 (A1 (y))) ≡ J1 (A1 (y)). It is equivalent to the 8rst equation of (95). Let us multiply the second equation of (95) with P0; 0 on the left. As a result we obtain the condition P0; 0 [2P1; 0 (A2 (y; •)) + P0; 1 (A1 (y))]J1 (A1 (y)) = 0 ; for solution of Eqs. (95), because P0; 0 (1 − P0; 0 ) ≡ 0. If A1 (y) is a solution of the 8rst equation of (95), then this condition becomes an identity, and we can write the second equation of (95) in the form (1 − P0; 0 ) ×[J1 (A2 (y; y)) + J2 (A1 (y); A1 (y)) − (2P1; 0 (A2 (y; •)) + P0; 1 (A1 (y)))J1 (A1 (y))] = 0 :
(96)
It should be stressed, that the choice of projector 8eld Py (89) has impact only on the F(y) parametrization, whereas the invariant geometrical properties of solutions of (84) do not depend on projector 8eld if some transversality and analycity conditions hold. The conditions of thermodynamic structures preservation signi8cantly reduce ambiguousness of the projector choice. One of the most important condition is ker Py ⊂ ker Dx S, where x = F(y) and S is the entropy (see the section
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about the entropy below). The thermodynamic projector is the unique operator which transforms the arbitrary vector 8eld equipped with the given Lyapunov function into a vector 8eld with the same Lyapunov function on the arbitrary submanifold which is not tangent to the level of the Lyapunov function. For the thermodynamic projectors Py the entropy S(F(y)) conserves on solutions F(y; t) of Eq. (84) for any y ∈ W . If projectors Py in Eqs. (89)–(96) are thermodynamic, then P0; 0 is the orthogonal projector with respect to the entropic scalar product. 6 For orthogonal projectors the operator P1; 0 has a simple explicit form. Let A : L → E be an isomorphic injection (an isomorphism on the image), and P : E → E be the orthogonal projector on the image of A. The orthogonal projector on the image of perturbed operator A + A is P + P, P = (1 − P)AA−1 P + (AA−1 P)+ (1 − P) + o(A) ; P1; 0 (A(•)) = (1 − P)A(•)A−1 P + (A(•)A−1 P)+ (1 − P) :
(97)
Here, in (97), the operator A−1 is de8ned on im A, im A = im P, the operator A−1 P acts on E. Formula for P (97) follows from the three conditions: (P + P)(A + A) = A + A;
(P + P)2 = P + P;
(P + P)+ = P + P :
(98)
Every Ak is driven by A1 ; : : : ; Ak −1 . Stability of the germ of the positively invariant analytical manifold F(W ) at the point 0 (F(0) = x∗ ) is de8ned as stability of the solution of the corresponding equations sequence (88). Moreover, the notion of the k-jet stability can be useful: let’s call k-jet stable such a germ of positively invariant manifold F(M ) at the point 0 (F(0) = x∗ ), if the corresponding solution of the equations sequence (88) is stable for k = 1; : : : ; n. The simple “triangle” structure of the equation sequence (88) with form (92) of principal linear part makes the problem of jets stability very similar for all orders n ¿ 1. Let us demonstrate the stability conditions for the 1-jets in a n-dimensional space E. Let the Jacobian matrix J1 = Dx J (x)|x∗ be self-adjoint with a simple spectrum 1 ; : : : ; n , and the projector P0; 0 be orthogonal (this is a typical “thermodynamic” situation). Eigenvectors of J1 form a basis in E: {ei }ni=1 . Let a linear space of parameters L be a k-dimensional real space, k ¡ n. We shall study stability of a operator A01 which is a 8xed point for the 8rst equation of sequence (88). The operator A01 is a 8xed point of this equation, if im A01 is a J1 -invariant subspace in E. We discuss full-rank operators, so, for some order of {ei }ni=1 numbering, the matrix of A01 should have a form: a01ij = 0, if i ¿ k. Let us choose the basis in L: lj = (A01 )−1 ej ; (j = 1; : : : ; k). For this basis a01ij = ij ; (i = 1; : : : ; n; j = 1; : : : ; k, ij is the Kronecker symbol). The corresponding projectors P and 1 − P have the matrices: P = diag(1; : : : ; 1; 0; : : : ; 0); k
n− k
1 − P = diag(0; : : : ; 0; 1; : : : ; 1) ; k
(99)
n− k
where diag(11 ; : : : ; 1n ) is the n × n diagonal matrix with numbers 11 ; : : : ; 1n on the diagonal. 6
This scalar product is the bilinear form de8ned by the negative second di6erential of the entropy at the point x∗ , −D S(x). 2
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Equations of the linear approximation for the dynamics of the deviations A read: dA = diag(0; : : : ; 0; 1; : : : ; 1)[diag( 1 ; : : : ; n )A − Adiag( 1 ; : : : ; k )] : dt n− k
k
(100)
k
˙ The time derivative of A is orthogonal to A: for any y; z ∈ L the equality (A(y); A(x)) = 0 holds, hence, for the stability analysis it is necessary and suRcient to study A with im A01 ⊥ im A. The matrix for such a A has a form aij = 0
if i 6 k :
For i = k + 1; : : : ; n, j = 1; : : : ; k Eq. (100) gives daij = ( i − j )aij : dt
(101)
From Eq. (101), the stability condition follows:
i − j ¡ 0
for all i ¿ k; j 6 k :
(102)
This means that the relaxation towards im A (with the spectrum of relaxation times | i |−1 (i = k + 1; : : : ; n)) is faster, then the relaxation along im A (with the spectrum of relaxation times | j |−1 (j = 1; : : : ; k)). Let condition (102) holds. The relaxation time for the 8lm (in the 8rst approximation) is 4 = 1= min | i | − max | j | ; i¿k
j 6k
thus it depends on the spectral gap in the spectrum of the operator J1 = Dx J (x)|x∗ . It is the gap between spectra of two restrictions of the operator J1 , J1 and J1⊥ , respectively. The operator J1 is the restriction of J1 on the J1 -invariant subspace im A01 (it is the tangent space to the slow invariant manifold at the point x∗ ). The operator J1⊥ is the restriction of J1 on the orthogonal complement to im A01 . This subspace is also J1 -invariant, because J1 is selfadjoint. The spectral gap between spectra of these two operators is the spectral gap between relaxation towards the slow manifold and relaxation along this manifold. The stability condition (102) demonstrates that our formalization of the slowness of manifolds as the stability of 8xed points for the 8lm extension (84) of initial dynamics met the intuitive expectations. For the analysis of system (88) in the neighborhood of some manifold F0 (F0 (0) = x∗ ), the following parametrization can be convenient. Let’s consider F0 (y) = A1 (y) + · · · ; T0 = A1 (L) is a tangent space to F0 (W ) at the point x∗ , E = T0 ⊕ H is the direct sum decomposition. We shall consider analytical sub-manifolds in the form x = x∗ + (y; ;(y)) ;
(103)
where y ∈ W0 ⊂ T0 , W0 is neighborhood of zero in T0 , ;(y) is an analytical map of W0 in H , ;(0) = 0. Any analytical manifold close to F0 can be represented in this form.
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Let us de8ne the projector Py that corresponds to decomposition (103), as the projector on Ty parallel to H . Furthermore, let us introduce the corresponding decomposition of the vector 8eld J = Jy ⊕ Jz , Jy ∈ T0 , Jz ∈ H . Then Py (J ) = (Jy ; (Dy ;(y))Jy ) :
(104)
The corresponding equation of motion of 8lm (84) has the following form: d;(y) = Jz (y; ;(y)) − (Dy ;(y))Jy (y; ;(y)) : dt
(105)
If Jy and Jz depend analytically on their arguments, then from (105) one can easily obtain a hierarchy of equations of form (88) (of course, Jy (x∗ ) = 0; Jz (x∗ ) = 0). Using these notions, it is convenient to formulate the Lyapunov Auxiliary Theorem [144]. Let T0 = Rm , H = Rp , and in U an analytical vector 8eld is de8ned J (y; z) = Jy (y; z) ⊕ Jz (y; z), (y ∈ T0 ; z ∈ H ), and the following conditions are satis8ed: (1) J (0; 0) = 0; (2) Dz Jy (y; z)|(0; 0) = 0; (3) 0 ∈ conv{k1 ; : : : ; km }, where k1 ; : : : ; km are the eigenvalues of Dy Jy (y; z)|(0:0) , and conv{k1 ; : : : ; km } is the convex envelope of {k1 ; : : : ; km }; (4) the numbers ki and j are not related by any equation of the form m
m i ki = j ;
(106)
i=1
where j (j = 1; : : : ; p) are eigenvalues of Dz Jz (y; z)|(0; 0) , and mi ¿ 0 are integers, mi=1 mi ¿ 0. Let us consider analytical manifold (y; ;(y)) in U in the neighborhood of zero (;(0) = 0) and write for it the di6erential invariance equation with projector (104): (Dy ;(y))Jy (y; ;(y)) = Jz (y; ;(y)) :
(107)
Lyapunov auxiliary theorem. Given conditions 1–4, Eq. (103) has the unique analytical in the neighborhood of zero solution, satisfying condition ;(0) = 0. Recently various new applications of this theorem were developed [33,147–149]. Studying germs of invariant manifolds using Taylor expansion in a neighborhood of a 8xed point is de8nitely useful from the theoretical as well as from the practical perspective. But the well known diRculties pertinent to this approach, of convergence, of small denominators (connected with proximity to resonances (106)) and others call for development of di6erent methods. A hint can be found in the famous KAM theory: one should use iterative methods instead of the Taylor expansion [28–30]. Below we present two such methods: • The Newton method subject to incomplete linearization; • The relaxation method which is the Galerkin-type approximation to Newton’s method with projection on defect of invariance (78), i.e. on the right hand side of Eq. (84).
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5. Entropy, quasi-equilibrium and projectors -eld Projector operators Py contribute both to the invariance equation (77), and to the 8lm extension of dynamics (84). Limiting results, exact solutions, etc. only weakly depend on the particular choice of projectors, or do not depend at all on it. However, the validity of approximations obtained in each iteration step towards the limit does strongly depend on the choice of the projector. Moreover, if we want each approximate solution to be consistent with such physically crucial conditions as the second law of thermodynamics (the entropy of the isolated systems increases), then the choice of the projector becomes practically unique. In this section we consider the main ingredients for constructing the projector, based on the two additional structures: (a) the moment parameterization, and (b) the entropy and the entropic scalar product. 5.1. Moment parameterization Same as in the previous section, let a regular map (projection) is de8ned, L : U → W . We consider only maps F : W → U which satisfy L ◦ F = 1. We seek slow invariant manifolds among such maps. (A natural remark is in order here: sometimes one has to consider F which are de8ned not on the whole W but only on some subset of it.) In this case, the unique projector consistent with the given structure is the superposition of the di6erentials: Py J = (Dy F)y ◦ (Dx L)F(y) :
(108)
In the language of di6erential equations, formula (108) has the following signi8cance: First, Eq. (76) is projected, dy = (Dx L)F(y) J (F(y)) : (109) dt Second, the latter equation is lifted back to U with the help of F and its di6erential, dy d x = (Dy F)y ((Dx L)F(y) J (F(y))) = Py J : = (Dy F)y (110) x(t) = F(y(t)); dt dt x=F(y)
The most standard example of the construction just described is as follows: x is the distribution density, y = L(x) is the set of selected moments of this density, F : y → x is a “closure assumption”, which constructs a distribution density parameterized by the values of the moments y. Another standard example is relevant to problems of chemical kinetics: x is a detailed description of the reacting species (including all the intermediates and radicals), y are concentrations of stable reactants and products of the reaction. The moment parameterization and moment projectors (108) are often encountered in the applications. However, they have some shortcomings. In particular, it is by far not always happens that the moment projection transforms a dissipative system into another dissipative system. Of course, for invariant F(y) any projector transforms the dissipative system into a dissipative system. However, for various approximations to invariant manifolds (closure assumptions) this is not readily the case. 7 The property of projectors to preserve the type of the dynamics will be imposed below as one of the requirements. 7
See, e.g. a discussion of this problem for the Tamm–Mott–Smith approximation for the strong shock wave in [5].
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5.2. Entropy and quasi-equilibrium The dissipation properties of system (76) are described by specifying the entropy S, the distinguished Lyapunov function which monotonically increases along solutions of Eq. (76). In a certain sense, this Lyapunov function is more fundamental than system (76) itself. That is, usually, the entropy is known much better than the right hand side of Eq. (76). For example, in chemical kinetics, the entropy is obtained from the equilibrium data. The same holds for other Lyapunov functions, which are de8ned by the entropy and by speci8cation of the reaction conditions (the free energy, U − TS, for the isothermal isochoric processes, the free enthalpy, U − TH , for the isothermal isobaric processes, etc.). On physical grounds, all these entropic Lyapunov functions are proportional (up to additive constants) to the entropy of the minimal isolated system which includes the system under study [125]. In general, with some abuse of language, we term the Lyapunov functional S the entropy elsewhere below, although it is a di6erent functional for nonisolated systems. Thus, we assume that a concave functional S is de8ned in U , such that it takes maximum in an inner point x∗ ∈ U . This point is termed the equilibrium. For any dissipative system (76) under consideration in U , the derivative of S due to Eq. (76) must be nonnegative, dS = (Dx S)(J (x)) ¿ 0 ; (111) dt x
where Dx S is the linear functional, the di6erential of the entropy, while the equality in (111) is achieved only in the equilibrium x = x∗ . Most of the works on nonequilibrium thermodynamics deal with corrections to quasi-equilibrium approximations, or with applications of these approximations (with or without corrections). This viewpoint is not the only possible but it proves very eRcient for the construction of a variety of useful models, approximations and equations, as well as methods to solve them. 8 We shall now introduce the quasi-equilibrium approximation in the most general setting. 8 From time to time it is discussed in the literature, who was the 8rst to introduce the quasi-equilibrium approximations, and how to interpret them. At least a part of the discussion is due to a di6erent rˆole the quasi-equilibrium plays in the entropy-conserving and the dissipative dynamics. The very 8rst use of the entropy maximization dates back to the classical work of Gibbs [166], but it was 8rst claimed for a principle of informational statistical thermodynamics by Jaynes [153]. Probably the 8rst explicit and systematic use of quasi-equilibria to derive dissipation from entropy-conserving systems is due to the works of Zubarev. Recent detailed exposition is given in [155]. The method of nonequilibrium ensemble was developed also by Eu [173]. For dissipative systems, the use of the quasi-equilibrium to reduce description can be traced to the works of Grad on the Boltzmann equation [158]. A review of the ideas of the underlying method behind informational statistical thermodynamics was presented in Ref. [172]. The connection between entropy maximization and (nonlinear) Onsager formalism was also studied [171,127]. The viewpoint of two of the present authors (ANG and IVK) was in=uenced by the papers by L.I. Rozonoer and co-workers, in particular, [167–169]. A detailed exposition of the quasi-equilibrium approximation for Markov chains is given in the book [125] (Chapter 3, Quasi-equilibrium and entropy maximum, pp. 92–122), and for the BBGKY hierarchy in the paper Ref. [170]. We have applied maximum entropy principle to the description the universal dependence the 3-particle distribution function F3 on the 2-particle distribution function F2 in classical systems with binary interactions [174]. For a discussion the quasi-equilibrium moment closure hierarchies for the Boltzmann equation [168] see the papers [178,179,175]. A very general discussion of the maximum entropy principle with applications to dissipative kinetics is given in the review [176]. Recently the quasi-equilibrium approximation with some further correction was applied to description of rheology of polymer solutions [198,199] and of ferro=uids [200,201].
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A linear moment parameterization is a linear operator, L : E → L, where L = im L = E=ker L, ker L is a closed linear subspace of space E, and L is the projection of E onto factor-space L. Let us denote W = L(U ). quasi-equilibrium (or restricted equilibrium, or conditional equilibrium) is the embedding, F ∗ : W → U , which puts into correspondence to each y ∈ W the solution to the entropy maximization problem: S(x) → max;
L(x) = y :
(112) F ∗ (y) ∈ int U
We assume that, for each y ∈ int W , there exists the unique solution to problem ∗ (112). This solution, F (y), is called the quasi-equilibrium, corresponding to the value y of the macroscopic variables. The set of quasi-equilibria F ∗ (y), y ∈ W , forms a manifold in int U , parameterized by the values of the macroscopic variables y ∈ W . Let us specify some notations: E T is the adjoint to the E space. Adjoint spaces and operators will be indicated by T , whereas notation ∗ is earmarked for equilibria and quasi-equilibria. Furthermore, [l; x] is the result of application of the functional l ∈ E T to the vector x ∈ E. We recall that, for an operator A : E1 → E2 , the adjoint operator, AT : E1T → E2T is de8ned by the following relation: For any l ∈ E2T and x ∈ E1 , [l; Ax] = [AT l; x] : Next, Dx S(x) ∈ E T is the di6erential of the entropy functional S(x), Dx2 S(x) is the second di6erential of the entropy functional S(x). The corresponding quadratic functional Dx2 S(x)(z; z) on E is de8ned by the Taylor formula, S(x + z) = S(x) + [Dx S(x); z] + 12 Dx2 S(x)(z; z) + o( z 2 ) :
(113)
We keep the same notation for the corresponding symmetric bilinear form, Dx2 S(x)(z; p), and also for the linear operator, Dx2 S(x) : E → E T , de8ned by the formula [Dx2 S(x)z; p] = Dx2 S(x)(z; p) : In the latter formula, on the left hand side, there is an operator, on the right hand side there is a bilinear form. Operator Dx2 S(x) is symmetric on E, Dx2 S(x)T = Dx2 S(x). Concavity of the entropy S means that for any z ∈ E, the inequality holds, Dx2 S(x)(z; z) 6 0 ; in the restriction onto the aRne subspace parallel to ker L we assume the strict concavity, Dx2 S(x)(z; z) ¡ 0
if z ∈ ker L and if z = 0 :
In the remainder of this subsection we are going to construct the important object, the projector onto the tangent space of the quasi-equilibrium manifold. Let us compute the derivative Dy F ∗ (y). For this purpose, let us apply the method of Lagrange multipliers: There exists such a linear functional M(y) ∈ (L)T , that Dx S(x)|F ∗ (y) = M(y) · L;
L(F ∗ (y)) = y
(114)
or Dx S(x)|F ∗ (y) = LT · M(y);
L(F ∗ (y)) = y :
(115)
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From Eq. (115) we get L(Dy F ∗ (y)) = 1L ;
(116)
where we have indicated the space in which the unit operator acts. Next, using the latter expression, we transform the di6erential of Eq. (114), 1 T −1 Dy M = (L(Dx2 S)− ; F ∗ (y) L )
(117)
and, consequently, 1 T 2 −1 T −1 : Dy F ∗ (y) = (Dx2 S)− F ∗ (y) L (L(Dx S)F ∗ (y) L )
(118)
Notice that, elsewhere in Eq. (118), operator (Dx2 S)−1 acts on the linear functionals from LT . These functionals are precisely those which become zero on ker L or, that is the same, those which can be represented as linear functionals of macroscopic variables. The tangent space to the quasi-equilibrium manifold at the point F ∗ (y) is the image of the operator Dy F ∗ (y): 1 T 2 −1 im(Dy F ∗ (y)) = (Dx2 S)− F ∗ (y) L = (Dx S)F ∗ (y) Ann(ker L)
(119)
where Ann(ker L) is the set of linear functionals which become zero on ker L. Another way to write Eq. (119) is the following: x ∈ im(Dy F ∗ (y)) ⇔ (Dx2 S)F ∗ (y) (z; p) = 0;
p ∈ ker L :
(120)
This means that im(Dy F ∗ (y)) is the orthogonal completement of ker L in E with respect to the scalar product, z|pF ∗ (y) = −(Dx2 S)F ∗ (y) (z; p) :
(121)
The entropic scalar product (121) appears often in the constructions below. (Usually, it becomes the scalar product indeed after the conservation laws are excluded). Let us denote as Ty = im(Dy F ∗ (y)) the tangent space to the quasi-equilibrium manifold at the point F ∗ (y). Important role in the construction of quasi-equilibrium dynamics and its generalizations is played by the quasi-equilibrium projector, an operator which projects E on Ty parallel to ker L. This is the orthogonal projector with respect to the entropic scalar product, Py∗ : E → Ty : Py∗ = Dy F ∗ (y) · L = (Dx2 S|F ∗ (y) )−1 LT (L(Dx2 S|F ∗ (y) )−1 LT )−1 L :
(122)
It is straightforward to check the equality Py∗2 = Py∗ , and the self-adjointness of Py∗ with respect to the entropic scalar product (121). Thus, we have introduced the basic constructions: the quasi-equilibrium manifold, the entropic scalar product, and the quasi-equilibrium projector. The construction of the quasi-equilibrium allows for the following generalization: Almost every manifold can be represented as a set of minimizers of the entropy under linear constrains. However, in contrast to the standard quasi-equilibrium, these linear constrains will depend, generally speaking, on the point on the manifold. So, let the manifold 8 = F(W ) ⊂ U be given. This is a parametric set of distribution functions. However, now macroscopic variables y are not functionals on R or U but just parameters de8ning points on the manifold. The problem is how to extend the de8nitions of y onto a neighborhood of
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F(W ) in such a way that F(W ) will appear as the solution to the variational problem: S(x) → max;
L(x) = y :
(123)
For each point F(y), we identify Ty ∈ E, the tangent space to the manifold 8 in Fy , and the subspace Yy ⊂ E, which depends smoothly on y, and which has the property, Yy ⊕ Ty = E. Let us de8ne L(x) in the neighborhood of F(W ) in such a way, that L(x) = y
if x − F(y) ∈ Yy :
(124)
The point F(y) is the solution of the quasi-equilibrium problem (123) if and only if Dx S(x)|F(y) ∈ Ann Yy :
(125)
That is, if and only if Yy ⊂ ker Dx S(x)|F(y) . It is always possible to construct subspaces Yy with the properties just speci8ed, at least locally, if the functional Dx S|F(y) is not identically equal to zero on Ty . The construction just described allows to consider practically any manifold as a quasi-equilibrium. This construction is required when one seeks the induced dynamics on a given manifold. Then the vector 8elds are projected on Ty parallel to Yy , and this preserves intact the basic properties of the quasi-equilibrium approximations. Let us return to the usual linear moment parametrization. quasi-equilibrium entropy S(y) is a functional on W . It is de8ned as the value of the entropy on the corresponding quasi-equilibrium x = F ∗ (y): S(y) = S(F ∗ (y))
(126)
Quasi-equilibrium dynamics is a dynamics on W , de8ned by Eq. (109) for the quasi-equilibrium F ∗ (y): dy (127) = LJ (F ∗ (y)) : dt Here L is constant linear operator (in the general case (109), it may become nonlinear). The corresponding quasi-equilibrium dynamics on the quasi-equilibrium manifold F ∗ (W ) is de8ned using the projector (108): dx (128) = Py∗ |x=F ∗ (y) J (x) = (Dy F ∗ )x=F ∗ (y) LJ (x); x ∈ F ∗ (W ) : dt The orthogonal projector Py∗ on the right hand side of Eq. (128) can be explicitly written using the second derivative of S and the operator L (122). Let us remind that the only distinguished scalar product in E is the entropic scalar product (121): z; px = −(Dx2 S)x (z; p)
(129)
It depends on the point x ∈ U . This dependence |x endows U with the structure of a Riemann space. The most important property of the quasi-equilibrium system (127), (128) is highlighted by the conservation of the dynamics type theorem: if for the original dynamic system (76) dS=dt ¿ 0, then for the quasi-equilibrium dynamics dS=dt ¿ 0. If for the original dynamic system (76) dS=dt = 0 (conservative system), then for the quasi-equilibrium dynamics dS=dt = 0 as well.
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5.3. Thermodynamic projector without a priori parameterization Quasi-equilibrium manifolds is a place where the entropy and the moment parameterization meet each other. The projectors Py for a quasi-equilibrium manifold is nothing but the orthogonal with respect to the entropic scalar product |x projector (122). The quasi-equilibrium projector preserves the type of dynamics. Note that in order to preserve the type of dynamics we needed only one condition to be satis8ed, ker Py ⊂ ker(Dx S)x=F(y) :
(130)
Let us require that the 8eld of projectors, P(x; T ), is de8ned for any x and T , if T ⊂ ker Dx S :
(131)
It follows immediately from these conditions that in the equilibrium, P(x∗ ; T ) is the orthogonal projector onto T (ortogonality is with respect to the entropic scalar product |x∗ ). The 8eld of projectors is constructed in the neighborhood of the equilibrium based on the requirement of the maximal smoothness of P as a function of gx = Dx S and x. It turns out that to the 8rst order in the deviations x − x∗ and gx − gx∗ , the projector is de8ned uniquely. Let us 8rst describe the construction of the projector, and next discuss its uniqueness. Let the subspace T ⊂ E, the point x, and the di6erential of the entropy at this point, g = Dx S, be de8ned in such a way that the transversality condition (131) is satis8ed. Let us de8ne T0 =T ∩ker gx . By condition (131), T0 = T . Let us denote, eg = eg (T ) ∈ T the vector in T , such that eg is orthogonal to T0 , and is normalized by the condition g(eg ) = 1. The vector eg is de8ned unambiguously. The projector PS; x = P(x; T ) is de8ned as follows: For any z ∈ E, PS; x (z) = P0 (z) + eg gx (z) ;
(132)
where P0 is the orthogonal projector on T0 (orthogonality is with respect to the entropic scalar product |x ). The entropic projector (132) depends on the point x through the x-dependence of the scalar product |x , and also through the di6erential of S in x, the functional gx . Obviously, P(z) = 0 implies g(z) = 0, that is, the thermodynamicity requirement (130) is satis8ed. Uniqueness of the thermodynamic projector (132) is supported by the requirement of the maximal smoothness (analyticity) [10] of the projector as a function of gx and |x , and is done in two steps which we sketch here (detailed proof is given in Ref. [177]): 1. Considering the expansion of the entropy in the equilibrium up to the quadratic terms, one demonstrates that in the equilibrium the thermodynamic projector is the orthogonal projector with respect to the scalar product |x∗ . 2. For a given g, one considers auxiliary dissipative dynamic systems (76), which satisfy the condition: For every x ∈ U , it holds, gx (J (x )) = 0, that is, gx de8nes an additional linear conservation law for the auxiliary systems. For the auxiliary systems, the point x is the equilibrium. Eliminating the linear conservation law gx , and using the result of the previous point, we end up with formula (132).
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Thus, the entropic structure de8nes unambiguously the 8eld of projectors (132), for which the dynamics of any dissipative system (76) projected on any closure assumption remains dissipative. Example 1: Quasi-equilibrium projector and defect of invariance for the local Maxwellians manifold of the Boltzmann equation The Boltzmann equation is one of the everlasting equations. It remains the most inspiring source for the model reduction problems. With this subsection we start a series of examples for the Boltzmann equation. DiJculties of classical methods of the Boltzmann equation theory. As was mentioned above, the 8rst systematic and (at least partially) successful method of constructing invariant manifolds for dissipative systems was the celebrated Chapman–Enskog method [51] for the Boltzmann kinetic equation. The main diRculties of the Chapman–Enskog method [51] are “nonphysical” properties of high-order approximations. This was stated by a number of authors and was discussed in detail in [87]. In particular, as it was noted in [53], the Burnett approximation results in a short-wave instability of the acoustic spectra. This fact contradicts the H -theorem (cf. in [53]). The Hilbert expansion contains secular terms [87]. The latter contradicts the H -theorem. The other diRculties of both of these methods are: the restriction upon the choice of initial approximation (the local equilibrium approximation), the demand for a small parameter, and the usage of slowly converging Taylor expansion. These diRculties never allow a direct transfer of these methods on essentially nonequilibrium situations. The main diRculty of the Grad method [158] is the uncontrollability of the chosen approximation. An extension of the list of moments can result in a certain success, but it cannot also give anything. DiRculties of moment expansion in the problems of shock waves and sound propagation can be seen in [87]. Many attempts were made to re8ne these methods. For the Chapman–Enskog and Hilbert methods these attempts are based in general on some “good” rearrangement of expansions (e.g. neglecting high-order derivatives [87], reexpanding [87], Pade approximations and partial summing [25,178,165], etc.). This type of work with formal series is wide spread in physics. Sometimes the results are surprisingly good—from the renormalization theory in quantum 8elds to the Percus–Yevick equation and the ring-operator in statistical mechanics. However, one should realize that success could not be guaranteed. Moreover, rearrangements never remove the restriction upon the choice of the initial local equilibrium approximation. Attempts to improve the Grad method are based on quasi-equilibrium approximations [167,168]. It was found in [168] that the Grad distributions are linearized versions of appropriate quasi-equilibrium approximations (see also the late papers [178,179,175]). A method which treats =uxes (e.g. moments with respect to collision integrals) as independent variables in a quasi-equilibrium description was introduced in [190,178,192,179]. An important feature of quasi-equilibrium approximations is that they are always thermodynamic, i.e. they are concordant with the H -theorem due to their construction. However, quasi-equilibrium approximations do not remove the uncontrollability of the Grad method. Boltzmann Equation (BE). The phase space E consists of distribution functions f(C; x) which depend on the spatial variable x and on velocity variable C. The variable x spans an open domain 8x3 ⊆ Rx , and the variable C spans the space RC3 . We require that f(C; x) ∈ F are nonnegative
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functions, and also that the following integrals are 8nite for every x ∈ 8x (the existence of moments and of the entropy): (i1 i2 i3 ) (f) = v1i1 v2i2 v3i3 f(C; x) d 3 C; i1 ¿ 0; i2 ¿ 0; i3 ¿ 0 ; (133) Ix Hx (f) = f(C; x)(ln f(C; x) − 1) d 3 C; H (f) = Hx (f) d 3 x : (134) Here and below integration in C is made over RC3 , and it is made over 8x in x. For every 8xed x ∈ 8x , Ix(···) and Hx might be treated as functionals de8ned in F. We write BE in the form of (76) using standard notations [87]: 9f = J (f); 9t
J (f) = −vs
9f + Q(f; f) : 9xs
(135)
Here and further a summation in two repeated indices is assumed, and Q(f; f) stands for the Boltzmann collision integral [1]. The latter represents the dissipative part of the vector 8eld J (f) (135). In this paper we consider the case when boundary conditions for Eq. (135) are relevant to the local with respect to x form of the H -theorem. 3 For every 8xed x, we denote as Hx0 (f) the space of linear functionals 4i=0 ai (x) i (C)f(C; x) d C, 2 where i (C) represent summational invariants of a collision [1,2] ( 0 = 1; i = vi ; i = 1; 2; 3; 4 = v ). We write (mod Hx0 (f)) if an expression is valid within the accuracy of adding a functional from Hx0 (f). The local H -theorem states: for any functional (136) Hx (f) = f(C; x)(ln f(C; x) − 1) d 3 C (mod Hx0 (f)) the following inequality is valid: dHx (f)=dt ≡ Q(f; f)|f=f(C; x) ln f(C; x) d 3 C 6 0 :
(137)
Expression (137) is equal to zero if and only if ln f = 4i=0 ai (x) i (C). Although all functionals (136) are equivalent in the sense of the H -theorem, it is convenient to deal with the functional Hx (f) = f(C; x)(ln f(C; x) − 1) d 3 C : All what was said in the previous sections can be applied to BE (135). Now we will discuss some speci8c points. Local manifolds. Although the general description of manifolds 8 ⊂ F (Section 2.1) holds as well for BE, a speci8c class of manifolds might be de8ned due to the di6erent character of spatial and of velocity dependencies in BE vector 8eld (135). These manifolds will be called local manifolds, and they are constructed as follows. Denote as Floc the set of functions f(C) with 8nite integrals (a) (i1 i2 i3 ) (f) = v1i1 v2i2 v3i3 f(C) d 3 C; i1 ¿ 0; i2 ¿ 0; i3 ¿ 0 ; I
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(b)
f(C)ln f(C) d 3 C :
H (f) =
251
(138)
In order to construct a local manifold in F, we, 8rstly, consider a manifold in Floc . Namely, we de8ne a domain A ⊂ B, where B is a linear space, and consider a smooth immersion A → Floc : a → f(a; C). The set of functions f(a; C) ∈ Floc , where a spans the domain A, is a manifold in Floc . Secondly, we consider all bounded and suRciently smooth functions a(x): 8x → A, and we de8ne the local manifold in F as the set of functions f(a(x); C). Roughly speaking, the local manifold is a set of functions which are parameterized with x-dependent functions a(x). A local manifold will be called a locally 1nite-dimensional manifold if B is a 8nite-dimensional linear space. Locally 8nite-dimensional manifolds are a natural source of initial approximations for constructing dynamic invariant manifolds in BE theory. For example, the Tamm–Mott–Smith (TMS) approximation gives us locally two-dimensional manifold {f(a− ; a+ )} which consists of distributions f(a− ; a+ ) = a− f− + a+ f+ :
(139)
Here a− and a+ (the coordinates on the manifold 8TMS ={f(a− ; a+ )}) are nonnegative real functions of the position vector x, and f− and f+ are 8xed Maxwellians. Next example is locally 8ve-dimensional manifold {f(n; u; T )} which consists of local Maxwellians (LM). The LM manifold consists of distributions f0 which are labeled with parameters n; u, and T : 2,kB T −3=2 m(C − u)2 : (140) exp − f0 (n; u; T ) = n m 2kB T Parameters n; u, and T in (140) are functions depending on x. In this section we will not indicate this dependency explicitly. Distribution f0 (n; u; T ) is the unique solution of the variational problem: H (f) = f ln f d 3 C → min for
M0 (f) = Mi (f) =
1 · f d3 C ; vi f d 3 C = nui ;
i = 1; 2; 3 ;
3nkB T + nu2 : (141) m Hence, the LM manifold is a quasi-equilibrium manifold. Considering n; u, and T as 8ve scalar parameters (see the remark on locality in Section 3), we see that LM manifold is parameterized with the values of Ms (f); s = 0; : : : ; 4; which are de8ned in the neighborhood of LM manifold. It is sometimes convenient to consider the variables Ms (f0 ), s = 0; : : : ; 4; as new coordinates on LM manifold. The relationship between the sets {Ms (f0 )} and {n; u; T } is m n = M0 ; ui = M0−1 Mi ; i = 1; 2; 3; T = M −1 (M4 − M0−1 Mi Mi ) : (142) 3kB 0 This is the standard moment parametrization of the quasi-equilibrium manifold. M4 (f) =
v2 f d 3 C =
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Thermodynamic quasi-equilibrium projector. Thermodynamic quasi-equilibrium projector Pf0 (n;u;T ) (J ) onto the tangent space Tf0 (n;u;T ) is de8ned as 4 9f0 (n; u; T ) 3 Pf0 (n;u;T ) (J ) = (143) sJ d C : 9M s s=0 Here we have assumed that n; u, and T are functions of M0 ; : : : ; M4 (see relationship (142)), and 0
= 1;
i
= vi ;
i = 1; 2; 3;
4
= v2 :
(144)
Calculating derivatives in (143), and next returning to variables n; u, and T , we obtain m(C − u)2 1 mu2 T 3 mui Pf0 (n;u;T ) (J ) = f0 (n; u; T ) − − − (vi − ui ) + n nkB T 3nkB n 2kB T 2 2T m 2mui m(C − u)2 3 × 1 · J d3 C + (vi − ui ) − − 2 nkB T 3nkB 2kB T 2T m(C − u)2 m 3 × vi J d 3 C + C2 J d 3 C : − 2 3nkB 2kB T 2T It is sometimes convenient to rewrite (145) as 4 (s) (s) 3 Pf0 (n;u;T ) (J ) = f0 (n; u; T ) f0 (n;u;T ) f0 (n;u;T ) J d C
(145)
(146)
s=0
Here (0) f0 (n;u;T )
= n−1=2 ;
(4) f0 (n;u;T )
(i) f0 (n;u;T )
= (2=n)1=2 ci ;
= (2=3n)1=2 (c2 − (3=2));
It is easy to check that f0 (n; u; T ) f(k) 0 (n;u;T )
(l) f0 (n;u;T )
i = 1; 2; 3 ;
ci = (m=2kB T )1=2 (vi − ui ) :
d 3 C = kl :
(147)
(148)
Here kl is the Kronecker delta. Defect of invariance for the LM manifold. The defect of invariance for the LM manifold at the point f0 (n; u; T ) for the BE is 9f0 (n; u; T ) G(f0 (n; u; T )) = Pf0 (n;u;T ) −(vs − us ) + Q(f0 (n; u; T )) 9xs 9f0 (n; u; T ) + Q(f0 (n; u; T )) − −(vs − us ) 9xs 9f0 (n; u; T ) 9f0 (n; u; T ) + (vs − us ) (149) = Pf0 (n;u;T ) −(vs − us ) 9xs 9xs
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Substituting (145) into (149), we obtain m(C − u)2 5 9 ln T − (vi − ui ) G(f0 (n; u; T )) = f0 (n; u; T ) 2kB T 2 9xi 9us m 1 2 (vi − ui )(vs − us ) − is (C − u) : + kB T 3 9xi
253
(150)
The LM manifold is not a dynamic invariant manifold of the Boltzmann equation and defect (150) is not identical to zero. Example 2: Scattering rates versus moments: alternative Grad equations In this subsection scattering rates (moments of collision integral) are treated as new independent variables, and as an alternative to moments of the distribution function, to describe the rare8ed gas near local equilibrium. A version of entropy maximum principle is used to derive the Grad-like description in terms of a 8nite number of scattering rates. New equations are compared to the Grad moment system in the heat nonconductive case. Estimations for hard spheres demonstrate, in particular, some 10% excess of the viscosity coeRcient resulting from the scattering rate description, as compared to the Grad moment estimation. In 1949, Harold Grad [158] has extended the basic assumption behind the Hilbert and Chapman– Enskog methods (the space and time dependence of the normal solutions is mediated by the 8ve hydrodynamic moments [51]). A physical rationale behind the Grad moment method is an assumption of the decomposition of motion (i). During the time of order 4, a set of distinguished moments M (which include the hydrodynamic moments and a subset of higher-order moment) does not change signi8cantly as compared to the rest of the moments M (the fast evolution) (ii). Towards the end of the fast evolution, the values of the moments M become unambiguously determined by the values of the distinguished moments M , and (iii). On the time of order 4, dynamics of the distribution function is determined by the dynamics of the distinguished moments while the rest of the moments remains to be determined by the distinguished moments (the slow evolution period). Implementation of this picture requires an ansatz for the distribution function in order to represent the set of states visited in the course of the slow evolution. In Grad’s method, these representative sets are 8nite-order truncations of an expansion of the distribution functions in terms of Hermit velocity tensors: N fC (M ; C) = fLM (%; u; E; C) 1 + a1 (M )H(1) (C − u) ; (151) (1)
where H(1) (C − u) are various Hermit tensor polynomials, orthogonal with the weight fLM , while coeRcient a(1) (M ) are known functions of the distinguished moments M , and N is the highest order of M . Other moments are functions of M : M = M (fC (M )). Slow evolution of distinguished moments is found upon substitution of Eq. (151) into the Boltzmann equation and 8nding the moments of the resulting expression (Grad’s moment equations).
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Following Grad, this extremely simple approximation can be improved by extending the list of distinguished moments. The most well known is Grad’s thirteen-moment approximation where the set of distinguished moments consists of 8ve hydrodynamic moments, 8ve components of the traceless stress tensor ij = m[(v − ui )(vj − uj ) − ij (C − u)2 =3]f dC, and of the three components of the heat i =ux vector qi = (vi − ui )m(C − u)2 =2f dC. The time evolution hypothesis cannot be evaluated for its validity within the framework of Grad’s approach. It is not surprising therefore that Grad’s methods failed to work in situations where it was (unmotivatedly) supposed to, primarily, in the phenomena with sharp time-space dependence such as the strong shock wave. On the other hand, Grad’s method was quite successful for describing transition between parabolic and hyperbolic propagation, in particular the second sound e6ect in massive solids at low temperatures, and, in general, situations slightly deviating from the classical Navier–Stokes–Fourier domain. Finally, the Grad method has been important background for development of phenomenological nonequilibrium thermodynamics based on hyperbolic 8rst-order equation, the so-called EIT (extended irreversible thermodynamics [180]). Important generalization of the Grad moment method is the concept of quasi-equilibrium approximations already mentioned above. The quasi-equilibrium distribution function for a set of distinguished moment M maximizes the entropy density S for 8xed M . The quasi-equilibrium manifold 83 (M ) is the collection of the quasi-equilibrium distribution functions for all admissible values of M . The quasi-equilibrium approximation is the simplest and extremely useful (not only in the kinetic theory itself) implementation of the hypothesis about the decomposition: If M are considered as slow variables, then states which could be visited in the course of rapid motion in the neighbored of 8∗ (M ) belong to the planes 9M = {f | m (f − f∗ (M )) = 0}. In this respect, the thermodynamic construction in the method of invariant manifold is a generalization of the quasi-equilibrium approximation where the given manifold is equipped with a quasi-equilibrium structure by choosing appropriately the macroscopic variables of the slow motion. In contrast to the quasi-equilibrium, the macroscopic variables thus constructed are not obligatory moments. A text book example of the quasi-equilibrium approximation is the generalized Gaussian func tion for M = {%; %u; P} where Pij = vi vj f dC is the pressure tensor. The quasi-equilibrium approximation does not exist if the highest order moment is an odd polynomial of velocity (therefore, there exists no quasi-equilibrium for thirteen Grad’s moments). Otherwise, the Grad moment approximation is the 8rst-order expansion of the quasi-equilibrium around the local Maxwellian. The classical Grad moment method [158] provides an approximate solution to the Boltzmann equation, and leads to a closed system of equations where hydrodynamic variables %, u, and P (density, mean =ux, and pressure) are coupled to a 8nite set of nonhydrodynamic variables. The latter are usually the stress tensor and the heat =ux q constituting 10 and 13 moment Grad systems. The Grad method was originally introduced for diluted gases to describe regimes beyond the normal solutions [51], but later it was used, in particular, as a prototype of certain phenomenological schemes in nonequilibrium thermodynamics [180]. However, the moments do not constitute the unique system of nonhydrodynamic variables, and the exact dynamics might be equally expressed in terms of other in8nite sets of variables (possibly, of a nonmoment nature). Moreover, as long as one shortens the description to only a 8nite subset of variables, the advantage of the moment description above other systems is not obvious.
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Nonlinear functionals instead of moments in the closure problem. Here we consider a new system of nonhydrodynamic variables, scattering rates M w (f): w Mi1 i2 i3 (f) = @i1 i2 i3 Qw (f) dC ; @i1 i2 i3 = mv1i1 v2i2 v3i3 ;
(152) w
which, by de8nition, are the moments of the Boltzmann collision integral Q (f): w Q (f) = w(C ; C1 ; C; C1 ){f(C )f(C1 ) − f(C)f(C1 )} dC dC1 dC1 : Here w is the probability density of a change of the velocities, (C; C1 ) → (C ; C1 ), of the two particles after their encounter, and w is de8ned by a model of pair interactions. The description in w terms of the scattering rates M (152) is alternative to the usually treated description in terms of the moments M : Mi1 i2 i3 (f) = @i1 i2 i3 f dC. A reason to consider scattering rates instead of the moments is that M w (152) re=ect features of the interactions because of the w incorporated in their de8nition, while the moments do not. For this reason we can expect that, in general, a description with a 1nite number of scattering rates will be more informative than a description provided by the same number of their moment counterparts. To come to the Grad-like equations in terms of the scattering rates, we have to complete the following two steps: (i) To derive a hierarchy of transport equations for %, u, P, and Miw1 i2 i3 in a neighborhood of the local Maxwell states f0 (%; u; P). (ii) To truncate this hierarchy, and to come to a closed set of equations with respect to %, u, P, and a 8nite number of scattering rates. In step (i), we derive a description with in8nite number of variables, which is formally equivalent both to the Boltzmann equation near the local equilibrium, and to the description with an in8nite number of moments. The approximation comes into play in step (ii) where we reduce the description to a 8nite number of variables. The di6erence between the moment and the alternative description occurs at this point. Program (i) and (ii) is similar to what is done in the Grad method [158], with the only exception (and this is important) that we should always use scattering rates as independent variables and not to expand them into series in moments. Consequently, we will use a method of a closure in the step (ii) that does not refer to the moment expansions. Major steps of the computation will be presented below. Linearization. To complete step (i), we represent f as f0 (1+’), where f0 is the local Maxwellian, and we linearize the scattering rates (152) with respect to ’: YMiw1 i2 i3 (’) = Y@iw1 i2 i3 f0 ’ dC ; Y@iw1 i2 i3 = Lw (@i1 i2 i3 ) : w
(153) w
Here L is the usual linearized collision integral, divided by f0 . Though YM are linear in ’, they are not moments because their microscopic densities, Y@w , are not velocity polynomials for a general case of w.
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It is not diRcult to derive the corresponding hierarchy of transport equations for variables YMiw1 i2 i3 , %, u, and P (we will further refer to this hierarchy as to the alternative chain): one has to calculate the time derivative of the scattering rates (152) due to the Boltzmann equation, in the linear approximation (153), and to complete the system with the 8ve known balance equations for the hydrodynamic moments (scattering rates of the hydrodynamic moments are equal to zero due to conservation laws). The structure of the alternative chain is quite similar to that of the usual moment transport chain, and for this reason we do not reproduce it here (details of calculations can be found in [181]). One should only keep in mind that the stress tensor and the heat =ux vector in the balance equations for u and P are no more independent variables, and they are expressed in terms of YMiw1 i2 i3 , %, u, and P. Truncating the chain. To truncate the alternative chain (step (ii)), we have, 8rst, to choose a 8nite set of “essential” scattering rates (153), and, second, to obtain the distribution functions which depend parametrically only on %, u, P, and on the chosen set of scattering rates. We will restrict our consideration to a single nonhydrodynamic variable, ijw , which is the counterpart of the stress tensor ij . This choice corresponds to the polynomial mvi vj in expressions (152) and (153), and the resulting equations will be alternative to the 10 moment Grad system. 9 For a spherically symmetric interaction, the expression for ijw may be written as ijw (’) = Y@ijw f0 ’ dC ; Y@ijw
P 1 w 2 2 S (c ) ci cj − ij c : = L (mvi vj ) = w P0 (T ) 3 w
(154)
Here Pw0 (T ) is the 8rst Sonine polynomial approximation of the Chapman–Enskog viscosity coef 8cient (VC) [51], and, as usual, c = m=2kT (C − u). The scalar dimensionless function S w depends only on c2 , and its form depends on the choice of interaction w. Entropy maximization. Next, we 8nd the functions f∗ (%; u; P; ijw )=f0 (%; u; P)(1+’∗ (%; u; P; ijw )) which maximize the Boltzmann entropy S(f) in a neighborhood of f0 (the quadratic approximation to the entropy is valid within the accuracy of our consideration), for 8xed values of ijw . That is, ’∗ is a solution to the following conditional variational problem: kB f0 ’2 dC → max; YS(’) = − 2 (155) (i) Y@ijw f0 ’ dC = ijw ; (ii) {1; C; v2 }f0 ’ dC = 0 : The second (homogeneous) condition in (155) re=ects that a deviation ’ from the state f0 is due only to nonhydrodynamic degrees of freedom, and it is straightforwardly satis8ed for Y@ijw (154). Notice, that if we turn to the usual moment description, then condition (i) in (155) would 8x the stress tensor ij instead of its scattering counterpart ijw . Then the resulting function f∗ (%; u; P; ij ) will be exactly the 10 moment Grad approximation. It can be shown that a choice of any 8nite set of higher moments as the constraint (i) in (155) results in the corresponding Grad approximation. 9
To get the alternative to the 13 moment Grad equations, one should take into account the scattering counterpart of the heat =ux, qiw = m vi (v2 =2)Qw (f) dC.
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In that sense our method of constructing f∗ is a direct generalization of the Grad method onto the alternative description. The Lagrange multipliers method gives straightforwardly the solution to the problem (155). After the alternative chain is closed with the functions f∗ (%; u; P; ijw ), the step (ii) is completed, and we arrive at a set of equations with respect to the variables %, u, P, and ijw . Switching to the variable Qij = n−1 ijw , we have 9t n + 9i (nui ) = 0 ;
(156)
Pw0 (T )n Qik = 0 ; %(9t uk + ui 9i uk ) + 9k P + 9i 2r w P w 3 5 P0 (T )n (9t P + ui 9i P) + P9i ui + Qik 9i uk = 0 ; 2 2 2r w P 2 9t Qik + 9s (us Qik ) + Qks 9s ui + Qis 9s uk − ik Qrs 9s ur 3 2"w P2 2 w 9i uk + 9k ui − ik 9s us + > − w Qik 9s us − w r P0 (T )n 3 −
1w P Qik = 0 : r w Pw0 (T )
(157) (158)
(159)
Here 9t = 9=9t; 9i = 9=9xi , summation in two repeated indices is assumed, and the coeRcients r w , "w , and 1w are de8ned with the help of the function S w (154) as follows: ∞ 8 2 w e−c c6 (S w (c2 ))2 dc ; r = √ 15 , 0 ∞ 8 dS w (c2 ) 2 w dc ; " = √ e−c c6 S w (c2 ) d(c2 ) 15 , 0 ∞ 8 2 w 1 = √ e−c c6 S w (c2 )Rw (c2 ) dc : (160) 15 , 0 The function Rw (c2 ) in the last expression is de8ned due to the action of the operator Lw on the function S w (c2 )(ci cj − 13 ij c2 ): P w 2 1 1 2 w w 2 2 : (161) R (c ) ci cj − ij c = L S (c ) ci cj − ij c 3 3 Pw0 Finally, the parameter >w in (156)–(160) re=ects the temperature dependence of the VC: w T dP0 (T ) 2 w 1− w : > = 3 P0 (T ) dT The set of 10 equations (156)–(160) is alternative to the 10-moment Grad equations.
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A new determination of molecular dimensions (revisited). The 8rst observation to be made is that for Maxwellian molecules we have: S MM ≡ 1, and PMM ˙ T ; thus >MM = "MM = 0, r MM = 1MM = 12 , 0 and (156)–(160) becomes the 10-moment Grad system under a simple change of variables Qij = ij , where is the proportionality coeRcient in the temperature dependence of PMM 0 . w These properties (the function S is a constant, and the VC is proportional to T ) are true only for Maxwellian molecules. For all other interactions, the function S w is not identical to one, and the VC Pw0 (T ) is not proportional to T . Thus, the shortened alternative description is not equivalent indeed to the Grad moment description. In particular, for hard spheres, the exact expression for the function S HS (154) reads √ √ 5 2 1 HS exp(−c2 t 2 )(1 − t 4 )(c2 (1 − t 2 ) + 2) dt; PHS T : (162) S = 0 ˙ 16 0 Thus, >HS = 13 , and "HS =r HS ≈ 0:07, and the equation for the function Qik (160) contains a nonlinear term HS
Qik 9s us ;
(163)
where HS ≈ 0:19. This term is missing in the Grad 10 moment equation. Finally, let us evaluate the VC which results from the alternative description (156)–(160). Following Grad’s arguments [158], we see that, if the relaxation of Qik is fast compared to the hydrodynamic variables, then the two last terms in the equation for Qik (156)–(160) become dominant, and the equation for u casts into the standard Navier–Stokes form with an e6ective VC Pwe6 : 1 (164) Pwe6 = w Pw0 : 21 For Maxwellian molecules, we easily derive that the coeRcient 1w in Eq. (164) is equal to 12 . Thus, as one expects, the e6ective VC (164) is equal to the Grad value, which, in turn, is equal to the exact value in the frames of the Chapman–Enskog method for this model. For all interactions di6erent from the Maxwellian molecules, the VC Pwe6 (164) is not equal to w P0 . For hard spheres, in particular, a computation of the VC (164) requires information about the function RHS (161). This is achieved upon a substitution of the function S HS (162) into Eq. (161). Further, we have to compute the action of the operator LHS on the function S HS (ci cj − 13 ij c2 ), which is rather complicated. However, the VC PHS e6 can be relatively easily estimated by using a function SaHS = √12 (1 + 17 c2 ), instead of the function S HS , in Eq. (161). Indeed, the function SaHS is tangent to the function S HS at c2 = 0, and is its majorant (see Fig. 3). Substituting SaHS into Eq. (161), and computing the action of the collision integral, we 8nd the approximation RHS a ; thereafter we evaluate HS the integral 1 (160), and 8nally come to the following expression: 75264 HS P ≈ 1:12PHS (165) PHS 0 : e6 ¿ 67237 0 Thus, for hard spheres, the description in terms of scattering rates results in the VC of more than 10% higher than in the Grad moment description. A discussion of the results concerns the following two items. 1. Having two not equivalent descriptions which were obtained within one method, we may ask: which is more relevant? A simple test is to compare characteristic times of an approach to hydroHS dynamic regime. We have 4G ∼ PHS 0 =P for 10-moment description, and 4a ∼ Pe6 =P for alternative
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1.6
1.4
1.2
1
0.8 0
2
4
6
8
10
c2
Fig. 3. Approximations for hard spheres: bold line—function S HS , solid line—approximation SaHS , dotted line—Grad moment approximation. Table 1 Three virial coeRcients: experimental Bexp , classical B0 [182], and reduced Be6 for three gases at T = 500 K
Argon Helium Nitrogen
Bexp
B0
Be6
8.4 10.8 168
60.9 21.9 66.5
50.5 18.2 55.2
description. As 4a ¿ 4G , we see that scattering rate decay slower than corresponding moment, hence, at least for rigid spheres the alternative description is more relevant. For Maxwellian molecules both the descriptions are, of course, equivalent. HS 2. The VC PHS e6 (165) has the same temperature dependence as P0 , and also the same dependence on a scaling parameter (a diameter of the sphere). In the classical book [51] (pp. 228–229), “sizes” of molecules are presented, assuming that a molecule is represented with an equivalent sphere and HS VC is estimated as PHS 0 . Since our estimation of VC di6ers only by a dimensionless factor from P0 , it is straightforward to conclude that e6ective sizes of molecules will be reduced by the factor b, where HS b = PHS 0 =Pe6 ≈ 0:94 : Further, it is well known that sizes of molecules estimated via viscosity in [51] disagree with the estimation via the virial expansion of the equation of state. In particular, in the book [182, p. 5] the measured second virial coeRcient Bexp was compared with the calculated B0 , in which the diameter of the sphere was taken from the viscosity data. The reduction of the diameter by factor b gives Be6 = b3 B0 . The values Bexp and B0 [182] are compared with Be6 in Table 1 for three gases at T = 500 K. The results for argon and helium are better for Be6 , while for nitrogen Be6 is worth than B0 . However, both B0 and Be6 are far from the experimental values.
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Hard spheres is, of course, an oversimpli8ed model of interaction, and the comparison presented does not allow for a decision between PHS and PHS 0 e6 . However, this simple example illustrates to what extend the correction to the VC can a6ect a comparison with experiment. Indeed, as it is well known, the 8rst-order Sonine polynomial computation for the Lennard–Jones (LJ) potential gives a very good 8t of the temperature dependence of the VC for all noble gases [183], subject to a proper choice of the two unknown scaling parameters of the LJ potential. 10 We may expect that a dimensionless correction of the VC for the LJ potential might be of the same order as above for rigid spheres. However, the functional character of the temperature dependence will not be a6ected, and a 8t will be obtained subject to a di6erent choice of the molecular parameters of the LJ potential. There remains, however, a general question how the estimation of the VC (164) responds to the exact value [51,184]. Since the analysis performed above does not immediately appeal to the exact Chapman–Enskog expressions just mentioned, this question remains open for a further work. 6. Newton method with incomplete linearization Let us come back to the invariance equation (78), Gy = (1 − Py )J (F(y)) = 0 : One of the most eRcient methods to solve this equation is the Newton method with incomplete linearization. Let us linearize the vector 8eld J around F(y): J (F(y) + F(y)) = J (F(y)) + (DJ )F(y) F(y) + o(F(y)) :
(166)
Equation of the Newton method with incomplete linearization makes it possible to determine F(y): Py F(y) = 0 ; (1 − Py )(DJ )F(y) F(y) = (1 − Py )J (F(y)) :
(167)
The crucial point here is that the same projector Py is used as in Eq. (78), that is, without computing the variation of the projector P (hence, the linearization of Eq. (78) is incomplete). We recall that projector Py depends on the tangent space Ty = Im(DF)y . If the thermodynamic projector (132) is used here, then Py depends also on |F(y) and on g = (DS)F(y) . Equations of the Newton method with incomplete linearization (167) are not di6erential equations in y anymore, they do not contain derivatives of the unknown F(y) with respect to y (which would be the case if the variation of the projector P has been taken into account). The absence of the derivatives in Eq. (167) signi8cantly simpli8es its solving. However, even this is not the main advantage of the incomplete linearization. More essential is the fact that iterations of the Newton method with incomplete linearization are expected to converge to slow invariant manifolds, unlike the usual Newton method. This has been demonstrated in [6] in the linear approximation. In order to illustrate the nature of Eq. (167), let us consider the case of linear manifolds for linear systems. Let a linear evolution equation be given in the 8nite-dimensional real space: x˙ = Ax, where 10
A comparison of molecular parameters of the LJ potential, as derived from the viscosity data, to those obtained from independent sources, can be found elsewhere, e.g. in Ref. [51, p. 237].
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261
A is negatively de8nite symmetric matrix with a simple spectrum. Let us further assume quadratic Lyapunov function, S(x) = x; x. The manifolds we consider are lines, l(y) = ye, where e is the unit vector, and y is a scalar. The invariance equation for such manifolds reads: ee; Ae − Ae = 0, and is simply the eigenvalue problem for the operator A. Solutions to the latter equation are eigenvectors ei , corresponding to eigenvalues i . Assume that we have chosen a line, l0 = ye0 , de8ned by the unit vector e0 , and that e0 is not an eigenvector of A. We seek another line, l1 = ae1 , where e1 is another unit vector, e1 = x1 = x1 , x1 = e0 + x. The additional condition in (167) reads: Py F(y) = 0, i.e. e0 ; x = 0. Then Eq. (167) becomes [1 − e0 e0 ; ·]A[e0 + x] = 0. Subject to the additional condition, the unique solution is as follows: e0 + x = ; A−1 e0 −1 A−1 e0 . Rewriting the latter expression in the eigenbasis of A, we e0 − have: e0 + y ˙ i i 1 ei ei ; e0 . The leading term in this sum corresponds to the eigenvalue with the minimal absolute value. The example indicates that the method (167) seeks the direction of the slowest relaxation. For this reason, the Newton method with incomplete linearization (167) can be recognized as the basis of iterative construction of the manifolds of slow motions. In an attempt to simplify computations, the question which always can be asked is as follows: to what extend is the choice of the projector essential in Eq. (167)? This question is a valid one, because, if we accept that iterations converge to a relevant slow manifold, and also that the projection on the true invariant manifold is insensible to the choice of the projector, then should one care of the projector on each iteration? In particular, for the moment parameterizations, can one use in Eq. (167) the projector (108)? Experience gained from some of the problems studied by this method indicates that this is possible. However, in order to derive physically meaningful equations of motion along the approximate slow manifolds, one has to use the thermodynamic projector (132). Otherwise we are not guaranteed from violating the dissipation properties of these equations of motion. Example 3: Nonperturbative correction of Local Maxwellian manifold and derivation of nonlinear hydrodynamics from Boltzmann equation (1D) This section is a continuation of Example 1. Here we apply the method of invariant manifold to a particular situation when the initial manifold consists of local Maxwellians (140) (the LM manifold). This manifold and its corrections play the central role in the problem of derivation of hydrodynamics from BE. Hence, any method of approximate investigation of BE should be tested with the LM manifold. Classical methods (the Chapman–Enskog and Hilbert methods) use Taylor-type expansions into powers of a small parameter (the Knudsen number expansion). However, as we have mentioned above, the method of invariant manifold, generally speaking, assumes no small parameters, at least in its formal part where convergency properties are not discussed. We will develop an appropriate technique to consider the invariance equation of the 8rst iteration. This involves ideas of parametrix expansions of the theory of pseudodi6erential and Fourier integral operators [193,194]. This approach will make it possible to reject the restriction of using small parameters. We search for a correction to the LM manifold as f1 (n; u; T ) = f0 (n; u; T ) + f1 (n; u; T ) :
(168)
We will use the Newton method with incomplete linearization for obtaining the correction f1 (n; u; T ), because we search for a manifold of slow (hydrodynamic) motions. We introduce the
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representation: f1 (n; u; T ) = f0 (n; u; T )’(n; u; T ) :
(169)
Positivity and normalization. When searching for a correction, we should be ready to face two problems that are typical for any method of successive approximations in BE theory. Namely, the 8rst of this problems is that the correction f8k+1 = f8k + f8k+1 obtained from the linearized invariance equation of the (k +1)th iteration may be not a nonnegatively de8ned function and thus it cannot be used directly to de8ne the thermodynamic projector for the (k + 1)th approximation. In order to overcome this diRculty, we can treat the procedure as a process of correcting the dual variable @f = Df H (f) rather than the process of immediate correcting the distribution functions. The dual variable @f is @f |f=f(x; C) = Df H (f)|f=f(x; C) = Df Hx (f)|f=f(x; C) = ln f(C; x) :
(170)
Then, at the (k + 1)th iteration, we search for new dual variables @f |8k+1 : @f |8k+1 = @f |8k + @f |8k+1 :
(171)
Due to the relationship @f ↔ f, we have @f |8k+1 = ’8k+1 + O(f82k+1 );
’8k+1 = f8−k 1 f8k+1 :
(172)
Thus, solving the linear invariance equation of the kth iteration with respect to the unknown function f8k+1 , we 8nd a correction to the dual variable ’8k+1 (172), and we derive the corrected distributions f8k+1 as f8k+1 = exp(@f |8k + ’8k+1 ) = f8k exp(’8k+1 ) :
(173)
Functions (173) are positive, and they satisfy the invariance equation and the additional conditions within the accuracy of ’8k+1 . However, the second diRculty which might occur is that functions (173) might have no 8nite integrals (134). In particular, this diRculty can be a result of some approximations used in solving equations. Hence, we have to “regularize” functions (173). A sketch of an approach to make this regularization might be as follows: instead of f8k+1 (173), we consider functions = f8k exp(’8k+1 + ’reg (")) : f8(") k+1
(174)
Here ’reg (") is a function labeled with " ∈ B, and B is a linear space. Then we derive "∗ from the condition of coincidence of macroscopic parameters. Further consideration of this procedure [6] is out of frames of this paper. The two diRculties mentioned here are not speci8c for the approximate method developed. For example, corrections to the LM distribution in the Chapman–Enskog method [51] and the 13-moment Grad approximation [158] are not nonnegatively de8ned functions, while the 13-moment quasi-equilibrium approximation [168] has no 8nite integrals (133) and (134).
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Galilean invariance of invariance equation. In some cases, it is convenient to consider BE 8eld in a reference system which moves with the =ow velocity. In this reference system, we the BE vector 8eld as df df 9f 9f 9f = Ju (f); = + ux; s (f) ; Ju (f) = −(vs − ux; s (f)) + Q(f; f) : dt dt 9t 9xs 9xs Here ux; s (f) stands for the sth component of the =ow velocity: 1 3 (f) v f(C; x) d C; n (f) = f(C; x) d 3 C : ux; s (f) = n− s x x
263
vector de8ne (175)
(176)
In particular, this form of BE vector 8eld is convenient when the initial manifold 80 consists of functions f80 which depend explicitly on (C−ux (f)) (i.e., if functions f80 ∈ 80 do not change under velocity shifts: C → C + c, where c is a constant vector). Substituting Ju (f) (175) instead of J (f) (135) into all expressions which depend on the BE vector 8eld, we transfer all procedures developed above into the moving reference system. In particular, we obtain the following analog of the invariance equation of the 8rst iteration: 0∗ (Pa(x) (·) − 1)Ju;0 lin; a(x) (f1 (a(x); C)) + G(f0 (a(x); C)) = 0 ; −1 0 Ju; lin; a(x) (g) = nx (f0 (a(x))) vs g d 3 C
−1
+ ux; s (f0 (a(x)))nx (f0 (a(x)))
3
gd C
9f0 (a(x); C) 9xs
9g + Lf0 (a(x);C) (g) ; 9xs ∗ G(f0 (a(x); C)) = (Pa(x) (·) − 1)Ju (f0 (a(x); C)) : − (vs − ux; s (f0 (a(x))))
(177)
Additional conditions do not depend on the vector 8eld, and thus they remain valid for Eq. (177). The equation of the 1rst iteration. The equation of the 8rst iteration in the form of (172) for the correction ’(n; u; T ) is 9f0 (n; u; T ) + f0 (n; u; T )Lf0 (n;u;T ) (’) {Pf0 (n;u;T ) (·) − 1} −(vs − us ) 9xs 9(f0 (n; u; T )’) −1 − (vs − us ) − n (f0 (n; u; T )) vs f0 (n; u; T )’ d 3 C 9xs 9f0 (n; u; T ) 3 + us (f0 (n; u; T )) f0 (n; u; T )’ d C =0 : (178) 9xs Here f0 (n; u; T )Lf0 (n;u;T ) (’) is the linearized Boltzmann collision integral: f0 (n; u; T )Lf0 (n;u;T ) (’) = w(v ; C | C; C1 )f0 (n; u; T ) ×{’ + ’1 − ’1 − ’} d 3 v d 3 v1 d 3 C1 :
(179)
and w(v ; C1 | C; C1 ) is the kernel of the Boltzmann collision integral, standard notations label the velocities before and after collision.
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Additional condition for Eq. (178) has the form Pf0 (n;u;T ) (f0 (n; u; T )’) = 0 : In detail notation: 1 · f0 (n; u; T )’ d 3 C = 0;
(180)
vi f0 (n; u; T )’ d 3 C = 0;
i = 1; 2; 3 ;
C2 f0 (n; u; T )’ d 3 C = 0 :
(181)
Eliminating in (178) the terms containing vs f0 (n; u; T )’ d 3 C and f0 (n; u; T )’ d 3 C with the aid of (181), we obtain the following form of Eq. (178): {Pf0 (n;u;T ) (·) − 1} 9f0 (n; u; T ) 9(f0 (n; u; T )’) =0 : + f0 (n; u; T )Lf0 (n;u;T ) (’) − (vs − us ) × −(vs − us ) 9xs 9xs (182) In order to consider the properties of Eq. (182), it is useful to introduce real Hilbert spaces Gf0 (n;u;T ) with scalar products: (183) (’; )f0 (n;u;T ) = f0 (n; u; T )’ d 3 C : Each Hilbert space is associated with the corresponding LM distribution f0 (n; u; T ). The projector Pf0 (n;u;T ) (146) is associated with a projector Lf0 (n;u;T ) which acts in the space Gf0 (n;u;T ) : Lf0 (n;u;T ) (’) = f0−1 (n; u; T )Pf0 (n;u;T ) (f0 (n; u; T )’) :
(184)
It is an orthogonal projector, because Lf0 (n;u;T ) (’) =
4 s=0
(s) (s) f0 (n;u;T ) ( f0 (n;u;T ) ; ’)f0 (n;u;T )
:
(185)
Here f(s) are given by expression (147). 0 (n;u;T ) We can rewrite the equation of the 8rst iteration (182) in the form Lf0 (n;u;T ) (’) + Kf0 (n;u;T ) (’) = Df0 (n;u;T ) :
(186)
Notations used here are: Df0 (n;u;T ) = f0−1 (n; u; T )G(f0 (n; u; T )) ; Kf0 (n;u;T ) (’) = {Lf0 (n;u;T ) (·) − 1}f0−1 (n; u; T )(vs − us )
9(f0 (n; u; T )’) : 9xs
(187)
The additional condition for Eq. (186) is (
(s) f0 (n;u;T ) ; ’)f0 (n;u;T )
= 0;
s = 0; : : : ; 4 :
(188)
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265
Now we will list the properties of Eq. (186) for usual collision models [51]: (a) The linear integral operator Lf0 (n;u;T ) is selfadjoint with respect to the scalar product (·; ·)f0 (n;u;T ) , and the quadratic form (’; Lf0 (n;u;T ) (’)) is negatively de8ned in Imf0 (n;u;T ) . (b) The kernel of Lf0 (n;u;T ) does not depend on f0 (n; u; T ), and it is the linear envelope of the polynomials 0 = 1, i = vi , i = 1; 2; 3; and 4 = v2 . (c) The RHS Df0 (n;u;T ) is orthogonal to ker Lf0 (n;u;T ) in the sense of the scalar product (·; ·)f0 (n;u;T ) . (d) The projecting operator Lf0 (n;u;T ) is the selfadjoint projector onto ker Lf0 (n;u;T ) : Lf0 (n;u;T ) (’) ∈ ker Lf0 (n;u;T ) :
(189)
Projector Lf0 (n;u;T ) projects orthogonally. (e) The image of the operator Kf0 (n;u;T ) is orthogonal to ker Lf0 (n;u;T ) . (f) Additional condition (188) requires the solution of Eq. (186) to be orthogonal to ker Lf0 (n;u;T ) . These properties result in the necessity condition for solving Eq. (186) with the additional constraint (188). This means the following: Eq. (186), provided with constraint (188), satis8es the necessary condition for to have an unique solution in Im Lf0 (n;u;T ) . Remark. Because of the diFerential part of the operator Kf0 (n;u;T ) , we are not able to apply the Fredholm alternative to obtain the necessary and suJcient conditions for solvability of Eq. (188). Thus, the condition mentioned here is, rigorously speaking, only the necessity condition. Nevertheless, we will still develop a formal procedure for solving Eq. (186). To this end, we paid no attention to the dependency of all functions, spaces, operators, etc, on x. It is useful to rewrite once again Eq. (186) in order to separate the local in x operators from those di6erential. Furthermore, we shall replace the subscript f0 (n; u; T ) with the subscript x in all expressions. We represent (186) as 9 Aloc (x; C)’ − Adi6 x; ; C ’ = −D(x; C) ; 9x Aloc (x; C)’ = −{Lx (C)’ + (Lx (C) − 1)rx ’} ; 9 9 Adi6 x; ; C ’ = (Lx (·) − 1) (vs − us ) ’ ; 9x 9xs Lx (C)g =
4
(s) (s) x ( x ; g)
;
s=0 (0) x (4) x
= n−1=2 ;
(s) x
= (2=n)1=2 cs (x; C);
s = 1; 2; 3 ;
= (2=3n)1=2 (c2 (x; C) − 3=2); ci (x; C) = (m=2kB T (x))1=2 (vi − ui (x)) ; 9 ln n m 9ui m(C − u)2 3 9 ln T (vi − ui ) − rx = (vs − us ) ; + + 9xs kB T 9xs 2kB T 2 9xs
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m(C − u)2 5 9 ln T (vi − ui ) D(x; C) = − 2kB T 2 9xi 9us m 1 2 (vi − ui )(vs − us ) − is (C − u) : + kB T 3 9xi
(190)
Here we have omitted the dependence on x in the functions n(x), ui (x), and T (x). Further, if no discrepancy might occur, we will always assume this dependence, and we will not indicate it explicitly. The additional condition for this equation is Lx (’) = 0 :
(191)
Eq. (190) is linear in ’. However, the main diRculty in solving this equation is caused by the di6erential in x operator Adi6 which does not commutate with the local in x operator Aloc . Parametrix expansion. In this subsection we introduce a procedure to construct approximate solutions of Eq. (189). This procedure involves an expansion similar to the parametrix expansion in the theory of pseudo-di6erential (PDO) and Fourier integral operators (FIO). Considering ’ ∈ Im Lx , we write a formal solution of Eq. (190) as −1 9 ’(x; C) = Aloc (x; C) − Adi6 x; ; C (−D(x; C)) : (192) 9x It is useful to extract the di6erential operator 9=9x from the operator Adi6 (x; 9=9x; C): 9 −1 ’(x; C) = 1 − Bs (x; C) ’loc (x; C) : 9xs
(193)
Notations used here are: 1 ’loc (x; C) = A− loc (x; C)(−D(x; C))
= [ − Lx (C) − (Lx (C) − 1)rx ]−1 (−D(x; C)) ; 1 Bs (x; C) = A− loc (x; C)(Lx (C) − 1)(vs − us )
= [ − Lx (C) − (Lx (C) − 1)rx ]−1 (Lx (C) − 1)(vs − us ) :
(194)
We will now discuss in more details the character of expressions in (194). For every x, the function ’loc (x; C), considered as a function of C, is an element of the Hilbert space Gx . It gives a solution to the integral equation: − Lx (C)’loc − (Lx (C) − 1)(rx ’loc ) = (−D(x; C))
(195)
This latter linear integral equation has an unique solution in Im Lx (C). Indeed, + ker A+ loc (x; C) = ker(Lx (C) + (Lx (C) − 1)rx )
= ker(Lx (C))+ ∩ ker((Lx (C) − 1)rx )+ = ker(Lx (C))+ ∩ ker(rx (Lx (C) − 1))
and
Gx ∩ Lx (C)Gx = {0} :
(196)
Thus, the existence of the unique solution of Eq. (195) follows from the Fredholm alternative.
A.N. Gorban et al. / Physics Reports 396 (2004) 197 – 403
Let us consider the operator R(x; 9=9x; C): 9 9 −1 R x; ; C = 1 − Bs (x; C) : 9x 9xs One can represent it as a formal series: ∞ 9 m 9 Bs (x; C) : R x; ; C = 9x 9xs m=0 Here
9 m 9 9 Bs (x; C) = Bs1 (x; C) · · · Bsm (x; C) : 9xs 9xs1 9xsm
267
(197)
(198)
(199)
Every term of type (199) can be represented as a 8nite sum of operators which are superpositions of the following two operations: of the integral in C operations with kernels depending on x, and of di6erential in x operations. Our goal is to obtain an explicit representation of the operator R(x; 9=9x; C) (197) as an integral operator. If the operator Bs (x; C) would not depend on x (i.e., if no dependence on spatial variables would occur in kernels of integral operators, in Bs (x; C)), then we could reach our goal via usual Fourier transformation. However, operators Bs (x; C) and 9=9xk do not commutate, and thus this elementary approach does not work. We will develop a method to obtain the required explicit representation using the ideas of PDO and IOF technique. We start with representation (198). Our strategy is to transform every summand (199) in order to place integral in C operators Bs (x; C) left to di6erential operators 9=9xk . The transposition of every pair 9=9xk Bs (x; C) yields an elementary transform: 9 9 9 : (200) Bs (x; C) → Bs (x; C) − Bs (x; C); 9xk 9xk 9xk Here [M; N ] = MN − NM denotes the commutator of operators M and N . We can represent (199) as 9 m 9 9 9 : (201) Bs (x; C) = Bs1 (x; C) · · · Bsm (x; C) ··· + O Bsi (x; C); 9xs 9xs1 9xsm 9xsk Here O([Bsi (x; C); 9=9xsk ]) denotes the terms which contain one or more pairs of brackets [·; ·]. The 8rst term in (201) contains none of these brackets. We can continue this process of selection and extract the 8rst-order in the number of pairs of brackets terms, the second-order terms, etc. Thus, we arrive at the expansion into powers of commutator of expressions (199). In this paper we will consider explicitly the zeroth-order term of this commutator expansion. Neglecting all terms with brackets in (201), we write 9 m 9 9 Bs (x; C) = Bs1 (x; C) · · · Bsm (x; C) ··· : (202) 9xs 0 9xs1 9xsm Here the subscript zero indicates the zeroth order with respect to the number of brackets.
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We now substitute expressions [Bs (x; C)9=9xs ]m0 (202) instead of expressions [Bs (x; C)9=9xs ]m (199) into the series (198): ∞ 9 9 m R0 x; ; C = Bs (x; C) : (203) 9x 9x s 0 m=0 The action of every summand (202) might be de8ned via the Fourier transform with respect to spatial variables. Denote as F the direct Fourier transform of a function g(x; C): (204) Fg(x; C) ≡ g(k; ˆ C) = g(x; C)exp(−iks xs ) d p x : Here p is the spatial dimension. Then the inverse Fourier transform is ˆ C)exp(iks xs ) d p k : ˆ C) = (2,)−p g(k; g(x; C) ≡ F −1 g(k;
(205)
The action of operator (202) on a function g(x; C) is de8ned as 9 m g(x; C) Bs (x; C) 9xs 0 9 9 −p g(k; ˆ C)eiks xs d p k (2,) ··· = Bs1 (x; C) · · · Bsm (x; C) 9xs1 9xsm −p exp(iks xs )[ikl Bl (x; C)]m g(k; ˆ C) d p k : =(2,)
(206)
The account of (206) in formula (203) yields the following de8nition of the operator R0 : ˆ C) d p k : R0 g(x; C) = (2,)−p eiks xs (1 − ikl Bl (x; C))−1 g(k;
(207)
This is the Fourier integral operator (note that the kernel of this integral operator depends on k and on x). The commutator expansion introduced above is a version of the parametrix expansion [193,194], while expression (207) is the leading term of this expansion. The kernel (1−ikl Bl (x; C))−1 is called the main symbol of the parametrix. The account of (207) in formula (193) yields the zeroth-order term of parametrix expansion ’0 (x; C): ’0 (x; C) = F −1 (1 − ikl Bl (x; C))−1 F’loc : In detail notation: −p
’0 (x; C) = (2,)
(208)
exp(iks (xs − ys ))
×(1 − iks [ − Lx (C) − (Lx (C) − 1)rx ]−1 (Lx (C) − 1)(vs − us (x)))−1 ×[ − Ly (C) − (Ly (C) − 1)ry ]−1 (−D(y; C)) d p y d p k : We now will list the steps to calculate the function ’0 (x; C) (209).
(209)
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269
Step 1: Solve the linear integral equation [ − Lx (C) − (Lx (C) − 1)rx ]’loc (x; C) = −D(x; C) : and obtain the function ’loc (x; C). Step 2: Calculate the Fourier transform ’ˆ loc (k; C): ’ˆ loc (k; C) = ’loc (y; C)exp(−iks ys ) d p y :
(210)
(211)
Step 3: Solve the linear integral equation ˆ k; C) ; [ − Lx (C) − (Lx (C) − 1)(rx + iks (vs − us (x)))]’ˆ 0 (x; k; C) = −D(x; ˆ k; C) = [ − Lx (C) − (Lx (C) − 1)rx ]’ˆ loc (k; C) : −D(x; and obtain the function ’ˆ 0 (x; k; C). Step 4: Calculate the inverse Fourier transform ’0 (x; C): −p ’ˆ 0 (x; k; C)exp(iks xs ) d p k : ’0 (x; C) = (2,)
(212)
(213)
Completing these four steps, we obtain an explicit expression for the zeroth-order term of parametrix expansion ’0 (x; C) (208). As we have already mentioned above, Eq. (210) of Step 1 has a unique solution in Im Lx (C). ˆ k; C) Eq. (212) of Step 3 has the same property. Indeed, for every k, the right hand side −D(x; is orthogonal to Im Lx (C), and thus the existence and the uniqueness of formal solution ’ˆ 0 (x; k; C) follows again from the Fredholm alternative. Thus, in Step 3, we obtain the unique solution ’ˆ 0 (x; k; C). For every k, this is a function which belongs to Im Lx (C). Accounting that f0 (x; C)=f0 (n(x); u(x); T (x); C) exposes no explicit dependency on x, we see that the inverse Fourier transform of Step 4 gives ’0 (x; C) ∈ Im Lx (C). Eqs. (210)–(213) provide us with the scheme of constructing the zeroth-order term of parametrix expansion. Finishing this section, we will outline brie=y the way to calculate the 8rst-order term of this expansion. Consider a formal operator R = (1 − AB)−1 . Operator R is de8ned by a formal series: ∞ (AB)m : (214) R= m=0
In every term of this series, we want to place operators A left to operators B. In order to do this, we have to commutate B with A from left to right. The commutation of every pair BA yields the elementary transform BA → AB − [A; B] where [A; B] = AB − BA. Extracting the terms with no commutators [A; B] and with a single commutator [A; B], we arrive at the following representation: R = R0 + R1 + (terms with more than two brackets) : Here R0 =
∞
Am B m ;
(215) (216)
m=0
∞ ∞ iAm−i [A; B]Ai−1 Bi−1 Bm−i : R1 = − m=2 i=2
(217)
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Operator R0 (216) is the zeroth-order term of parametrix expansion derived above. Operator R1 (the 1rst-order term of parametrix expansion) can be represented as follows: ∞ ∞ ∞ m i i m R1 = − mA [A; B] AB B =− mAm CBm ; C = [A; B]R0 : (218) m=1
i=0
m=1
This expression can be considered as an ansatz for the formal series (214), and it gives the most convenient way to calculate R1 . Its structure is similar to that of R0 . Continuing in this manner, we can derive the second-order term R2 , etc. We will not discuss these questions in this paper. In the next subsection we will consider in more detail the 8rst-order term of parametrix expansion. Finite-dimensional approximations to integral equations. Dealing further only with the zeroth-order term of parametrix expansion (209), we have to solve two linear integral equations, (210) and (212). These equations satisfy the Fredholm alternative, and thus they have unique solutions. The problem we face here is exactly of the same level of complexity as that of the Chapman–Enskog method [51]. The usual approach is to replace integral operators with some appropriate 8nite-dimensional operators. First we will recall standard objectives of 8nite-dimensional approximations, considering Eq. (210). Let pi (x; C), where i = 1; 2; : : :, be a basis in Im Lx (C). Every function ’(x; C) ∈ Im Lx (C) might be represented in this basis as ∞ ai (x)pi (x; C); ai (x) = (’(x; C); pi (x; C))x : (219) ’(x; C) = i=1
Eq. (210) is equivalent to an in8nite set of linear algebraic equations with respect to unknowns ai (x): ∞ mki (x)ai (x) = dk (x); k = 1; 2; : : : : (220) i=1
Here mki (x) = (pk (x; C); Aloc (x; C)pi (x; C))x ; dk (x) = −(pk (x; C); D(x; C))x :
(221)
For a 8nite-dimensional approximation of Eq. (220) we use a projection onto a 8nite number of basis elements pi (x; C), i = i1 ; : : : ; in . Then, instead of (219), we search for the function ’8n : n ais (x)pis (x; C) : (222) ’8n (x; C) = s=1
In8nite set of Eqs. (220) is replaced with a 8nite set of linear algebraic equations with respect to ais (x), where s = 1; : : : ; n: n mis il (x)ail (x) = dis (x); s = 1; : : : ; n : (223) l=1
There are no a priori restrictions upon the choice of the basis, as well as upon the choice of its 8nite-dimensional approximations. In this paper we use the standard basis of unreducible Hermite
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tensors (see, for example, [87,158]). The simplest appropriate version of a 8nite-dimensional approximation occurs if the 8nite set of Hermite tensors is chosen as: pk (x; C) = ck (x; C)(c2 (x; C) − (5=2)); pij (x; C) = ci (x; C)cj (x; C) −
k = 1; 2; 3 ;
1 ij c2 (x; C); 3
1 ci (x; C) = C− T (x)(vi − ui (x));
i; j = 1; 2; 3 ;
vT (x) = (2kB T (x)=m)1=2 :
(224)
It is important to stress here that “good” properties of orthogonality of Hermite tensors, as well as of other similar polynomial systems in BE theory, have the local in x character, i.e. when these functions are treated as polynomials in c(x; C) rather than polynomials in C. For example, functions pk (x; C) and pij (x; C) (224) are orthogonal in the sense of the scalar product (·; ·)x : 2 (225) (pk (x; C); pij (x; C))x ˙ e−c (x;C) pk (x; C)pij (x; C) d 3 c(x; C) = 0 : On contrary, functions pk (y; C) and pij (x; C) are not orthogonal neither in the sense of the scalar product (·; ·)y , nor in the sense of the scalar product (·; ·)x , if y = x. This distinction is important for constructing the parametrix expansion. Further, we will omit the dependencies on x and C in the dimensionless velocity ci (x; C) (224) if no misunderstanding might occur. In this section we will consider the case of one-dimensional in x equations. We assume that: u1 (x) = u(x1 );
u2 = u3 = 0;
T (x) = T (x1 );
n(x) = n(x1 ) :
(226)
We write x instead of x1 below. Finite-dimensional approximation (224) requires only two functions: 1 p3 (x; C) = c12 (x; C) − c2 (x; C); p4 (x; C) = c1 (x; C)(c2 (x; C) − (5=2)) ; 3 c1 (x; C) = vT−1 (x)(v1 − u(x));
c2; 3 (x; C) = vT−1 (x)v2; 3 :
(227)
Now we will make a step-by-step calculation of the zeroth-order term of parametrix expansion, in the one-dimensional case, for the 8nite-dimensional approximation (227). Step 1. Calculation of ’loc (x; C) from Eq. (210). We search for the function ’loc (x; C) in approximation (227) as ’loc (x; C) = aloc (x)(c12 − (1=3)c2 ) + bloc (x)c1 (c2 − (5=2)) :
(228)
Finite-dimensional approximation (223) of integral equation (210) in basis (227) yields m33 (x)aloc (x) + m34 (x)bloc (x) = 1loc (x) ; m43 (x)aloc (x) + m44 (x)bloc (x) = "loc (x) : Notations used are: 11 9u 27 9u ; m44 (x) = n(x) 4 (x) + ; 9 9x 4 9x vT (x) 9 ln n 11 9 ln T + ; m34 (x) = m43 (x) = 3 9x 2 9x m33 (x) = n(x) 3 (x) +
(229)
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1
3; 4 (x) = − 3=2 , 1loc (x) = −
e−c
2 9u ; 3 9x
2
(x;C)
p3; 4 (x; C)Lx (C)p3; 4 (x; C) d 3 c(x; C) ¿ 0 ;
5 9 ln T "loc (x) = − vT (x) : 4 9x
(230)
Parameters 3 (x) and 4 (x) are easily expressed via Enskog integral brackets, and they are calculated in [51] for a wide class of molecular models. Solving Eq. (229), we obtain coeRcients aloc (x) and bloc (x) in expression (228): aloc =
Aloc (x) ; Z(x; 0)
bloc =
Bloc (x) ; Z(x; 0)
Z(x; 0) = m33 (x)m44 (x) − m234 (x) ;
Aloc (x) = 1loc (x)m44 (x) − "loc (x)m34 (x) ; Bloc (x) = "loc (x)m33 (x) − 1loc (x)m34 (x) ; aloc =
bloc =
− 23 9u=9x(n 4 + (n 3 +
11 9
27 4
9u=9x) +
9u=9x)(n 4 +
− 54 vT 9 ln T=9x(n 3 + (n 3 +
11 9
11 9
27 4
5 12
vT2 9 ln T=9x(9 ln n=9x +
9u=9x) −
vT2 9
(9 ln n=9x +
11 2
11 2
9 ln T=9x)2
9u=9x) + 29 vT 9u=9x(9 ln n=9x +
9u=x)(n 4 +
27 4
9u=9x) −
vT2 9
(9 ln n=x +
11 2
9 ln T=9x)
11 2
;
9 ln T=9x)
9 ln T=x)2
:
(231)
These expressions complete Step 1. Step 2. Calculation of Fourier transform of ’loc (x; C) and its expression in the local basis. In this step we make two operations: (i) The Fourier transformation of the function ’loc (x; C): +∞ exp(−iky)’loc (y; C) dy : ’ˆ loc (k; C) = −∞
(232)
(ii) The representation of ’ˆ loc (k; C) in the local basis {p0 (x; C); : : : ; p4 (x; C)}: p0 (x; C) = 1;
p1 (x; C) = c1 (x; C);
p3 (x; C) = c12 (x; C) − (1=3)c2 (x; C);
p2 (x; C) = c2 (x; C) − (3=2) ; p4 (x; C) = c1 (x; C)(c2 (x; C) − (5=2)) :
(233)
Operation (ii) is necessary for completing Step 3 because there we deal with x-dependent operators. Obviously, the function ’ˆ loc (k; C) (232) is a 8nite-order polynomial in C, and thus the operation (ii) is exact. We obtain in (ii): ’ˆ loc (x; k; C) ≡ ’ˆ loc (x; k; c(x; C)) =
4 i=0
hˆi (x; k)pi (x; C) :
(234)
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Here 2 ˆ loc (k; C); pi (x; C))x : hˆi (x; k) = (pi (x; C); pi (x; C))− x (’
(235)
Let us introduce notations: # ≡ #(x; y) = (T (x)=T (y))1=2 ;
> ≡ >(x; y) =
u(x) − u(y) : vT (y)
(236)
CoeRcients hˆi (x; k) (235) have the following explicit form: +∞ ˆ exp(−iky)hi (x; y) dy; hi (x; y) = Z −1 (y; 0)gi (x; y) ; hi (x; k) = −∞
5 2 2 g0 (x; y) = Bloc (y) > + >(# − 1) + Aloc (y)>2 ; 2 3 5 4 g1 (x; y) = Bloc (y) 3#>2 + #(#2 − 1) + Aloc (y)#> ; 2 3 g2 (x; y) =
3
5 Bloc (y)#2 > ; 3
g3 (x; y) = Bloc (y)2#> + Aloc (y)#2 ; g4 (x; y) = Bloc (y)#3 :
(237)
Here Z(y; 0); Bloc (y) and Aloc (y) are the functions de8ned in (231). Step 3: Calculation of the function ’ˆ 0 (x; k; C) from Eq. (212). Linear integral equation (212) has a form similar to that of Eq. (210). We search for the function ’ˆ 0 (x; k; C) in basis (227) as ’ˆ 0 (x; k; C) = aˆ0 (x; k)p3 (x; C) + bˆ0 (x; k)p4 (x; C) :
(238)
Finite-dimensional approximation of the integral Eq. (212) in basis (227) yields the following equations for unknowns aˆ0 (x; k) and bˆ0 (x; k): 1 m33 (x)aˆ0 (x; k) + m34 (x) + ikvT (x) bˆ0 (x; k) = 1ˆ0 (x; k) ; 3 1 m43 (x) + ikvT (x) aˆ0 (x; k) + m44 (x)bˆ0 (x; k) = "ˆ0 (x; k) : (239) 3 Notations used here are 1ˆ0 (x; k) = m33 (x)hˆ3 (x; k) + m34 (x)hˆ4 (x; k) + sˆ1 (x; k) ; "ˆ0 (x; k) = m43 (x)hˆ3 (x; k) + m44 (x)hˆ4 (x; k) + sˆ" (x; k) ;
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sˆ1; " (x; k) =
+∞
exp(−iky)s1; " (x; y) dy ;
−∞
1 s1 (x; y) = vT (x) 3 5 s" (x; y) = vT (x) 4
9 ln T 9 ln n +2 9x 9x
h1 (x; y) +
2 9u (h0 (x; y) + 2h2 (x; y)) ; 3 9x
9 ln n 9 ln T 29u h2 (x; y) + (3h2 (x; y) + h0 (x; y)) + h1 (x; y) : 9x 9x 39x
(240)
(241)
Solving Eqs. (239), we obtain functions aˆ0 (x; k) and bˆ0 (x; k) in (238): aˆ0 (x; k) =
1ˆ0 (x; k)m44 (x) − "ˆ0 (x; k)(m34 (x) + 13 ikvT (x)) ; Z(x; 13 ikvT (x))
bˆ0 (x; k) =
"ˆ0 (x; k)m33 (x) − 1ˆ0 (x; k)(m34 (x) + 13 ikvT (x)) : Z(x; 13 ikvT (x))
(242)
Here
1 k 2 vT2 (x) 2 + ikvT (x)m34 (x) Z x; ikvT (x) = Z(x; 0) + 3 9 3 vT2 (x) 9 ln n 119 ln T 2 279u n 4 + − + 49x 9 9x 29x k 2 vT2 (x) 2 2 9 ln n 119 ln T + ikvT (x) + : + 9 9 9x 29x
119u = n 3 + 99x
(243)
Step 4: Calculation of the inverse Fourier transform of the function ’ˆ 0 (x; k; C). The inverse Fourier transform of the function ’ˆ 0 (x; k; C) (238) yields: ’0 (x; C) = a0 (x)p3 (x; C) + b0 (x)p4 (x; C) :
(244)
Here 1 a0 (x) = 2, b0 (x) =
1 2,
+∞
−∞
+∞
−∞
exp(ikx)aˆ0 (x; k) d k ; exp(ikx)bˆ0 (x; k) d k :
(245)
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Taking into account expressions (231), (240)–(243), and (237), we obtain the explicit expression for the 1nite-dimensional approximation of the zeroth-order term of parametrix expansion (244): +∞ +∞ 1 1 dy d k exp(ik(x − y))Z −1 x; ikvT (x) a0 (x) = 2, −∞ 3 −∞ × Z(x; 0)h3 (x; y) + [s1 (x; y)m44 (x) − s" (x; y)m34 (x)] 1 − ikvT (x)[m34 (x)h3 (x; y) + m44 (x)h4 (x; y) + s" (x; y)] ; 3 +∞ +∞ 1 1 −1 x; ikvT (x) b0 (x) = dy d k exp(ik(x − y))Z 2, −∞ 3 −∞ × Z(x; 0)h4 (x; y) + [s" (x; y)m33 (x) − s1 (x; y)m34 (x)]
1 − ikvT (x)[m34 (x)h4 (x; y) + m33 (x)h3 (x; y) + s1 (x; y)] 3
:
(246)
Hydrodynamic equations. Now we will discuss brie=y the utility of obtained results for hydrodynamics. The correction to LM functions f0 (n; u; T ) (140) obtained has the form f1 (n; u; T ) = f0 (n; u; T )(1 + ’0 (n; u; T )) :
(247)
Here the function ’0 (n; u; T ) is given explicitly with expressions (244)–(246). The usual form of closed hydrodynamic equations for n; u, and T , where the traceless stress tensor ik and the heat =ux vector qi are expressed via hydrodynamic variables, will be obtained if we substitute function (247) into balance equations of the density of the momentum, and of the energy. For LM approximation, these balance equations result in Euler equation of the nonviscid liquid (i.e. ik (f0 ) ≡ 0; and qi (f0 ) ≡ 0). For the correction f1 (247), we obtain the following expressions of = xx (f1 ) and q=qx (f1 ) (all other components are equal to zero in the one-dimensional situation under consideration): =
1 na0 ; 3
q=
5 nb0 : 4
(248)
Here a0 and b0 are given by expression (246). From the geometrical viewpoint, hydrodynamic equations with the stress tensor and the heat =ux vector (248) have the following interpretation: we take the corrected manifold 81 which consists of functions f1 (247), and we project the BE vectors Ju (f1 ) onto the tangent spaces Tf1 using the LM projector Pf0 (145). Although a detailed investigation of these hydrodynamic equations is a subject of a special study and it is not the goal of this paper, some points should be mentioned. Nonlocality. Expressions (246) expose a nonlocal spatial dependency, and, hence, the corresponding hydrodynamic equations are nonlocal. This nonlocality appears through two contributions. The
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8rst of these contributions might be called a frequency-response contribution, and it comes through explicit nonpolynomial k-dependency of integrands in (246). This latter dependency has the form +∞ A(x; y) + ikB(x; y) exp(ik(x − y)) d k : (249) 2 −∞ C(x; y) + ikD(x; y) + k E(x; y) Integration over k in (249) can be completed via auxiliary functions. The second nonlocal contribution might be called correlative, and it is due to relationships via (u(x) − u(y)) (the di6erence of =ow velocities in points x and y) and via T (x)=T (y) (the ratio of temperatures in points x and y). Acoustic spectra. The purely frequency-response contribution to hydrodynamic equations is relevant to small perturbations of equilibria. The stress tensor and the heat =ux q (248) are: 2 2 9u 2 9 T =− n0 T0 R 2! ; − 3! 3 9S 9S2 5 9T 8 2 92 u 3=2 T0 n0 R 3! ! q=− : (250) − 4 9S 5 9S2 Here
2 2 92 R= 1− ! −1 : 5 9S2
(251)
In (250), we have expressed parameters 3 and 4 via the viscosity coeRcient @ of the Chapman– Enskog method [51] (it is easy to see from (230) that 3 = 4 ˙ @−1 for spherically symmetric models of a collision), and we have used the following notations: T0 and n0 are the equilibrium temperature and density, S=(PT01=2 )−1 n0 x is the dimensionless coordinate, P=@(T0 )=T0 , u =T0−1=2 u, T = T=T0 , n = n=n0 , and u, T , n are the deviations of the =ux velocity, of the temperature and of the density from their equilibrium values u = 0, T = T0 and n = n0 . We also use the system of units with kB = m = 1. In the linear case, the parametrix expansion degenerates, and its zeroth-order term (213) gives the solution of Eq. (190). The dispersion relationship for approximation (250) is !3 + (23k 2 =6D)!2 + {k 2 + (2k 4 =D2 ) + (8k 6 =5D2 )}! + (5k 4 =2D) = 0 ; D = 1 + ( 45 )k 2 :
(252)
Here k is the wave vector. Acoustic spectra given by dispersion relationship (252) contains no nonphysical short-wave instability characteristic to the Burnett approximation (Fig. 4). The regularization of the Burnett approximation [25,26] has the same feature. Both of these approximations predict a limit of the decrement Re! for short waves. Nonlinearity. Nonlinear dependency on 9u=9x, on 9 ln T=9x, and on 9 ln n=9x appears already in the local approximation ’loc (231). In order to outline some peculiarities of this nonlinearity, we represent the zeroth-order term of the expansion of aloc (231) into powers of 9 ln T=9x and 9 ln n=9x: 2 9u 11 9u −1 9 ln T 9 ln n n 3 + ; : (253) +O aloc = − 3 9x 9 9x 9x 9x
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Fig. 4. Acoustic dispersion curves for approximation 250 (solid line), for second (the Burnett) approximation of the Chapman–Enskog expansion [53] (dashed line) and for the regularization of the Burnett approximation via partial summing of the Chapman–Enskog expansion [25,26] (punctuated dashed line). Arrows indicate an increase of k 2 .
This expression describes the asymptotic of the “purely nonlinear” contribution to the stress tensor (248) for a strong divergency of a =ow. The account of nonlocality yields instead of (250): +∞ +∞ 1 2 9u 11 9u −1 n 3 + dy d k exp(ik(x − y)) a0 (x) = − 2, −∞ 3 9y 9 9y −∞ −1 k 2 vT2 11 9u 27 9u 11 9u 27 9u n 4 + + n 4 + n 3 + × n 3 + 9 9x 4 9x 9 9 9x 4 9x 27 9u 9u −2 2 9u 4 n 4 + vT (u(x) − u(y))2 − ik (u(x) − u(y)) + 9 4 dy 9x 3 9x 9 ln T 9 ln n ; : (254) +O 9x 9x Both expressions, (253) and (254) become singular when ∗ 9u 9n 3 9u : =− → 9y 9y 11
(255)
Hence, the stress tensor (249) becomes in8nite if 9u=9y tends to 9u∗ =9y in any point y. In other words, the =ow becomes in8nitely viscid when 9u=9y approaches the negative value −9n 3 =11. This in8nite viscosity threshold prevents a transfer of the =ow into nonphysical region of negative viscosity if 9u=9y ¿ 9u∗ =9y because of the in8nitely strong dumping at 9u∗ =9y. This peculiarity was detected in [25,26] as a result of partial summing of the Chapman–Enskog expansion. In particular,
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partial summing for the simplest nonlinear situation [27,178] yields the following expression for the stress tensor : −1 5 9u 4 92 92 1 − !2 2 = IR + IIR ; IR = − ! + !2 2 ; = T + n ; 3 3 9S 9S 9S 7 9u −1 92 u 28 1+ ! : (256) IIR = 9 3 9S 9S2 Notations here follow (250) and (251). Expression (256) might be considered as a “rough draft” of the “full” stress tensor de8ned by a0 (246). It accounts both the frequency-response and the nonlinear contributions ( IR and IIR , respectively) in a simple form of a sum. However, the superposition of these contributions in (246) is more complicated. Moreover, the explicit correlative nonlocality of expression (246) was never detected neither in [27], nor in numerous examples of partial summing [178]. Nevertheless, approximation (256) contains the peculiarity of viscosity similar to that in (253) and (254). In dimensionless variables and ! = 1, expression (256) predicts the in8nite threshold at 9 velocity divergency equal to −( 37 ), rather than −( 11 ) in (253) and (254). Viscosity tends to zero as the divergency tends to positive in8nity in both approximations. Physical interpretation of these phenomena was given in [27]: large positive values of 9u=9x means that the gas diverges rapidly, and the =ow becomes nonviscid because the particles retard to exchange their momentum. On contrary, 9 its negative values (such as −( 37 ) for (256) and −( 11 )) for (253) and (254)) describe a strong compression of the =ow. Strong deceleration results in “solid =uid” limit with an in8nite viscosity (Fig. 5). Thus, hydrodynamic equations for approximation (247) are both nonlinear and nonlocal. This result is not surprising, accounting the integro-di6erential character of Eq. (190). It is important that no small parameters were used neither when we were deriving Eq. (190) nor when we were obtaining correction (247). Example 4: Nonperturbative derivation of linear hydrodynamics from Boltzmann equation (3D) Using the Newton method instead of power series, a model of linear hydrodynamics is derived from the Boltzmann equation for regimes where the Knudsen number is of order unity. The model demonstrates no violation of stability of acoustic spectra in contrast to Burnett hydrodynamics. The Knudsen number ! (a ratio between the mean free path, lc , and the scale of hydrodynamic =ows, lh ) is a recognized order parameter when hydrodynamics is derived from the Boltzmann equation [183]. The Chapman–Enskog method [51] establishes the Navier–Stokes hydrodynamic equations as the 8rst-order correction to Euler hydrodynamics at ! → 0, and it also derives formal corrections of order !2 , !3 ; : : : (known as Burnett and super-Burnett corrections). These corrections are important outside the strictly hydrodynamic domain !1, and has to be considered for an extension of hydrodynamic description into a highly nonequilibrium domain ! 6 1. Not much is known about high-order in ! hydrodynamics, especially in a nonlinear case. Nonetheless, in a linear case, some de8nite information can be obtained. On the one hand, experiments on sound propagation in noble gases are considerably better explained with the Burnett and the super-Burnett hydrodynamics rather than with the Navier–Stokes approximation alone [185]. On the other hand, a direct calculation shows
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Fig. 5. Dependency of viscosity on compression for approximation (253) (solid line), for partial summing (256) (punctuated dashed line), and for the Burnett approximation [27,178] (dashed line). The latter changes the sign at a regular point and, hence, nothing prevents the =ow to transfer into the nonphysical region.
nonphysical behavior of the Burnett hydrodynamics for ultra-short waves: acoustic waves increase instead of decay [53]. The latter failure of the Burnett approximation cannot be rejected on a basis that for such regimes they might be not applicable because for the Navier–Stokes approximation, which is formally still less valid, no such violation is observed. These two results indicate that, at least in a linear regime, it makes sense to consider hydrodynamics at ! 6 1, but Enskog way of deriving such hydrodynamics is problematic. The problem of constructing solutions to the Boltzmann equation valid when ! is of order unity is one of the main open problems of classical kinetic theory [183]. In this example we suggest a new approach to derive hydrodynamics at ! 6 1. The main idea is to pose a problem of a 8nding a correction to the Euler hydrodynamics in such a fashion that expansions in ! do not appear as a necessary element of analysis. This will be possible by using the Newton method instead of Taylor expansions to get such correction. We restrict our consideration to a linear case. Resulting hydrodynamic equations do not exhibit the mentioned violation. The starting point is the set of local Maxwell distribution functions (LM) f0 (n; u; T ; C), where C is the particle’s velocity, and n, u, and T are local number density, average velocity, and temperature. We write the Boltzmann equation as df = J (f); dt
J (f) = −(v − u)i · 9i f + Q(f) ;
(257)
where d=dt = 9=9t + ui · 9i is the material derivative, 9i = 9=9xi , while Q is the Boltzmann collision integral [183].
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On the one hand, calculating r.h.s. of Eq. (257) in LM-states, we obtain J (f0 ), a time derivative of LM-states due to Boltzmann equation. On the other hand, calculating a time derivative of LM-states due to Euler dynamics, we obtain P0 J (f0 ), where P0 is a projector operator onto the LM manifold (see [6]): f0 2 2 3 3 2 c − c − J dc ; (258) J dc + 2ci · ci J dc + P0 J = n 3 2 2 Since the LM functions are not solutions to the Boltzmann equation (257) (except for constant n, u, and T ), di6erence between J (f0 ) and P0 J (f0 ) is not equal to zero: 9i T 1 5 2 2 ci c − : (259) G(f0 ) = J (f0 ) − P0 J (f0 ) = −f0 2(9i uk ) ci ck − ik c + vT 3 T 2 here c =vT−1 (C−u), and vT = 2kB T=m is the thermal velocity. Note that the latter expression gives a complete discrepancy of the linearized local Maxwell approximation, and it is neither big nor small in itself. An unknown hydrodynamic solution of Eq. (257), f∞ (n; u; T ; C), satis8es the following equation: G(f∞ ) = J (f∞ ) − P∞ J (f∞ ) = 0 ;
(260)
where P∞ is an unknown projecting operator. Both P∞ and f∞ are unknown in Eq. (260), but, nontheless, one is able to consider a sequence of corrections {f1 ; f2 ; : : :}, {P1 ; P2 ; : : :} to the initial approximation f0 and P0 . A method to deal with equations of a form (260) was developed in [6] for a general case of dissipative systems. In particular, it was shown, how to ensure the H -theorem on every step of approximations by choosing appropriate projecting operators Pn . In the present illustrative example we will not consider projectors other than P0 , rather, we will use an iterative procedure to 8nd f1 . Let us apply the Newton method with incomplete linearization to Eq. (260) with f0 as initial approximation for f∞ and with P0 as an initial approximation for P∞ . Writing f1 = f0 + f, we get the 8rst Newton iterate: L(f=f0 ) + (P0 − 1)(v − u)i 9i f + G(f0 ) = 0 ; where L is a linearized collision integral. L(g) = f0 (C) w(C1 ; C ; C1 ; C)f0 (C1 ){g(C1 ) + g(C ) − g(C1 ) − g(C)} dC1 dC dC1 :
(261)
(262)
Here w is a probability density of velocities change, (C; C1 ) ↔ (C ; C1 ), of a pair of molecules after their encounter. When deriving (261), we have accounted P0 L=0, and an additional condition which 8xes the same values of n, u, and T in states f1 as in LM states f0 : P0 f = 0 :
(263)
Eq. (261) is basic in what follows. Note that it contains no Knudsen number explicitly. Our strategy will be to treat Eq. (261) in such a way that the Knudsen number will appear explicitly only at the latest stage of computations.
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The two further approximations will be adopted. The 8rst concerns a linearization of Eq. (261) about a global equilibria F0 . The second concerns a 8nite-dimensional approximation of integral operator in (261) in velocity space. It is worthwhile noting here that none of these approximations concerns an assumption about the Knudsen number. Following the 8rst of the approximations mentioned, denote as n, u, and T deviations of hydrodynamic variables from their equilibrium values n0 , u0 =0, and T0 . Introduce also nondimensional variables Yn = n=n0 , Yu = u=vT0 , and YT = T=T0 , where vT0 is a heat velocity in equilibria, and a nondimensional relative velocity = C=vT0 . Correction f1 in the approximation, linear in deviations from F0 reads: f1 = F0 (1 + ’0 + ’1 ) ; where ’0 = Yn + 2Yui Si + YT (S2 − 3=2) is a linearized deviation of LM from F0 , and ’1 is an unknown function. The latter is to be obtained from a linearized version of Eq. (261). Following the second approximation, we search for ’1 in a form 5 1 + Bik (x) Si Sk − ik S2 + · · · ; ’1 = Ai (x)Si S2 − (264) 2 3 where dots denote terms of an expansion of ’1 in velocity polynomials, orthogonal to Si (S2 − 5=2) and Si Sk − 1=3ik S2 , as well as to 1, to , and to S2 . These terms do not contribute to shear stress tensor and heat =ux vector in hydrodynamic equations. Independency of functions A and B from S2 amounts to the 8rst Sonine polynomial approximation of viscosity and heat transfer coeRcients. Put another way, we consider a projection onto a 8nite-dimensional subspace spanned by Si (S2 − 5=2) and Si Sk − 1=3ik S2 . Our goal is to derive functions A and B from a linearized version of Eq. (261). Knowing A and B, we get the following expressions for shear stress tensor and heat =ux vector q: 5 p0 vT0 A ; (265) 4 where p0 is equilibrium pressure of ideal gas. Linearizing Eq. (261) near F0 , using an ansatz for ’1 cited above, and turning to Fourier transform in space, we derive = p0 B;
q=
5p0 5 ai (k) + ivT0 bij (k)kj = − ivT0 ki 4(k) ; 3P0 2 p0 bij (k) + ivT0 ki aj (k) = −2ivT0 ki >j (k) ; (266) P0 √ where i = −1, k is the wave vector, P0 is the 8rst Sonine polynomial approximation of shear viscosity coeRcient, a(k), b(k), 4(k) and (k) are Fourier transforms of A(x), B(x), YT (x), and Yu(x), respectively, and the over-bar denotes a symmetric traceless dyad: ai bj = 2ai bj −
2 ij as bs : 3
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Introducing a dimensionless wave vector f = [(vT0 P0 )=(p0 )]k, solution to Eq. (266) may be written as 10 i>l (k)fj [(5=3) + (1=2)f2 ]−1 3
blj (k) = −
5 15 i(>s (k)fs )fl fj [(5=3) + (1=2)f2 ]−1 [5 + 2f2 ]−1 − 4(k)fl fj [5 + 2f2 ]−1 ; 3 2
+ al (k) = −
15 ifl 4(k)[5 + 2f2 ]−1 2
− [5 + 2f2 ]−1 [(5=3) + (1=2)f2 ]−1 [(5=3)fl (>s (k)fs ) + >l (k)f2 (5 + 2f2 )] :
(267)
Considering z-axis as a direction of propagation and denoting kz as k, > as >z , we obtain from (266) the k-dependence of a = az and b = bzz : p0−1 P0 vT0 ik4(k) + 45 p0−2 P20 (vT0 )2 k 2 >(k)
3
a(k) = − 2
;
1 + 25 p0−2 P20 (vT0 )2 k 2
4
b(k) = − 3
p0−1 P0 vT0 ik>(k) + p0−2 P20 (vT0 )2 k 2 4(k) 1 + 25 p0−2 P20 (vT0 )2 k 2
:
(268)
Using expressions for and q cited above, and also using (268), it is an easy matter to close the linearized balance equations (given in Fourier terms): 1 9t J(k) + ik>k = 0 ; vT0 2 9t >(k) + ik(4(k) + J(k)) + ikb(k) = 0 ; vT0 3 5 94 + ik>(k) + ika(k) = 0 : 4 2vT0
(269)
Eqs. (269), together with expressions (268), complete our derivation of hydrodynamic equations. To this end, the Knudsen number was not penetrating our derivations. Now it is worthwhile to introduce it. The Knudsen number will appear most naturally if we turn to dimensionless form of Eq. (268). Taking lc = vT0 P0 =p0 (lc is of order of a mean free path), and introducing a hydrodynamic scale lh , so that k = T=lh , where T is a nondimensional wave vector, we obtain in (268): 3
a(T) = − 2
i!T4(T) + 45 !2 T2 >T ; 1 + 25 !2 T2
4
b(T) = − 3
i!T>(T) + !2 T2 4(T) ; 1 + 25 !2 T2
(270)
where ! = lc =lh . Considering the limit ! → 0 in (270), we come back to the familiar Navier–Stokes NS expressions: zz = − 43 P0 9z uz , qzNS = − 0 9z T , where 0 = 15kB P0 =4m is the 8rst Sonine polynomial approximation of heat conductivity coeRcient.
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283
3
Re w
2
1
0
2
4
6
8
10
12
14
16
18
20 k2
-1
Fig. 6. Attenuation rate of sound waves. Dotts: Burnett approximation. Bobylev’s instability occurs when the curve intersects the horizontal axis. Solid: First iteration of the Newton method on the invariance equation.
Since we were not assuming smallness of the Knudsen number ! while deriving (270), we are completely legal to put ! = 1. With all the approximations mentioned above, Eqs. (269) and (268) (or, equivalently, (269) and (270)) may be considered as a model of a linear hydrodynamics at ! of order unity. The most interesting feature of this model is a nonpolynomial dependence on T. This amounts to that share stress tensor and heat =ux vector depend on spatial derivatives of u and of T of an arbitrary high order. To 8nd out a result of nonpolynomial behavior (270), it is most informative to calculate a dispersion relation for planar waves. It is worthwhile introducing dimensionless frequency = !lh =vT0 , where ! is a complex frequency of a wave ∼ exp(!t + ikz) (Re! is a damping rate, and Im ! is a circular frequency). Making use of Eqs. (269) and (270), writing ! = 1, we obtain the following dispersion relation (T): 12(1 + 25 T2 )2 3 + 23T2 (1 + 25 T2 ) 2 + 2T2 (5 + 5T2 + 65 T4 ) +
15 2
T4 (1 + 25 T2 ) = 0 :
(271)
Fig. 6 presents a dependence Re (T2 ) for acoustic waves obtained from (271) and for the Burnett approximation [53]. The violation in the latter occurs when the curve overcomes the horizontal axis. In contrast to the Burnett approximation [53], the acoustic spectrum (271) is stable for all T. Moreover, Re (T2 ) demonstrates a 8nite limit, as T2 → ∞. A discussion of results concerns the following two items: 1. The approach used avoids expansion into powers of the Knudsen number, and thus we obtain a hydrodynamics valid (at least formally) for moderate Knudsen numbers as an immediate correction to Euler hydrodynamics. This is in contrast to usual treatment of high-order hydrodynamics as “(the well established) Navier–Stokes approximation + high-order terms”. The Navier–Stokes hydrodynamics is recovered a posteriori, as a limiting case, but not as a necessary intermediate step of computations. 2. Linear hydrodynamics derived is stable for all k, same as the Navier–Stokes hydrodynamics alone. The (1 + 1k 2 )−1 “cut-o6”, as in (268) and (270), was earlier found in a “partial summing” of Enskog series [25,24].
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Thus, we come to the following two conclusions: 1. A preliminary positive answer is given to the question of whether is it possible to construct solutions of the Boltzmann equation valid for the Knudsen number of order unity. 2. Linear hydrodynamics derived can be used as a model for ! = 1 with no danger to get a violation of acoustic spectra at large k. Example 5: Dynamic correction to moment approximations Dynamic correction or extension of the list of variables? Considering the Grad moment ansatz as a suitable 8rst approximation to a closed 8nite-moment dynamics, the correction is derived from the Boltzmann equation. The correction consists of two parts, local and nonlocal. Locally corrected 13-moment equations are demonstrated to contain exact transport coeRcients. Equations resulting from the nonlocal correction give a microscopic justi8cation to some phenomenological theories of extended hydrodynamics. A considerable part of the modern development of nonequilibrium thermodynamics is based on the idea of extension of the list of relevant variables. Various phenomenological and semi-phenomenological theories in this domain are known under the common title of the extended irreversible thermodynamics (EIT) [180]. With this, the question of a microscopic justi8cation of the EIT becomes important. Recall that a justi8cation for some of the versions of the EIT was found within the well known Grad moment method [158]. Originally, the Grad moment approximation was introduced for the purpose of solving the Boltzmann-like equations of the classical kinetic theory. The Grad method is used in various kinetic problems, e.g., in plasma and in phonon transport. We mention also that Grad equations assist in understanding asymptotic features of gradient expansions, both in linear and nonlinear domains [178,163,162,23,24]. The essence of the Grad method is to introduce an approximation to the one-particle distribution function f which would depend only on a 8nite number N of moments, and, subsequently, to use this approximation to derive a closed system of N moment equations from the kinetic equation. The number N (the level at which the moment transport hierarchy is truncated) is not speci8ed in the Grad method. One particular way to choose N is to obtain an estimation of the transport coeRcients (viscosity and heat conductivity) suRciently close to their exact values provided by the Chapman– Enskog method (CE) [51]. In particular, for the 13-moment (13M) Grad approximation it is well known that transport coeRcients are equal to the 8rst Sonine polynomial approximation to the exact CE values. Accounting for higher moments with N ¿ 13 can improve this approximation (good for neutral gases but poor for plasmas [176]). However, what should be done, starting with the 13M approximation, to come to the exact CE transport coeRcients is an open question. It is also well known [161] that the Grad method provides a poorly converging approximation when applied to strongly nonequilibrium problems (such as shock and kinetic layers). Another question comes from the approximate character of the Grad equations, and is discussed in frames of the EIT: while the Grad equations are strictly hyperbolic at any level N (i.e., predicting a 8nite speed of propagation), whether this feature will be preserved in the further corrections. These two questions are special cases of a more general one, namely, how to derive a closed description with a given number of moments? Such a description is sometimes called mesoscopic
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[195] since it occupies an intermediate level between the hydrodynamic (macroscopic) and the kinetic (microscopic) levels of description. Here we aim at deriving the mesoscopic dynamics of 13 moments [9] in the simplest case when the kinetic description satis8es the linearized Boltzmann equation. Our approach will be based on the two assumptions: (i). The mesoscopic dynamics of thirteen moments exists, and is invariant with respect to the microscopic dynamics, and (ii). The 13M Grad approximation is a suitable 8rst approximation to this mesoscopic dynamics. The assumption (i) is realized as the invariance equation for the (unknown) mesoscopic distribution function. Following the assumption (ii), we solve the invariance equation iteratively, taking the 13M Grad approximation for the input approximation, and consider the 8rst iteration (further we refer to this as to the dynamic correction, to distinguish from constructing another ansatz). We demonstrate that the correction results in the exact CE transport coeRcients. We also demonstrate how the dynamic correction modi8es the hyperbolicity of the Grad equations. A similar viewpoint on derivation of hydrodynamics was earlier developed in [6] (see previous examples). We will return to a comparison below. Invariance equation for 13M parameterization. We denote as n0 , u0 = 0, and p0 the equilibrium values of the hydrodynamic parameters (n is the number density, u is the average velocity, and p = nkB T is the pressure). The global Maxwell distribution function F is F = n0 (vT )−3 ,−3=2 exp(−c2 ) ; where vT = 2kB T0 m−1 is the equilibrium thermal velocity, and c = C=vT is the peculiar velocity of a particle. The near-equilibrium dynamics of the distribution function, f = F(1 + ’), is due to the linearized Boltzmann equation: ˆ ; 9t ’ = Jˆ’ ≡ −vT ci 9i ’ + L’ ˆ L’ = wF(C1 )[’(C1 ) + ’(C ) − ’(C1 ) − ’(C)] dC1 dC dC1 ; where Lˆ is the linearized collision operator, and w is the probability density of pair encounters. Furthermore, 9i = 9=9xi , and summation convention in two repeated indices is assumed. Let n = n=n0 , u = u=vT , p = p=p0 (p = n + T; T = T=T0 ), be dimensionless deviations of the hydrodynamic variables, while ==p0 and q=q=(p0 vT ) are dimensionless deviations of the stress tensor , and of the heat =ux q. The linearized 13M Grad distribution function is f0 = F(c)[1 + ’0 ], where ’0 = ’ 1 + ’ 2 ; ’1 = n + 2ui ci + T [c2 − (3=2)] ; ’2 = ik ci ck + (4=5)qi ci [c2 − (5=2)] :
(272)
The overline denotes a symmetric traceless dyad. We use the following convention: ai bk = ai bk + ak bi − 23 ik al bl ; 9i fk = 9i fk + 9k fi − 23 ik 9l fl : The 13M Grad’s equations are derived in two steps: 8rst, the 13M Grad’s distribution function (272) is inserted into the linearized Boltzmann equation to give a formal expression, 9t ’0 = Jˆ’0 ,
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second, projector P0 is applied to this expression, where P0 = P1 + P2 , and operators P1 and P2 act as follows: F X0 X0 J dC + Xi Xi J dC + X4 X4 J dC ; P1 J = n0 F P2 J = Yik Yik J dC + Zi Zi J dC : (273) n0 √ √ Here X0 = 1, Xi = 2ci , where i = 1; 2; 3, X4 = 2=3(c2 − 32 ), Yik = 2ci ck , and Zi = √25 ci (c2 − 52 ). The resulting equation, P0 [F9t ’0 ] = P0 [F Jˆ’0 ] ; is a compressed representation for the 13M Grad equations for the macroscopic variables M13 = {n; u; T; ; q}. Now we turn to the main purpose of this paper, and derive the dynamic correction to the 13M distribution function (272). Assumption (i) [existence of closed dynamics of thirteen moments] implies ˜ 13 ; c) = F[1 + ’(M the invariance equation for the true mesoscopic distribution function, f(M ˜ 13 ; c)], where we have stressed that this function depends parametrically on the same 13 macroscopic ˜ 13 ; c) reads [6] parameters, as the original Grad approximation. The invariance condition for f(M ˜ (1 − P)[F Jˆ’] ˜ =0 ; (274) ˜ Generally speaking, the projector P˜ depends on the where P˜ is the projector associated with f. distribution function f˜ [6,176]. In the following, we use the projector P0 (273) which will be consistent with our approximate treatment of Eq. (274). Following assumption (ii) [13M Grad’s distribution function (272) is a good initial approximation], the Grad’s function f0 , and the projector P0 , are chosen as the input data for solving Eq. (274) iteratively. The dynamic correction amounts to the 8rst iterate. Let us consider these steps in a more detail. Substituting ’0 (272) and P0 (273) instead of ’ and P in the Eq. (274), we get: (1−P0 )[F Jˆ’0 ] ≡ G0 = 0, which demonstrates that (272) is not a solution to Eq. (274). Moreover, G0 splits in two nloc natural pieces: G0 = Gloc 0 + G0 , where ˆ Gloc 0 = (1 − P2 )[F L’2 ] ; Gnloc = (1 − P0 )[ − vT Fci 9i ’0 ] : (275) 0 ˆ = 0, and L’ ˆ 1 = 0. The 8rst piece of Eq. (275), Gloc Here we have accounted for P1 [F L’] 0 , can be termed local because it does not account for spatial gradients. Its origin is twofold. In the 8rst place, recall that we are performing our analysis in a nonlocal-equilibrium state (the 13M approximation ˆ 0 = 0). In the second place, is not a zero point of the Boltzmann collision integral, hence L’ specializing to the linearized case under consideration, functions cc and c[c2 − (5=2)], in general, are ˆ 0 ] = F L’ ˆ 0 , resulting in not the eigenfunctions of the linearized collision integral, and hence P2 [F L’ 11 Gloc = 0. 0 ˆ 0 = 0 but P2 [F L’ ˆ G ] = F L’ ˆ 0 . Same Except for Maxwellian molecules (interaction potential U ∼ r −4 ) for which L’ goes for the relaxation time approximation of the collision integral (Lˆ = −4−1 ). 11
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The nonlocal part may be written as Gnloc = −vT F(L1|krs 9k rs + L2|ik 9k qi + L3 9k qk ) ; 0
(276)
where L are velocity polynomials: L1|krs = ck [cr cs − (1=3)rs c2 ] − (2=5)ks cr c2 ; L2|ik = (4=5)[c2 − (7=2)][ci ck − (1=3)ik c2 ] ; L3 = (4=5)[c2 − (5=2)][c2 − (3=2)] − c2 : We seek the dynamic correction of the form f = F[1 + ’0 + V] : Substituting ’ = ’0 + V, and P = P0 , into Eq. (274), we derive an equation for the correction V: ˆ 2 + V)] = (1 − P0 )[vT Fci 9i (’0 + V)] : (1 − P2 )[F L(’
(277)
Eq. (277) should be supplied with the additional condition, P0 [FV] = 0. Solution of the invariance equation. Let us apply the usual ordering to solve Eq. (277), introducing a small parameter j, multiplying the collision integral Lˆ with j−1 , and expanding V = n jn V(n) . Subject to the additional condition, the resulting sequence of linear integral equations is uniquely soluble. Let us consider the 8rst two orders in j. 0 Because Gloc 0 = 0, the leading correction is of the order j , i.e. of the same order as the initial approximation ’0 . The function V(0) is due the following equation: ˆ 2 + V(0) )] = 0 ; (1 − P2 )[F L(’
(278)
subject to the condition, P0 [FV(0) ] = 0. Eq. (278) has the unique solution: ’2 + V(0) = ik Yik(0) + qi Zi(0) , where functions, Yik(0) and Zi(0) , are solutions to the integral equations: ˆ ik(0) = bYik ; LY
ˆ i(0) = aZi ; LZ (0)
(279) (0)
subject to the conditions, P1 [FY ] = 0 and P1 [FZ ] = 0. Factors a and b are: 2 −3=2 ˆ i(0) dc; e−c Zi(0) LZ a=, b = ,−3=2
ˆ ik(0) dc : e−c Yik(0) LY 2
Now we are able to notice that Eq. (279) coincides with the CE equations [51] for the exact transport coeJcients (viscosity and temperature conductivity). Emergency of these well known equations in the present context is important and rather unexpected: when the moment transport equations are closed with the locally corrected function floc = F(1 + ’0 + V(0) ), we come to a closed set of thirteen equations containing the exact CE transport coeJcients. Let us analyze the next order (j1 ), where Gnloc comes into play. To simplify matters, we neglect 0 the di6erence between the exact and the approximate CE transport coeRcients. The correction V(1) is due to the equation, ˆ (1) ] + Gnloc =0 ; (1 − P2 )[F LV 0
(280)
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the additional condition is: P0 [FV(1) ] = 0. Problem (280) reduces to three integral equations of a familiar form ˆ 1|krs = L1|krs ; L:
ˆ 2|ik = L2|ik ; L:
ˆ 3 = L3 ; L:
(281)
subject to conditions: P1 [F:1|krs ] = 0, P1 [F:2|ik ] = 0, and P1 [F:3 ] = 0. Integral equations (281) are of the same structure as are the integral equations appearing in the CE method, and the methods to handle them are well developed [51]. In particular, a reasonable and simple approximation is to take :1|::: = −A1 L1|::: . Then V(1) = −vT (A1 L1|krs 9k rs + A2 L2|ik 9k qi + A3 L3 9k qk ) ;
(282)
where A1 are the approximate values of the kinetic coeRcients, and which are expressed via matrix elements of the linearized collision integral: −1 ˆ 1|::: dc ¿ 0 : (283) A1 ˙ − exp(−c2 )L1|::: LL The estimation can be extended to a computational scheme for any given molecular model (e.g., for the Lennard–Jones potential), in the manner of the transport coeRcients computations in the CE method. Corrected 13M equations. To summarize the results of the dynamic correction, we quote 8rst the unclosed equations for the variables M13 = M13 = {n; u; T; ; q}: (1=vT0 )9t n + 9i ui = 0 ;
(284)
(2=vT0 )9t ui + 9i (T + n) + 9k ik = 0 ;
(285)
(1=vT0 )9t T + (2=3)9i ui + (2=3)9i qi = 0 ;
(286)
(1=vT0 )9t ik + 29i uk − (2=3)9i qk + 9l hikl = Rik ;
(287)
(2=vT )9t qi − (5=2)9i p − (5=2)9k ik + 9k gik = Ri :
(288)
Terms spoiling the closure are: the higher moments of the distribution function, 2 −3=2 e−c ’ci ck cl dc ; hikl = 2, gik = 2,
−3=2
2
e−c ’ci ck c2 dc ;
and the “moments” of the collision integral, 2 −3=2 2 ˆ dc; Rik = e−c ci ck L’ , vT 2 −3=2 2 ˆ dc : e−c ci c2 L’ , Ri = vT
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The 13M Grad’s distribution function (272) provides the zeroth-order closure approximation to both the higher-order moments and the “moments” of the collision integral: −1 R(0) ik = −@0 ik ;
−1 R(0) i = − 0 qi ;
9l h(0) ikl = (2=3)ik 9l ql + (4=5)9i qk ; (0) 9l glk = (5=2)9k (p + T ) + (7=2)9l lk ;
(289)
where @0 and 0 are the 8rst Sonine polynomial approximations to the viscosity and the temperature conductivity coeRcients [51], respectively. The local correction improves the closure of the “moments” of collision integral: −1 Rik = −@CE ik ;
−1 Ri = − CE qi ;
(290)
where index CE corresponds to exact Chapman–Enskog values of the transport coeRcients. The nonlocal correction adds the following terms to the higher moments: (0) 9l glk = 9l glk − A 3 9 k 9l ql − A 2 9l 9l qk ;
9l hikl = 9l h(0) ikl − A1 9l 9l ik ;
(291)
where Ai are the kinetic coeRcients derived above. In order to illustrate what changes in Grad equations with the nonlocal correction, let us consider a model with two scalar variables, T (x; t) and q(x; t) (a simpli8ed case of the one-dimensional corrected 13M system where one retains only the variables responsible for heat conduction): 9t T + 9x q = 0;
9t q + 9x T − a92x q + q = 0 :
(292)
Parameter a ¿ 0 controls “turning on” the nonlocal correction. Using {q(k; !); T (k; !)}exp(!t +ikx), we come to a dispersion relation for the two roots !1; 2 (k). Without the correction (a = 0), there are two domains of k: for 0 6 k ¡ k− , dispersion is di6usion-like (Re!1; 2 (k) 6 0; Im !1; 2 (k) = 0), while as k ¿ k− , dispersion is wave-like (!1 (k) = !2∗ (k); Im !1 (k) = 0). For a between 0 and 1, the dispersion modi8es in the following way: The wave-like domain becomes bounded, and exists for k ∈ ]k− (a); k+ (a)[, while the di6usion-like domain consists of two pieces, k ¡ k− (a) and k ¿ k+ (a). The dispersion relation for a = 1=2 is shown in Fig. 7. As a increases to 1, the boundaries of the wave-like domain, k− (a) and k+ (a), move towards each other, and collapse at a = 1. For a ¿ 1, the dispersion relation becomes purely di6usive (Im !1; 2 = 0) for all k. Discussion: transport coeJcients, destroying of the hyperbolicity, etc. (i) Considering the 13M Grad ansatz as a suitable approximation to the closed dynamics of thirteen moments, we have found that the 8rst correction leads to exact Chapman–Enskog transport coeRcients. Further, the nonlocal part of this correction extends the Grad equations with terms containing spatial gradients of the heat =ux and of the stress tensor, destroying the hyperbolic nature of the former. Corresponding kinetic coeRcients are explicitly derived for the Boltzmann equation. (ii) Extension of Grad equations with terms like in (291) was mentioned in many versions of the EIT [196]. These derivations were based on phenomenological and semi-phenomenological argument. In particular, the extension of the heat =ux with appealing to nonlocality e6ects in dense =uids. Here
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0
1
0
k 2
3
k+
4
k-
-2
-4 Re -6
Fig. 7. Attenuation Re!1; 2 (k) (lower pair of curves), frequency Im !1; 2 (k) (upper pair of curves). Dashed lines—Grad case (a = 0), drawn lines—dynamic correction (a = 0:5).
we have derived the similar contribution from the simplest (i.e. dilute gas) kinetics, in fact, from the assumption about existence of the mesoscopic dynamics. The advantage of using the simplest kinetics is that corresponding kinetic coeRcients (283) become a matter of a computation for any molecular model. This computational aspect will be discussed elsewhere, since it a6ects the dilute gas contribution to dense =uids 8ts. Here we would like to stress a formal support of relevancy of the above analysis: the nonlocal peace of dynamic correction is intermediated by the local correction, improving the 13M Grad estimation to the ordinary transport coeRcients. (iii) When the invariance principle is applied to derive hydrodynamics (closed equations for the variables n, u and T ) then [6] the local Maxwellian flm is chosen as the input distribution function for the invariance equation. In the linear domain, flm = F[1 + ’1 ], and the projector is Plm = P1 , see Eqs. (272) and (273). When the latter expressions are substituted into the invariance equation 2 loc (274), we obtain Glm = Gnloc lm = −vT F{29i uk ci ck + 9i Tci [c − (5=2)]}, while Glm ≡ 0 because the local Maxwellians are zero points of the Boltzmann collision integral. Consequently, the dynamic correction begins with the order j, and the analog of Eq. (280) reads: 2 ˆ (1) LV lm = vT {29i uk ci ck + 9i Tci [c − (5=2)]} ;
subject to a condition, P1 [FV(1) lm ] = 0. The latter is the familiar Chapman–Enskog equation, resulting in the Navier–Stokes correction to the Euler equations [51]. Thus, the nonlocal dynamic correction is related to the 13M Grad equations entirely in the same way as the Navier–Stokes are related to the Euler equations. As the 8nal comment to this point, it was recently demonstrated with simple examples [24] that the invariance principle, as applied to derivation of hydrodynamics, is equivalent to the summation of the Chapman–Enskog expansion. (iv) Let us discuss brie=y the further corrections. The 8rst local correction (the functions Y1 and Z1 in Eq. (279)) is not the limiting point of our iterational procedure. When the latter is continued, the ˆ n+1 = bn+1 Yn , and LZ ˆ n+1 = an+1 Zn . subsequent local corrections are found from integral equations, LY ˆ ˆ ∞ = a ∞ Z∞ , Thus, we are led to the following two eigenvalue problems: LY∞ = b∞ Y∞ , and LZ
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291
where, in accord with general argument [6], a∞ and b∞ are the closest to zero eigenvalues among all the eigenvalue problems with the given tensorial structure [192]. (v) Approach of this example [9] can be extended to derive dynamic corrections to other (nonmoment) approximations of interest in the kinetic theory. The above analysis has demonstrated, in particular, the importance of the local correction, generically relevant to an approximation which is not a zero point of the collision integral. Very recently, this approach was successfully applied to improve the nonlinear Grad’s 13-moment equations [197]. 7. Decomposition of motions, nonuniqueness of selection of fast motions, self-adjoint linearization, Onsager -lter and quasi-chemical representation In Section 5 we used the second law of thermodynamics—existence of the entropy—in order to equip the problem of constructing slow invariant manifolds with a geometric structure. The requirement of the entropy growth (universally, for all the reduced models) restricts signi8cantly the form of projectors (132). In this section we introduce a di6erent but equally important argument—the micro-reversibility (T -invariance), and its macroscopic consequences, the reciprocity relations. As 8rst discussed by Onsager in 1931 [150], the implication of the micro-reversibility is the self-adjointness of the linear approximation of system (76) in the equilibrium x∗ : (Dx J )x∗ z|px∗ ≡ z|(Dx J )x∗ px∗ :
(293)
The main idea in the present section is to use the reciprocity relations (293) for the fast motions. In order to appreciate this idea, we should mention that the decomposition of motions into fast and slow is not unique. Requirement (293) for any equilibrium point of fast motions means the selection (8ltration) of the fast motions. We term this Onsager 1lter. Equilibrium points of fast motions are all the points on manifolds of slow motions. There exists a trivial way to symmetrization, linear operator A is decomposed into symmetric and skew-symmetric parts, A = 12 (A + A† ) + 12 (A − A† ). Here A† is adjoint to A with respect to a 8xed scalar product (entropic scalar product in present context). However, replacement of an operator with its symmetric part can lead to catastrophic (from the physical standpoint) consequences such as, for example, loss of stability. In order to construct a sensible Onsager 8lter, we shall use the quasi-chemical representation. The formalism of the quasi-chemical representation is one of the most developed means of modelling, it makes it possible to “assemble” complex processes out of elementary processes. There exist various presentations of the quasi-chemical formalism. Our presentation here is a generalization of the approach suggested 8rst by Feinberg [187] (see also [186,188,59]). Symbol Ai (“quasi-substance”) is put into correspondence to each variable xi . The elementary reaction is de8ned accordingly to the stoichiometric equation, 1i A i "i Ai ; (294) i
i
where 1i (loss stoichiometric coeJcients) and "i (gain stoichiometric coeJcients) are real numbers. Apart from the entropy, one speci8es a monotonic function of one variable, :(a), : (a) ¿ 0. In particular, function :(a) = exp( a), = const, is frequently encountered in applications.
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Given the elementary reaction (294), one de8nes the rates of the direct and of the inverse reactions: 1i @i ; W + = w∗ : i
W
−
∗
=w :
"i @i
;
(295)
i
where @i = 9S=9xi , x∗ = const, x∗ ¿ 0. The rate of the elementary reaction is then de8ned as, W = W + − W −. The equilibrium of the elementary reaction (294) is given by the following equation: W+ = W− :
(296)
Thanks to the strict monotonicity of the function :, equilibrium of the elementary reaction is reached when the arguments of the functions coincide in Eq. (295), that is, whenever ("i − 1i )@i = 0 : (297) i
Vector with the components >i = "i − 1i is termed the stoichiometric vector of the reaction. Let x0 be a point of equilibrium of reaction (294). The linear approximation of the reaction rate has a particularly simple form: (298) W (x0 + ) = −w∗ : (a(x0 ))>|x0 + o() ; 0 0 0 where a(x ) = i 1i @i (x ) = i "i @i (x ), and |x0 is the entropic scalar product in the equilibrium. In other words, (Dx W )x0 = −w∗ : (a(x0 ))>| : Let us write down the kinetic equation for one elementary reaction: dx = >W (x) : dt Linearization of this equation in the equilibrium x0 has the following form: d = −w∗ : (a(x0 ))>>|x0 : dt That is, the matrix of the linear approximation has the form K = −k ∗ |>>| ;
(299) (300)
(301) (302)
where k ∗ = w∗ : (a(x0 )) ¿ 0 ; while the entropic scalar product of bra- and ket vectors is taken in the equilibrium point x0 . If there are several elementary reactions, then the stoichiometric vectors >r and the reaction rates Wr (x) are speci8ed for each reaction, while the kinetic equation is obtained by summing the right hand sides of Eq. (300) for individual reactions, dx r = > Wr (x) : (303) dt r
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293
Let us assume that under the inversion of motions, the direct reaction transforms into the inverse reaction. Thus, the T -invariance of the equilibrium means that it is reached in the point of the detailed balance, where all the elementary reaction come to equilibrium simultaneously: Wr+ (x∗ ) = Wr− (x∗ ) :
(304)
This assumption is nontrivial if vectors >r are linearly dependent (for example, if the number of reactions is greater than the number of species minus the number of conservation laws). In the detailed balance case, the linearization of Eq. (303) about x∗ has the following form (x = x∗ + ): d =− kr∗ >r >r |x∗ ; (305) dt r where kr∗ = wr∗ :r (a∗r ) ¿ 0; a∗r = 1ir @i (x∗ ) = "ir @i (x∗ ) : i
i
The following matrix of the linear approximation is obviously self-adjoint and stable: kr∗ |>r >r | : K =−
(306)
r
Note that matrix K is the sum of matrices of rank one. Let us now extract the self-adjoint part of form (306) in the arbitrary point x. After linearizing the reaction rate about x, we obtain W (x + ) = w∗ (: (a(x))1|x − : (b(x))"|x ) + o() ; where a(x) =
(307)
1i @i (x) ;
i
b(x) =
"i @i (x) :
i
Let us introduce notation, k SYM (x) = 12 w∗ (: (a(x)) + : (b(x))) ¿ 0 ; k A (x) = 12 w∗ (: (a(x)) − : (b(x))) : In terms of this notation, Eq. (307) may be rewritten as W (x + ) = −k SYM (x)>|x + k A (x)1 + "|x + o() :
(308)
The second term vanishes in the equilibrium (k A (x∗ ) = 0, due to detailed balance). Symmetric linearization (Onsager 8lter) consists in using only the 8rst term in the linearized vector 8eld (308) when analyzing the fast motion towards the (approximate) slow manifolds, instead
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of the full expression (307). Matrix K(x) of the linear approximation becomes then the form similar to Eq. (306): K(x) = − krSYM (x)|>r >r | ; (309) r
where krSYM (x) = 12 wr∗ (:r (a(x)) + :r (b(x))) ¿ 0 ; 1ir @i (x) ; ar (x) = i
br (x) =
"ir @i (x) ;
i
while the entropic scalar product |x is taken at the point x. For each index of the elementary reaction r, function krSYM (x) is positive. Thus, stability of the symmetric matrix (309) is evident. Symmetric linearization (309) is distinguished also by the fact that it preserves the rank of the elementary processes contributing to the complex mechanism: Same as in the equilibrium, matrix K(x) is the sum of rank one operators corresponding to each individual process. This is not the case of the standard symmetrization. Using the symmetric operator (309) in the above Newton method with incomplete linearization can be considered as a version of a heuristic strategy of “we act in such a way as if the manifold F(W ) were already slow invariant manifold”. If this were the case, then, in particular, the fast motions were described by the self-adjoint linear approximation. We describe the quasi-chemical formalism for 8nite-dimensional systems. In8nite-dimensional generalizations are almost obvious in many important cases, and are achieved by a mere replacement of summation by integration. The best example give us collisions in the Boltzmann equation: each velocity v corresponds to a quasi-substance Av , and a collision has a stoichiometric equation: A v + A w Av + A w : In the example to this section we consider the Boltzmann collision integral from this standpoint in more details. Example 6: Quasi-chemical representation and self-adjoint linearization of the Boltzmann collision operator A decomposition of motions near thermodynamically nonequilibrium states results in a linear relaxation towards this state. A linear operator of this relaxation is explicitly constructed in the case of the Boltzmann equation. An entropy-related speci8cation of an equilibrium state is due to the two points of view. From the 8rst, thermodynamic viewpoint, equilibria is a state in which the entropy is maximal. From the second, kinetic viewpoint, a quadratic form of entropy increases in a course of a linear regression towards this state. If an underlying microscopic dynamics is time-reversible, the kinetic viewpoint is realized due to known symmetric properties of a linearized kinetic operator.
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In most of near-equilibrium studies, a principle of a decomposition of motions into rapid and slow occupies a distinct place. In some special cases, decomposition of motions is taken into account explicitly, by introducing a small parameter into dynamic equations. More frequently, however, it comes into play implicitly, for example, through an assumption of a rapid decay of memory in projection operator formalism [154]. Even in presence of long-living dynamic e6ects (mode coupling), a decomposition of motions appears as a 8nal instance to get a closed set of equations for slow variables. However, for closed systems, there remains a question: whether and to what extend the two aforementioned entropy-related points of view are applicable to nonequilibrium states? Further, if an answer is positive, then how to make explicitly a corresponding speci8cation? This example is aimed at answering the questions just mentioned, and it is a straightforward continuation of results [4,6]. Namely, in [4,6], it was demonstrated that the principle of a decomposition of motions alone constitutes a necessary and suRcient condition for the thermodynamic speci8cation of a nonequilibrium state (this will be brie=y reviewed in the next section). However, in a general situation, one deals with states f other than f0 . A question is, whether these two ideas can be applied to f = f0 (at least approximately), and if so, then how to make the presentation explicit. A positive answer to this question was given partially in frames of the method of invariant manifolds [4–6]. Objects studied in [4–6] were manifolds in a space of distribution functions, and the goal was to construct iteratively a manifold that is tangent in all its points to a vector 8eld of a dissipative system (an invariant manifold), beginning with some initial manifold with no such property. It was natural to employ methods of KAM-theory (Newton-type linear iterations to improve the initial manifold). However, an extra idea of a decomposition of motions into rapid and slow near the manifold was strongly necessary to adapt KAM-theory to dissipative systems. A geometrical formulation of this idea [4–6] results in a de8nition of a hyperplane of rapid motion, 9f , associated with the state f, and orthogonal to the gradient of the entropy in f. In a physical interpretation, 9f contains all those states from a neighborhood of f, which come into f in the course of rapid relaxation (as if f were the 8nal state of rapid processes occurring in its neighborhood). Usually, 9f contains more states than can come into f in a rapid relaxation because of conservation of some macroscopic quantities (e.g. density, momentum, and energy, as well as, possibly, higher moments of f which practically do not vary in rapid processes). Extra states are eliminated by imposing additional restrictions, cutting out “thinner” linear manifolds, planes of rapid motions Pf , inside 9f . Extremal property of f on 9f is preserved on Pf as well (cf. [4–6]). Thus, decomposition of motions near a manifold results in the thermodynamical viewpoint: states f belonging to the manifold are described as unique points of maximum of entropy on corresponding hyperplanes of rapid motions 9f . This formulation de8nes a slow dynamics on manifolds in agreement with the H -theorem for the Boltzmann equation, or with its analogs for other systems (see [4–6] for details). As it was shown in [4–6], decomposition of motions in a neighborhood of f is a criteria (a necessary and suRcient condition) of an existence of the thermodynamic description of f. Newton iteration gives a correction, f + f, to states of a noninvariant manifold, while f is thought on 9f . Equation for f involves a linearization of the collision integral in state f. Here, if f = f0 , we come to a problem of how to perform a linearization of collision integral in concordance with the H -theorem (corrections to the manifold of local equilibrium states were studied in detail in [6]).
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Here we show that the aforementioned decomposition of motions results in the kinetic description of states on manifolds of slow motions, and that Onsager’s principle can be applied in a natural way to linearize the Boltzmann collision integral. Due to de8nition of 9f , the state f is the unique point of minimum of the H -function on 9f . In the 8rst nonvanishing approximation, we have the following expression for H in the states on 9f : H (f + f) ≈ H (f) + 12 f|ff Here−1·|·f denotes a scalar product generated by the second derivative of H in the state f: g1 |g2 f = f g1 g2 dC. Decomposition of motions means that quadratic form f|ff decays monotonically in the course of the linear relaxation towards the state f. It is natural, therefore, to impose the requirement that this linear relaxation should obey Onsager’s principle. Namely, the corresponding linear operator should be symmetric (formally self-adjoint) and nonpositively de8nite in scalar product ·|·f , and its kernel should consist of linear combinations of conserved quantities (1, C, and v2 ). In other words, decomposition of motions should give a picture of linear relaxation in a small neighborhood of f similar to that in a small neighborhood of f0 . Following this idea, we will now decompose the linearized collision integral Lf in two parts: LSYM (satisfying Onsager’s principle), and LAf f (nonthermodynamic part). In the state f, each direct encounter, (C; C1 ) → (C ; C1 ), together with the reverse encounter, (C ; C1 ) → (C; C1 ), contribute a rate, G(f) − L(f), to the collision integral, where (see Section 2): W (f) = W (C ; C1 ; C; C1 )exp{Df H |f=f(C) + Df H |f=f(C1 ) } ; W (f) = W (C ; C1 ; C; C1 )exp{Df H |f=f(C ) + Df H |f=f(C1 ) } ; A deviation f from the state f will change the rates of both the direct and the reverse processes. Resulting deviations of rates are: W = W (f){Df2 H |f=f(C) · f(C) + Df2 H |f=f(C1 ) · f(C1 )} ; W = W (f){Df2 H |f=f(C ) · f(C ) + Df2 H |f=f(C1 ) · f(C1 )} ; Symmetrization with respect to direct and reverse encounters will give a term proportional to a balanced rate, W SYM (f) = 12 (W (f) + W (f)), in both of the expressions W and W . Thus, we come to the decomposition of the linearized collision integral Lf = LSYM + LAf , where f f f1 + ff1 f f1 f1 f SYM dC1 dC dC1 ; Lf f = w + − − (310) 2 f f1 f1 f LAf f
=
f f1 − ff1 w 2
f f1 f1 f + + + f f1 f1 f
dC1 dC dC1 ;
(311)
f = f(C); f1 = f(C1 ); f = f(C ); f1 = f(C1 ); f = f(C); f1 = f(C1 ); f = f(C ); f1 = f(C1 ):
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Operator LSYM (310) has the complete set of the aforementioned properties corresponding to the f Onsager’s principle, namely: |g2 f = g2 |LSYM |g1 f (symmetry); (i) g1 |LSYM f f SYM (ii) g|Lf |gf 6 0 (local entropy production inequality); (conservation laws). (iii) f; Cf; v2 f ∈ ker LSYM f For an unspeci8ed f, nonthermodynamic operator LAf (311) satis8es none of these properties. If becomes the usual linearized collision integral f = f0 , then part (311) vanishes, while operator LSYM f0 due to the balance W (f0 ) = W (f0 ). Nonnegative de8nite form f|ff decays monotonically due to the equation of linear relaxation, 9t f=LSYM f, and the unique point of minimum, f=0, of f|ff corresponds to the equilibrium f f. point of vector 8eld LSYM f SYM Operator Lf describes the state f as the equilibrium state of a linear relaxation. Note that the method of extracting the symmetric part (310) is strongly based on the representation of direct and reverse processes, and it is not a simple procedure like, e.g., 12 (Lf + L+ f ). The latter expression cannot be used as a basis for Onsager’s principle since it would violate conditions (ii) and (iii). Thus, if motions do decompose into a rapid motion towards the manifold and a slow motion along the manifold, then states on this manifold can be described from both the thermodynamical and kinetic points of view. Our consideration results in an explicit construction of operator LSYM f (310) responsible for the rapid relaxation towards the state f. It can be used, in particular, for obtaining corrections to such approximations as the Grad moment approximations and the Tamm– Mott–Smith approximation, in frames of the method [4–6]. The nonthermodynamic part (311) is always present in Lf , when f = f0 , but if trajectories of an equation 9t f = Lf f are close to trajectories of an equation 9t f=LSYM f, then LSYM gives a good approximation to Lf . A conclusion f f on a closeness of trajectories depends on particular features of f, and normally it can be made on a base of a small parameter. On the other hand, the explicit thermodynamic and kinetic presentation of states on a manifold of slow motions (the extraction of LSYM performed above and construction f of hyper-planes 9f [4–6]) is based only the very idea of a decomposition of motions, and can be obtained with no consideration of a small parameter. Finally, though we have considered only the Boltzmann equation, the method of symmetrization can be applied to other dissipative systems with the same level of generality as the method [4–6]. 8. Relaxation methods Relaxation method is an alternative to the Newton iteration method described in Section 6: The initial approximation to the invariant manifold F0 is moved with the 8lm extension, Eq. (84), dFt (y) = (1 − Pt; y )J (Ft (y)) = GF(y) ; dt till a 8xed point is reached. Advantage of this method is a relative freedom in its implementation, because Eq. (84) needs not be solved exactly, one is interested only in 8nding 8xed points. Therefore, “large stepping” in the direction of the defect, GF(y) is possible, the termination point is de8ned by
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the condition that the vector 8eld becomes orthogonal to GF(y) . For simplicity, let us consider the procedure of termination in the linear approximation of the vector 8eld. Let F0 (y) be the initial approximation to the invariant manifold, and we seek the 8rst correction, F1 (y) = F0 (y) + 41 (y)GF0 (y) ; where function 4(y) has dimension of time, and is found from the condition that the linearized vector 8eld attached to the points of the new manifold is orthogonal to the initial defect, GF0 (y) |(1 − Py )[J (F0 (y)) + 41 (y)(Dx J )F0 (y) GF0 (y) ]F0 (y) = 0 :
(312)
Explicitly, 41 (y) = −
GF0 (y) |GF0 (y) F0 (y) : GF0 (y) |(Dx J )F0 (y) |GF0 (y) F0 (y)
(313)
Further steps 4k (y) are found in the same way. It is clear from the latter equations that the step of the relaxation method for the 8lm extension is equivalent to the Galerkin approximation for solving the step of the Newton method with incomplete linearization. Actually, the relaxation method was 8rst introduced in these terms in [12]. A partially similar idea of using the explicit Euler method to approximate the 8nite-dimensional invariant manifold on the basis of spectral decomposition was proposed earlier in Ref. [13]. An advantage of Eq. (313) is the explicit form of the size of the steps 4k (y). This method was successfully applied to the Fokker–Plank equation [12]. Example 7: Relaxation method for the Fokker–Planck equation Here we address the problem of closure for the FPE (31) in a general setting. First, we review the maximum entropy principle as a source of suitable quasi-equilibrium initial approximations for the closures. We also discuss a version of the maximum entropy principle, valid for a near-equilibrium dynamics, and which results in explicit formulae for arbitrary U and D. In this Example we consider the FPE of form (31): 9t W (x; t) = 9x · {D · [W 9x U + 9x W ]} :
(314)
Here W (x; t) is the probability density over the con8guration space x, at the time t, while U (x) and D(x) are the potential and the positively semi-de8nite (y · D · y ¿ 0) di6usion matrix. Quasi-equilibrium approximations for the Fokker–Planck equation. The quasi-equilibrium closures are almost never invariants of the true moment dynamics. For corrections to the quasiequilibrium closures, we apply the method of invariant manifold [6], which is carried out (subject to certain approximations explained below) to explicit recurrence formulae for one-moment near-equilibrium closures for arbitrary U and D. These formulae give a method for computing the lowest eigenvalue of the problem, and which dominates the near-equilibrium FPE dynamics. Results are tested with model potential, including the FENE-like potentials [113–115]. Let us denote as M the set of linearly independent moments {M0 ; M1 ; : ∗: : ; Mk }, where Mi [W ] = mi (x)W (x) d x, and where m0 =1. We assume that there exists a function W (M; x) which extremizes the entropy S (32) under the constrains of 8xed M . This quasi-equilibrium distribution function
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may be written
∗
W = Weq exp
k
299
Mi mi (x) − 1
;
(315)
i=0
where M = {M0 ; M1 ; : : : ; Mk } are Lagrange multipliers. Closed equations for moments M are derived in two steps. First, the quasi-equilibrium distribution (315) is substituted into the FPE (314) or (33) to give a formal expression: 9t W ∗ = Mˆ W ∗ (S=W )|W =W ∗ . Second, introducing a projector L∗ , k ∗ ∗ L •= (9W =9Mi ) m(x) • d x ; i=0
and applying L∗ on both sides of the formal expression, we derive closed equations for M in the quasi-equilibrium approximation. Further processing requires an explicit solution to the con strains, W ∗ (M; x)mi (x) d x = Mi , to get the dependence of Lagrange multipliers M on the moments M . Though typically the functions M(M ) are not known explicitly, one general remark about the moment equations is readily available. Speci8cally, the moment equations in the quasi-equilibrium approximation have the form M˙ i =
k
Mij∗ (M )(9S ∗ (M )=9Mj ) ;
(316)
j=0
where S ∗ (M ) = S[W ∗ (M )] is the macroscopic entropy, and where Mij∗ is an M -dependent (k + 1) × (k + 1) matrix: ∗ Mij = W ∗ (M; x)[9x mi (x)] · D(x) · [9x mj (x)] d x : The matrix Mij∗ is symmetric, positive semi-de8nite, and its kernel is the vector 0i . Thus, the quasi-equilibrium closure reproduces the GENERIC structure on the macroscopic level, the vector 8eld of macroscopic equations (316) is a metric transform of the gradient of the macroscopic entropy. The following version of the quasi-equilibrium closures makes it possible to derive more explicit results in the general case [190–192,178]: In many cases, one can split the set of moments M in two parts, MI = {M0 ; M1 ; : : : ; Ml } and MII = {Ml+1 ; : : : ; Mk }, in such a way that the quasi-equilibrium distribution can be constructed explicitly for MI as WI∗ (MI ; x). The full quasi-equilibrium problem for M = {MI ; MII } in the “shifted” formulation reads: extremize the functional S[WI∗ + YW ] with respect to YW , under the constrains MI [WI∗ + YW ] = MI and MII [WI∗ + YW ] = MII . Let us denote as YMII = MII − MII (MI ) deviations of the moments MII from their values in the MEP state WI∗ . For small deviations, the entropy is well approximated with its quadratic part YW 2 WI∗ 1 YS = − YW 1 + ln dx − dx : Weq 2 WI∗ Taking into account the fact that MI [WI∗ ] = MI , we come to the following maximizaton problem: YS[YW ] → max;
MI [YW ] = 0;
MII [YW ] = YMII :
(317)
The solution to problem (317) is always explicitly found from a (k + 1) × (k + 1) system of linear algebraic equations for Lagrange multipliers. This method was applied to systems of Boltzmann equations for chemical reacting gases [190,191], and for an approximate solution to the Boltzmann
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equation: scattering rates “moments of collision integral” are treated as independent variables, and as an alternative to moments of the distribution function, to describe the rare8ed gas near local equilibrium. Triangle version of the entropy maximum principle is used to derive the Grad-like description in terms of a 8nite number of scattering rates. The equations are compared to the Grad moment system in the heat nonconductive case. Estimations for hard spheres demonstrate, in particular, some 10% excess of the viscosity coeRcient resulting from the scattering rate description, as compared to the Grad moment estimation [192]. In the remainder of this section we deal solely with one-moment near-equilibrium closures: MI = M0 , (i.e. WI∗ = Weq ), and the set MII contains a single moment M = mW d x, m(x) = 1. We shall specify notations for the near-equilibrium FPE, writing the distribution function as W = Weq (1 + :), where the function : satis8es an equation: −1 ˆ J: ; 9t : = Weq
(318)
where Jˆ = 9x · [Weq D · 9x ]. The triangle one-moment quasi-equilibrium function reads: W (0) = Weq [1 + YMm(0) ]
(319)
where m(0) = [mm − m2 ]−1 [m − m] : (320) Here brackets : : : = Weq : : : d x denote equilibrium averaging. The superscript (0) indicates that the triangle quasi-equilibrium function (319) will be considered as the initial approximation to the procedure which we address below. Projector for the approximation (319) has the form m(0) (0) m(0) (x) d x : L • =Weq (0) (0) (321) m m Substituting function (319) into the FPE (318), and applying projector (321) on both the sides of the resulting formal expression, we derive the equation for M : M˙ = − 0 YM ;
(322)
where 1= 0 is an e6ective time of relaxation of the moment M to its equilibrium value, in the quasi-equilibrium approximation (319):
0 = m(0) m(0) −1 9x m(0) · D · 9x m(0) :
(323)
The invariance equation for the Fokker–Planck equation. Both the quasi-equilibrium and the triangle quasi-equilibrium closures are almost never invariants of the FPE dynamics. That is, the moments M of solutions to the FPE (314) vary in time di6erently from the solutions to the closed moment equations like (316), and these variations are generally signi8cant even for the near-equilibrium dynamics. Therefore, we ask for corrections to the quasi-equilibrium closures to 8nish with the invariant closures. This problem falls precisely into the framework of the method of invariant manifold [6], and we shall apply this method to the one-moment triangle quasi-equilibrium closing approximations.
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First, the invariant one-moment closure is given by an unknown distribution function W (∞) = Weq [1 + YMm(∞) (x)] which satis8es equation [1 − L(∞) ]Jˆm(∞) = 0 :
(324)
Here L(∞) is a projector, associated with an unknown function m(∞) , and which is also yet unknown. Eq. (324) is a formal expression of the invariance principle for a one-moment near-equilibrium closure: considering W (∞) as a manifold in the space of distribution functions, parameterized with the values of the moment M , we require that the microscopic vector 8eld Jˆm(∞) be equal to its projection, L(∞) Jˆm(∞) , onto the tangent space of the manifold W (∞) . Now we turn our attention to solving the invariance equation (324) iteratively, beginning with the triangle one-moment quasi-equilibrium approximation W (0) (319). We apply the following iteration process to Eq. (324): [1 − L(k) ]Jˆm(k+1) = 0 ;
(325)
where k = 0; 1; : : :, and where m(k+1) = m(k) + @(k+1) , and the correction satis8es the condition @(k+1) m(k) = 0. Projector is updated after each iteration, and it has the form m(k+1) (k+1) m(k+1) (x) d x : • =Weq (k+1) (k+1) (326) L m m Applying L(k+1) to the formal expression, Weq m(k+1) M˙ = YM [1 − L(k+1) ]m(k+1) ; we derive the (k + 1)th update of the e6ective time (323):
k+1 =
9x m(k+1) · D · 9x m(k+1) : m(k+1) m(k+1)
(327)
Specializing to the one-moment near-equilibrium closures, and following general argument [6], solutions to the invariance equation (324) are eigenfunctions of the operator Jˆ, while the formal limit of the iteration process (325) is the eigenfunction which corresponds to the eigenvalue with the minimal nonzero absolute value. Diagonal approximation. To obtain more explicit results, we shall now turn to an approximate solution to the problem (325) at each iteration. The correction @(k+1) satis8es the condition (k) m(k) @(k+1) = 0, and can be decomposed as follows: @(k+1) = 1k e(k) + eort . Here e(k) is the variance (k) − 1 (k) ˆ (k) (k) (k) of the kth approximation: e = Weq [1 − L ]J m = k m + R , where −1 ˆ (k) R(k) = Weq Jm :
(328)
(k) (k) (k) is orthogonal to both e(k) and m(k) (e(k) eort = 0, and m(k) eort = 0). The function eort (k) Our diagonal approximation (DA) consists in disregarding the part eort . In other words, we seek an improvement of the noninvariance of the kth approximation along its variance e(k) . Speci8cally, we consider the following ansatz at the kth iteration:
m(k+1) = m(k) + 1k e(k) :
(329)
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Table 2 Iterations k and the error k for U = x2 =2
Ex. 1
Ex. 2
0
1
4
8
12
16
20
1.99998 0:16 × 10−4
1.99993 0:66 × 10−4
1.99575 0:42 × 10−2
1.47795 0.24
1.00356 0:35 × 10−2
1.00001 0:13 × 10−4
1.00000 0:54 × 10−7
0
1
2
3
4
5
6
3.399 1.99
2.437 1.42
1.586 0.83
1.088 0.16
1.010 0:29 × 10−1
1.001 0:27 × 10−2
1.0002 0:57 × 10−3
Substituting ansatz (329) into Eq. (325), and integrating the latter expression with the function e(k) to evaluate the coeRcient 1k : 1k =
Ak − k2 ;
k3 − 2 k Ak + Bk
(330)
where parameters Ak and Bk represent the following equilibrium averages: Ak = m(k) m(k) −1 R(k) R(k) Bk = m(k) m(k) −1 9x R(k) · D · 9x R(k) :
(331)
Finally, putting together Eqs. (327)–(331), we arrive at the following DA recurrence solution, and which is our main result: m(k+1) = m(k) + 1k [ k m(k) + R(k) ] ;
(332)
k − (Ak − k2 )1k : 1 + (Ak − k2 )1k2
(333)
k+1 =
Notice that the stationary points of the DA process (333) are the true solutions to the invariance equation (324). What may be lost within the DA is the convergency to the true limit of the procedure (325), i.e. to the minimal eigenvalue. To test the convergency of the DA process (333) we have considered two potentials U in the FPE (314) with a constant di6usion matrix D. The 8rst test was with the square potential U = x2 =2, in the three-dimensional con8guration space, since for this potential the detail structure of the spectrum is well known. We have considered two examples of initial one-moment quasi-equilibrium closures with m(0) = x1 + 100(x2 − 3) (example 1), and m(0) = x1 + 100x6 x2 (example 2), in Eq. (320). The result of performance of the DA for k is presented in Table 2, together with the error k which was estimated as the norm of the variance at each iteration: k = e(k) e(k) =m(k) m(k) . In both examples, we see a good monotonic convergency to the minimal eigenvalue ∞ = 1, corresponding to the eigenfunction x1 . This convergency is even more striking in example 1, where the initial choice was very close to a di6erent eigenfunction x2 − 3, and which can be seen in the nonmonotonic behavior of the variance. Thus, we have an example to trust the DA approximation as converging to the proper object.
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Table 3 Iterations k for U = −50 ln(1 − x2 ) 0
1
2
3
4
5
6
7
8
Ex. 3
213.17
212.186
211.914
211.861
211.849
211.845
211.843
211.842
211.841
Ex. 4
216.586
213.135
212.212
211.998
211.929
211.899
211.884
211.876
211.871
For the second test, we have taken a one-dimensional potential U = −50 ln(1 − x2 ), the con8guration space is the segment |x| 6 1. Potentials of this type (so-called FENE potential) are used in applications of the FPE to models of polymer solutions [113–115]. Results are given in Table 3 for the two initial functions, m(0) = x2 + 10x4 − x2 + 10x4 (example 3), and m(0) = x2 + 10x8 − x2 + 10x8 (example 4). Both the examples demonstrate a stabilization of the k at the same value after some ten iterations. In conclusion, we have developed the principle of invariance to obtain moment closures for the Fokker–Planck equation (314), and have derived explicit results for the one-moment near-equilibrium closures, particularly important to get information about the spectrum of the FP operator.
9. Method of invariant grids Elsewhere above in this paper, we considered immersions F(y), and methods for their construction, without addressing the question of how to implement F in a constructive way. In most of the works (of us and of other people on similar problems), analytic forms were needed to represent manifolds (see, however, dual quasi-equilibrium integrators [198,199]). However, in order to construct manifolds of a relatively low dimension, grid-based representations of manifolds become a relevant option. The Method of invariant grids (MIG) was suggested recently in [10]. The principal idea of (MIG) is to 8nd a mapping of 8nite-dimensional grids into the phase space of a dynamic system. That is we construct not just a point approximation of the invariant manifold F ∗ (y), but an invariant grid. When re8ned, in the limit it is expected to converge, of course, to F ∗ (y), but it is a separate, independently de8ned object. Let’s denote L = Rn , G is a discrete subset of Rn . A natural choice would be a regular grid, but, this is not crucial from the point of view of the general formalism. For every point y ∈ G, a neighborhood of y is de8ned: Vy ⊂ G, where Vy is a 8nite set, and, in particular, y ∈ Vy . On regular grids, Vy includes, as a rule, the nearest neighbors of y. It may also include the points, next to the nearest. For our purposes, one should de8ne a grid di6erential operator. For every function, de8ned on the grid, also all derivatives are de8ned: 9f = qi (z; y)f(z); i = 1; : : : ; n : (334) 9yi y ∈G
z ∈Vy
where qi (z; y) are some coeRcients.
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Here we do not specify the choice of the functions qi (z; y). We just mention in passing that, as a rule, Eq. (334) is established using some interpolation of f in the neighborhood of y in Rn by some di6erentiable functions (for example, polynomial). This interpolation is based on the values of f at the points of Vy . For regular grids, qi (z; y) are functions of the di6erence z − y. For some ys which are close to the edges of the grid, functions are de8ned only on the part of Vy . In this case, the coeRcients in (334) should be modi8ed appropriately in order to provide an approximation using available values of f. Below we will assume this modi8cation is always done. We also assume that the number of points in the neighborhood Vy is always suRcient to make the approximation possible. This assumption restricts the choice of the grids G. Let’s call admissible all such subsets G, on which one can de8ne di6erentiation operator in every point. Let F be a given mapping of some admissible subset G ⊂ Rn into U . For every y ∈ V we de8ne tangent vectors: Ty = Lin{gi }n1 ;
(335)
where vectors gi (i = 1; : : : ; n) are partial derivatives (334) of the vector-function F: 9F = qi (z; y)F(z) ; gi = 9yi z∈V
(336)
y
or in the coordinate form 9Fj = qi (z; y)Fj (z) : (gi )j = 9yi z∈V
(337)
y
Here (gi )j is the jth coordinate of the vector (gi ), and Fj (z) is the jth coordinate of the point F(z). The grid G is invariant, if for every node y ∈ G the vector 8eld J (F(y)) belongs to the tangent space Ty (here J is the right hand site of the kinetic equations (76)). So, the de8nition of the invariant grid includes: (1) Finite admissible subset G ⊂ Rn ; (2) A mapping F of this admissible subset G into U (where U is the phase space for kinetic Eqs. (76)); (3) The di6erentiation formulae (334) with given coeRcients qi (z; y); The grid invariance equation has a form of inclusion: J (F(y)) ∈ Ty
for every y ∈ G ;
or a form of equation: (1 − Py )J (F(y)) = 0
for every y ∈ G ;
where Py is the thermodynamic projector (132). The grid di6erentiation formulae (334) are needed, in the 8rst place, to establish the tangent space Ty , and the null space of the thermodynamic projector Py in each node. It is important to realise that locality of construction of thermodynamic projector enables this without a need for a global parametrization.
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Basically, in our approach, the grid speci8cs are in: (a) di6erentiation formulae, (b) grid construction strategy (the grid can be extended, contracted, re8ned, etc.) The invariance equations (78), equations of the 8lm dynamics extension (84), the iteration Newton method (167), and the formulae of the relaxation approximation (313) do not change at all. For convenience, let us repeat all these formulae in the grid context. Let x=F(y) be position of a grid’s node y immersed into U . We have set of tangent vectors gi (x), de8ned in x (336), (337). Thus, the tangent space Ty is de8ned by (335). Also, one has entropy function S(x), the linear functional Dx S|x , and the subspace T0y = Ty ∩ ker Dx S|x in Ty . Let T0y = Ty . In this case we have a vector ey ∈ Ty , orthogonal to T0y , Dx S|x (ey ) = 1. Then, the thermodynamic projector is de8ned as Py • =P0y • +ey Dx S|x • ;
(338)
where P0y is the orthogonal projector on T0y with respect to the entropic scalar product |x . If T0y = Ty , then the thermodynamic projector is the orthogonal projector on Ty with respect to the entropic scalar product |x . For the Newton method with incomplete linearization, the equations for calculating new node position x = x + x are: Py x = 0 ; (1 − Py )(J (x) + DJ (x)x) = 0 :
(339)
Here DJ (x) is a matrix of derivatives of J , calculated in x. The self-adjoint linearization may be useful too (see Section 8). Eq. (339) is a system of linear algebraic equations. In practice, it is convenient to choose some orthonormal(with respect to the entropic scalar product) basis bi in ker Py . Let r = dim(ker Py ). Then x = ri=1 i bi , and the system looks like r
k bi | DJ (x)bk x = −J (x) | bi x ;
i = 1:::r :
(340)
k=1
Here |x is the entropic scalar product. This is the system of linear equations for adjusting the node position accordingly to the Newton method with incomplete linearization. For the relaxation method, one needs to calculate the defect Gx = (1 − Py )J (x), and the relaxation step 4(x) = −
Gx | Gx x : Gx | DJ (x)Gx x
(341)
Then, new node position x is calculated as x = x + 4(x)Gx : This is the equation for adjusting the node position according to the relaxation method.
(342)
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9.1. Grid construction strategy From all reasonable strategies of the invariant grid construction we will consider here the following two: growing lump and invariant Kag. 9.1.1. Growing lump In this strategy one chooses as initial the equilibrium point y∗ . The 8rst approximation is constructed as F(y∗ ) = x∗ , and for some initial V0 (Vy∗ ⊂ V0 ) one has F(y) = x∗ + A(y − y∗ ), where A is an isometric embedding (in the standard Euclidean metrics) of Rn in E. For this initial grid one makes a 8xed number of iterations of one of the methods chosen (Newton’s method with incomplete linearization or the relaxation method), and, after that, puts V1 = y∈V0 Vy and extends F from V0 onto V1 using linear extrapolation and the process continues. One of the possible variants of this procedure is to extend the grid from Vi to Vi+1 not after a 8xed number of iterations, but when invariance defect Gy becomes smaller than a given j (in a given norm, which is entropic, as a rule), for all nodes y ∈ Vi . The lump stops growing when it reaches the boundary and is within a given accuracy G ¡ j. 9.1.2. Invariant Kag For the invariant =ag one uses suRciently regular grids G, in which many points are situated on the coordinate lines, planes, etc. One considers the standard =ag R0 ⊂ R1 ⊂ R2 ⊂ · · · ⊂ Rn (every next space is constructed by adding one more coordinate). It corresponds to a succession of grids {y} ⊂ G 1 ⊂ G 2 · · · ⊂ G n , where {y∗ } = R0 , and G i is a grid in Ri . First, y∗ is mapped in x∗ and further F(y∗ )=x∗ . Then an invariant grid is constructed on V 1 ⊂ G 1 (up to the boundaries U and within a given accuracy G ¡ j). After the neighborhoods in G 2 are added to the points V 1 , and, using such extensions, the grid V 2 ⊂ G 2 is constructed (up to the boundaries and within a given accuracy) and so on, until V n ⊂ G n will be constructed. We must underline here that, constructing the kth grid V k ⊂ G k , the important role of the grids of smaller dimension V 0 ⊂ · · · ⊂ V k −1 ⊂ V k embedded in it, is preserved. The point F(y∗ ) = x∗ is preserved. For every y ∈ V q (q ¡ k) the tangent vectors g1 ; : : : ; gq are constructed, using the di6erentiation operators (334) on the whole V k . Using the tangent space Ty = Lin{g1 ; : : : ; gq }, the projector Py is constructed, the iterations are applied and so on. All this is done to obtain a succession of embedded invariant grids, given by the same map F. 9.1.3. Boundaries check and the entropy We construct grid mapping of F onto the 8nite set V ∈ G. The technique of checking if the grid still belongs to the phase space U of kinetic system U (F(V ) ⊂ U ) is quite straightforward: all the points y ∈ V are checked to belong to U . If at the next iteration a point F(y) leaves U , then it is returned inside by a homothety transform with the center in x∗ . Since the entropy is a concave function, the homothety contraction with the center in x∗ increases the entropy monotonously. Another variant is cutting o6 the points leaving U . By the way it was constructed, (132), the kernel of the entropic projector is annulled by the entropy di6erential. Thus, in the 8rst order, steps in the Newton method with incomplete linearization (167) as well as in the relaxation methods (312), (313) do not change the entropy. But, if the steps are
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307
Iteration 1 Iteration 2 Iteration 3 Iteration 4
Fig. 8. Grid instability. For small grid steps approximations in the calculation of grid derivatives lead to the grid instability e6ect. On the 8gure several successive iterations of the algorithm without adaptation of the time step are shown that lead to undesirable “oscillations”, which eventually destruct the grid starting from one of it’s ends.
quite large, then the increasing of the entropy can become essential and the points are returned on their entropy level by the homothety contraction with the center in the equilibrium point. 9.2. Instability of 1ne grids When one reduces the grid step (spacing between the nodes) in order to get a 8ner grid, then, starting from a de8nite step, it is possible to face the problem of the Courant instability [202–204]. Instead of converging, at the every iteration the grid becomes more entangled (see Fig. 8). The way to get rid o6 this instability is well-known. This is decreasing the time step. Instead of the real time step, we have a shift in the Newtonian direction. Formally, we can assign for one complete step in the Newtonian direction a value h = 1. Let us consider now the Newton method with an arbitrary h. For this, let us 8nd x =F(y) from (339), but we will change x proportionally to h: the new value of x n+1 = Fn+1 (y) will be equal to Fn+1 (y) = Fn (y) + hn Fn (y)
(343)
where the lower index n denotes the step number. One way to choose the h step value is to make it adaptive, controlling the average value of the hn plays a invariance defect Gy at every step. Another way is the convergence control: then role of time. Elimination of Courant instability for the relaxation method can be made quite analogously. Everywhere the step h is maintained as big as it is possible without convergence problems. 9.3. What space is the most appropriate for the grid construction? For the kinetics systems there are two distinguished representations of the phase space: • The densities space (concentrations, energy or probability densities, etc.). • The spaces of conjugate intensive quantities, potentials (temperature, chemical potentials, etc.).
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The density space is convenient for the construction of quasi-chemical representations. Here the balance relations are linear and the restrictions are in the form of linear inequalities (the densities themselves or some linear combinations of them must be positive). The conjugate variables space is convenient in the sense that the equilibrium conditions, given the linear restrictions on the densities, are in the linear form (with respect to the conjugate variables). In these spaces the quasi-equilibrium manifolds exist in the form of linear subspaces and, vise versa, linear balance equations turns out to be equations of the conditional entropy maximum. The duality we have just described is very well-known and studied in details in many works on thermodynamics and Legendre transformations [207,208]. This viewpoint of nonequilibrium thermodynamics uni8es many well-established mesoscopic dynamical theories, as for example the Boltzmann kinetic theory and the Navier–Stokes–Fourier hydrodynamics [209]. In the previous section, the grids were constructed in the density space. But the procedure of constructing them in the space of the conjugate variables seems to be more consistent. The principal argument for this is the speci8c role of quasi-equilibrium, which exists as a linear manifold. Therefore, linear extrapolation gives a thermodynamically justi8ed quasi-equilibrium approximation. Linear approximation of the slow invariant manifold in the neighborhood of the equilibrium in the conjugate variables space already gives the global quasi-equilibrium manifold, which corresponds to the motion separation (for slow and fast motions) in the neighborhood of the equilibrium point. For the mass action law, transition to the conjugate variables is simply the logarithmic transformation of the coordinates.
9.4. Carleman’s formulae in the analytical invariant manifolds approximations. First bene1t of analyticity: superresolution When constructing invariant grids, one must de8ne the di6erential operators (334) for every grid node. For calculating the di6erential operators in some point y, an interpolation procedure in the neighborhood of y is used. As a rule, it is an interpolation by a low-order polynomial, which is constructed using the function values in the nodes belonging to the neighbourhood of y in G. This approximation (using values in the closest nodes) is natural for smooth functions. But, we are looking for the analytical invariant manifold (see discussion in the section: “8lm extension: analyticity instead of the boundary conditions”). Analytical functions have much more “rigid” structure than the smooth ones. One can change a smooth function in the neighborhood of any point in such a way, that outside this neighborhood the function will not change. In general, this is not possible for analytical functions: a kind of “long-range” e6ect takes place (as is well known). The idea is to use this e6ect and to reconstruct some analytical function fG using function given on G. There is one important requirement: if these values on G are values (given at the points of G) of some function f which is analytical in the given neighborhood U , then if the G is re8ned “correctly”, one must have fG → f. The sequence of reconstructed function fG should converge to the “proper” function f. What is the “correct re8nement”? For smooth functions for the convergence fG → f it is necessary and suRcient that, in the course of re8nement, G would approximate the whole U with arbitrary accuracy. For analytical functions it is necessary only that, under the re8nement, G would
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approximate some uniqueness set 12 A ⊂ U . Suppose we have a sequence of grids G, each next is 8ner than previous, which approximates a set A. For smooth functions, using function values de8ned on the grids, one can reconstruct the function in A. For analytical functions, if the analyticity area U is known, and A is a uniqueness set in U , then one can reconstruct the function in U . The set U can be essentially bigger than A; because of this such extension was named as superresolution eFects [210]. There exist constructive formulae for construction of analytical functions fG for di6erent areas U , uniqueness sets A ⊂ U and for di6erent ways of discrete approximation of A by a sequence of 8ned grids G [210]. Here we provide only one Carleman’s formula which is the most appropriate for our purposes. Let area U = Q n ⊂ C n be a product of strips Q ⊂ C, Q = {z | Im z ¡ }. We will construct functions holomorphic in Q n . This is e6ectively equivalent to the construction of real analytical functions f in whole Rn with a condition on the convergence radius r(x) of the Taylor series for f as a function of each coordinate: r(x) ¿ in every point x ∈ Rn . The sequence of 8ned grids is constructed as follows: let for every l = 1; : : : ; n a 8nite sequence of distinct points Nl ⊂ Q be de8ned: Nl = {xlj | j = 1; 2; 3 : : :};
xlj = xli for i = j :
(344)
The uniqueness set A, which is approximated by a sequence of 8ned 8nite grids, has the form A = N1 × N2 × · · · × Nn = {(x1i1 ; x2i2 ; : : : ; x nin ) | i1; :::; n = 1; 2; 3; : : :} :
(345)
The grid Gm is de8ned as the product of initial fragments Nl of length m: Gm = {(x1i1 ; x2i2 : : : x nin ) | 1 6 i1; :::; n 6 m} :
(346)
Let’s denote = 2 =, ( is a half-width of the strip Q ). The key role in the construction of the Carleman’s formula is played by the functional !m (u; p; l) of three variables: u ∈ U = Q n , p is an integer, 1 6 p 6 m, l is an integer, 1 6 p 6 n. Further u will be the coordinate value at the point where the extrapolation is calculated, l will be the coordinate number, and p will be an element of multi-index {i1 ; : : : ; in } for the point (x1i1 ; x2i2 ; : : : ; x nin ) ∈ G: m (e xlp + e x]lp )(e u − e xlp ) (e xlp + e x]lj )(e u − e xlj )
!m (u; p; l) = (347)
(e u + e x]lp )(u − xlp )e xlp (e xlp − e xlj )(e u + e x]lj ) j=1 j =p
For real-valued xpk formula (347) becomes simpler: m e u − e xlp (e xlp + e xlj )(e u − e xlj ) !m (u; p; l) = 2
x
u
(e + e lp )(u − xlp ) (e xlp − e xlj )(e u + e xlj )
(348)
j=1 j =p
The Carleman’s formula for extrapolation from GM on U = Q n ( = , =2) has the form (z = (z1 ; : : : ; zn )): m n f(xk ) !m (zj ; kj ; j) ; (349) fm (z) = k1 ;:::; kn =1
j=1
where k = k1 ; : : : ; kn , xk = (x1k1 ; x2k2 ; : : : ; x nkn ). 12
Let’s remind to the reader that A ⊂ U is called uniqueness set in U if for analytical in U functions |A ≡ ’|A it follows = ’.
and ’ from
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There exists a theorem [210]: If f ∈ H 2 (Q n ), then f(z) = limm→∞ fm (z), where H 2 (Q n ) is the Hardy class of holomorphic in n Q functions. It is useful to present the asymptotics of (349) for big |Re zj |. For this we will consider the asymptotics of (349) for big |Reu|: m
x
x lp lj 2 e + e
|!m (u; p; l)| = + o(|Reu|−1 ) : (350)
xlp − e xlj
u e j=1 j=p From formula ! (349) one can see that for the 8nite m and |Re zj | → ∞ function |fm (z)| behaves like const · j | zj |−1 . This property (zero asymptotics) must be taken into account when using formula (349). When constructing invariant manifolds F(W ), it is natural to use (349) not for the immersion F(y), but for the deviation of F(y) from some analytical ansatz F0 (y) [211–214]. The analytical ansatz F0 (y) can be obtained using Taylor series, just as in the Lyapunov auxiliary theorem [144] (also see above in the sections about the 8lm extensions). Another variant is using Taylor series for the construction of Pade-approximations. It is natural to use approximations (349) in dual variables as well, since there exists for them (as the examples demonstrate) a simple and very e6ective linear ansatz for the invariant manifold. This is the slow invariant subspace Eslow of the operator of linearized system (76) in dual variables in the equilibrium point. This invariant subspace corresponds to the set of “slow” eigenvalues (with small |Re |, Re ¡ 0). In the initial space (of concentrations or densities) this invariant subspace is the quasi-equilibrium manifold. It consists of the maximal entropy points on the aRne manifolds of the x + Efast form, where Efast is the “fast” invariant subspace of the operator of linearized system (76) in the initial variables in the equilibrium point. It corresponds to the “fast” eigenvalues (big |Re |, Re ¡ 0). In the problem of invariant grids constructing we can use the Carleman’s formulae in two moments: 8rst, for the de8nition grid di6erential operators (334), second, for the analytical continuation the manifold from the grid. Example 8: Two-step catalytic reaction Let us consider a two-step four-component reaction with one catalyst A2 : A 1 + A 2 ↔ A3 ↔ A2 + A 4 :
4
(351)
eq i=1 ci [ln(ci =ci )
− 1]. The kinetic We assume the Lyapunov function of the form S = −G = − equation for the four-component vector of concentrations, c = (c1 ; c2 ; c3 ; c4 ), has the form c˙ = >1 W1 + >2 W2 :
(352)
Here >1; 2 are stoichiometric vectors, >1 = (−1; −1; 1; 0);
>2 = (0; 1; −1; 1) ;
(353)
while functions W1; 2 are reaction rates: W1 = k1+ c1 c2 − k1− c3 ;
W2 = k2+ c3 − k2− c2 c4 :
(354)
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0.2
c3
0.15 0.1 0.05 0 1 1 0.5
0.5
c4 0
c1
0
Fig. 9. One-dimensional invariant grid (circles) for two-dimensional chemical system. Projection into the 3d-space of c1 , c4 , c3 concentrations. The trajectories of the system in the phase space are shown by lines. The equilibrium point is marked by square. The system quickly reaches the grid and further moves along it.
Here k1;±2 are reaction rate constants. The system under consideration has two conservation laws, c 1 + c 3 + c 4 = B1 ;
c2 + c3 = B2 ;
(355)
or b1; 2 ; c = B1; 2 , where b1 = (1; 0; 1; 1) and b1 = (0; 1; 1; 0). The nonlinear system (351) is e6ectively two-dimensional, and we consider a one-dimensional reduced description. For our example, we chose the following set of parameters: k1+ = 0:3;
k1− = 0:15;
c1eq = 0:5;
c2eq = 0:1;
B1 = 1:0;
B2 = 0:2 :
k2+ = 0:8; c3eq = 0:1;
k2− = 2:0 ; c4eq = 0:4 ; (356)
In Fig. 9 one-dimensional invariant grid is shown in the (c1 ; c4 ; c3 ) coordinates. The grid was constructed by growing the grid, as described above. We used Newtonian iterations to adjust the nodes. The grid was grown up to the boundaries of the phase space. The grid derivatives for calculating tangent vectors g were taken as simple as g(xi ) = (xi+1 − xi−1 )= xi+1 −xi−1 for the internal nodes and g(x1 )=(x1 −x2 )= x1 −x2 , g(x n )=(x n −x n−1 )= x n −x n−1 for the grid’s boundaries. Here xi denotes the vector of the ith node position, n is the number of nodes in the grid. Close to the phase space boundaries we had to apply an adaptive algorithm for choosing the time step h: if, after the next growing step and applying N = 20 complete Newtonian steps, the grid did not converged, then we choose a new hn+1 = hn =2 and recalculate the grid. The 8nal value for h was h ≈ 0:001. The nodes positions are parametrized with entropic distance to the equilibrium point measured in the quadratic metrics given by Hc = − 92 S(c)=9ci 9cj in the equilibrium ceq . It means that every node is on a sphere in this quadratic metrics with a given radius, which increases linearly. On this 8gure the step of the increase is chosen to be 0.05. Thus, the 8rst node is on the distance 0.05 from the equilibrium, the second is on the distance 0.10 and so on. Fig. 10 shows several basic values
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Fig. 10. One-dimensional invariant grid for two-dimensional chemical system. (a) Values of the concentrations along the grid. (b) Values of the entropy and the entropy production (−dG=dt) along the grid. (c) Relation of the relaxation times “toward” and “along” the manifold. The nodes positions are parametrized with entropic distance measured in the quadratic metrics given by Hc = −92 S(c)=9ci 9cj in the equilibrium ceq . Zero corresponds to the equilibrium.
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which facilitate understanding of the object (invariant grid) extracted. The sign on the x-axis of the graphs at Fig. 10 is meaningless, since the distance is always positive, but in this situation it denotes two possible directions from the equilibrium point. Fig. 10a,b e6ectively represents the slow one-dimensional component of the dynamics of the system. Given any initial condition, the system quickly 8nds the corresponding point on the manifold and starting from this point the dynamics is given by a part of the graph on the Fig. 10a,b. One of the useful values is shown on the Fig. 10c. It is the relation between the relaxation times “toward” and “along” the grid ( 2 = 1 , where 1 ; 2 are the smallest and the second smallest by absolute value nonzero eigenvalue of the system, symmetrically linearized at the point of the grid node). It shows that the system is very sti6 close to the equilibrium point, and less sti6 (by one order of magnitude) on the borders. This leads to the conclusion that the reduced model is more adequate in the neighborhood of the equilibrium where fast and slow motions are separated by two orders of magnitude. On the very end of the grid which corresponds to the positive absciss values, our one-dimensional consideration faces with de8nite problems (slow manifold is not well-de8ned). Example 9: Model hydrogen burning reaction In this section we consider a more interesting illustration, where the phase space is six-dimensional, and the system is four-dimensional. We construct an invariant =ag which consists of one- and two-dimensional invariant manifolds. We consider chemical system with six species called (provisionally) H2 (hydrogen), O2 (oxygen), H2 O (water), H, O, OH (radicals). We assume the Lyapunov function of the form S = −G = − 6i=1 ci [ln(ci =cieq ) − 1]. The subset of the hydrogen burning reaction and corresponding (direct) rate constants have been taken as: 1: H2 ↔ 2H;
k1+ = 2 ;
2: O2 ↔ 2O;
k2+ = 1 ;
3: H2 O ↔ H + OH;
k3+ = 1 ;
4: H2 + O ↔ H + OH;
k4+ = 103 ;
5: O2 + H ↔ O + OH;
k5+ = 103 ;
6: H2 + O ↔ H2 O;
k6+ = 102 :
(357)
The conservation laws are: 2cH2 + 2cH2 O + cH + cOH = bH ; 2cO2 + cH2 O + cO + cOH = bO :
(358)
For parameter values we took bH = 2, bO = 1, and the equilibrium point: cHeq2 = 0:27
cOeq2 = 0:135
cHeq2 O = 0:7
cHeq = 0:05
cOeq = 0:02
eq cOH = 0:01 :
(359)
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Other rate constants ki− , i = 1::6 were calculated from ceq value and ki+ . For this system the stoichiometric vectors are: >1 = (−1; 0; 0; 2; 0; 0); >3 = (0; 0; −1; 1; 0; 1); >5 = (0; −1; 0; −1; 1; 1);
>2 = (0; −1; 0; 0; 2; 0) ; >4 = (−1; 0; 0; 1; −1; 1) ; >6 = (−1; 0; 1; 0; −1; 0) :
(360)
We stress here once again that the system under consideration is 8ctional in that sense that the subset of equations corresponds to the simpli8ed picture of this physical–chemical process and the constants do not correspond to any measured ones, but re=ect only basic orders of magnitudes of the real-world system. In this sense we consider here a qualitative model system, which allows us to illustrate the invariant grids method without excessive complication. Nevertheless, modeling of real systems di6ers only in the number of species and equations. This leads, of course, to computationally harder problems, but not the crucial ones, and the e6orts on the modeling of real-world systems are on the way. Fig. 11a presents a one-dimensional invariant grid constructed for the system. Fig. 11b shows the picture of reduced dynamics along the manifold (for the explanation of the meaning of the x-coordinate, see the previous subsection). In Fig. 11c the three smallest by absolute value nonzero eigenvalues of the symmetrically linearized system Asym have been shown. One can see that the two smallest values almost “exchange” on one of the grid end. It means that one-dimensional “slow” manifold has de8nite problems in this region, it is just not well de8ned there. In practice, it means that one has to use at least two-dimensional grids there. Fig. 12a gives a view onto the two-dimensional invariant grid, constructed for the system, using the “invariant =ag” strategy. The grid was grown starting from the 1D-grid constructed at the previous step. At the 8rst iteration for every node of the initial grid, two nodes (and two edges) were added. The direction of the step was chosen as the direction of the eigenvector of the matrix Asym (at the point of the node), corresponding to the second “slowest” direction. The value of the step was chosen to be j = 0:05 in terms of entropic distance. After several Newtonian iterations done until convergence, new nodes were added in the direction “orthogonal” to the 1D-grid. This time it is done by linear extrapolation of the grid on the same step j = 0:05. When some new nodes have one or several negative coordinates (the grid reaches the boundaries) they were cut o6. If a new node has only one edge, connecting it to the grid, it was excluded (since it does not allow calculating 2D-tangent space for this node). The process continues until the expansion is possible (after this, every new node has to be cut o6). The method for calculating tangent vectors for this regular rectangular 2D-grid was chosen to be quite simple. The grid consists of rows, which are co-oriented by construction to the initial 1D-grid, and columns that consist of the adjacent nodes in the neighboring rows. The direction of “columns” corresponds to the second slowest direction along the grid. Then, every row and column is considered as 1D-grid, and the corresponding tangent vectors are calculated as it was described before: grow (xk; i ) = (xk; i+1 − xk; i−1 )= xk; i+1 − xk; i−1 for the internal nodes and grow (xk; 1 ) = (xk; 1 − xk; 2 )= xk; 1 − xk; 2 ;
grow (xk; nk ) = (xk; nk − xk; nk −1 )= xk; nk − xk; nk −1
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for the nodes which are close to the grid’s edges. Here xk; i denotes the vector of the node in the kth row, ith column; nk is the number of nodes in the kth row. Second tangent vector gcol (xk; i ) is calculated completely analogously. In practice, it is convenient to orthogonalize grow (xk; i ) and gcol (xk; i ).
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Fig. 12. Two-dimensional invariant grid for the model hydrogen burning system. (a) Projection into the 3d-space of cH , cO , cOH concentrations. (b) Projection into the principal 3D-subspace. Trajectories of the system are shown coming out from the every grid node. Bold line denotes the one-dimensional invariant grid, starting from which the 2D-grid was constructed.
Since the phase space is four-dimensional, it is impossible to visualize the grid in one of the coordinate 3D-views, as it was done in the previous subsection. To facilitate visualization one can utilize traditional methods of multi-dimensional data visualization. Here we make use of the principal components analysis (see, for example, [206]), which constructs a three-dimensional linear subspace with maximal dispersion of the othogonally projected data (grid nodes in our case). In other words, method of principal components constructs in multi-dimensional space such a three-dimensional box inside which the grid can be placed maximally tightly (in the mean square distance meaning). After projection of the grid nodes into this space, we get more or less adequate representation of the two-dimensional grid embedded into the six-dimensional concentrations space (Fig. 12b). The
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disadvantage of the approach is that the axes now do not have explicit meaning, being some linear combinations of the concentrations. One attractive feature of two-dimensional grids is the possibility to use them as a screen, on which one can display di6erent functions f(c) de8ned in the concentrations space. This technology was exploited widely in the nonlinear data analysis by the elastic maps method [205]. The idea is to “unfold” the grid on a plane (to present it in the two-dimensional space, where the nodes form a regular lattice). In other words, we are going to work in the internal coordinates of the grid. In our case, the 8rst internal coordinate (let’s call it s1 ) corresponds to the direction, co-oriented with the one-dimensional invariant grid, the second one (let’s call it s2 ) corresponds to the second slow direction. By how it was constructed, s2 = 0 line corresponds to the one-dimensional invariant grid. Units of s1 and s2 are entropic distances in our case. Every grid node has two internal coordinates (s1 ; s2 ) and, simultaneously, corresponds to a vector in the concentration space. This allows us to map any function f(c) from the multi-dimensional concentration space to the two-dimensional space of the grid. This mapping is de8ned in a 8nite number of points (grid nodes), and can be interpolated (linearly, in the simplest case) in between them. Using coloring and isolines one can visualize the values of the function in the neighborhood of the invariant manifold. This is meaningful, since, by the de8nition, the system spends most of the time in the vicinity of the invariant manifold, thus, one can visualize the behavior of the system. As a result of applying the technology, one obtains a set of color illustrations (a stack of information layers), put onto the grid as a map. This allows applying all the methods, working with stack of information layers, like geographical information systems (GIS) methods, which are very well developed. In short words, the technique is a useful tool for exploration of dynamical systems. It allows to see simultaneously many di6erent scenarios of the system behavior, together with di6erent system’s characteristics. The simplest functions to visualize are the coordinates: ci (c) = ci . In Fig. 13 we displayed four colorings, corresponding to the four arbitrarily chosen concentrations functions (of H2 , O, H and OH; Fig. 13a–d). The qualitative conclusions that can be made from the graphs are that, for example, the concentration of H2 practically does not change during the 8rst fast motion (towards the 1D-grid) and then, gradually changes to the equilibrium value (the H2 coordinate is “slow”). The O coordinate is the opposite case, it is “fast” coordinate which changes quickly (on the 8rst stage of motion) to the almost equilibrium value, and then it almost does not change. Basically, the slope angles of the coordinate isolines give some presentation of how “slow” a given concentration is. Fig. 13c shows interesting behavior of the OH concentration. Close to the 1D grid it behaves like “slow coordinate”, but there is a region on the map where it has clear “fast” behavior (middle bottom of the graph). The next two functions which one can want to visualize are the entropy S and the entropy production (c) = −dG=dt(c) = i ln(ci =cieq )c˙i . They are shown in Fig. 14a,b. Finally, we visualize the relation between the relaxation times of the fast motion towards the 2D-grid and along it. This is given in Fig. 14c. This picture allows to make a conclusion that two-dimensional consideration can be appropriate for the system (especially in the “high H2 , high O” region), since the relaxation times “towards” and “along” the grid are de8nitely separated. One can compare this to Fig. 14d, where the relation between relaxation times towards and along the 1D-grid is shown.
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10. Method of natural projector Ehrenfest suggested in 1911 a model of dynamics with a coarse-graining of the original conservative system in order to introduce irreversibility [215]. The idea of Ehrenfest is the following: One partitions the phase space of the Hamiltonian system into cells. The density distribution of the ensemble over the phase space evolves in time according to the Liouville equation within the time segments n4 ¡ t ¡ (n + 1)4, where 4 is the 8xed coarse-graining time step. Coarse-graining is executed at discrete times n4, densities are averaged over each cell. This alternation of the regular =ow with the averaging describes the irreversible behavior of the system. The formally most general construction extending the Ehrenfest idea is given below. Let us stay with notation of Section 3, and let a submanifold F(W ) be de8ned in the phase space U . Furthermore, we assume a map (a projection) is de8ned, L : U → W , with the properties: L ◦ F = 1;
L(F(y)) = y :
(361)
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Fig. 14. Two-dimensional invariant grid as a screen for visualizing di6erent functions de8ned in the concentrations space. The coordinate axes are entropic distances (see the text for the explanations) along the 8rst and the second slowest directions on the grid. The corresponding 1D invariant grid is denoted by bold line, the equilibrium is denoted by square. (a) Entropy. (b) Entropy production. (c) 3 = 2 relation. (d) 2 = 1 relation.
In addition, one requires some mild properties of regularity, in particular, surjectivity of the di6erential, Dx L : E → L, in each point x ∈ U . Let us 8x the coarse-graining time 4 ¿ 0, and consider the following problem: Find a vector 8eld : in W , dy = :(y) ; (362) dt such that, for every y ∈ W , L(T4 F(y)) = W4 y ;
(363)
where T4 is the shift operator for system (76), and W4 is the (yet unknown!) shift operator for the system in question (362). Eq. (363) means that one projects not the vector 8elds but segments of trajectories. Resulting vector 8eld :(y) is called the natural projection of the vector 8eld J (x).
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Let us assume that there is a very sti6 hierarchy of relaxation times in system (76): The motions of the system tend very rapidly to a slow manifold, and next proceed slowly along it. Then there is a smallness parameter, the ratio of these times. Let us take F for the initial condition to the 8lm equation (84). If the solution Ft relaxes to the positively invariant manifold F∞ , then, in the limit of a very sti6 decomposition of motions, the natural projection of the vector 8eld J (x) tends to the usual in8nitesimal projection of the restriction of J on F∞ , as 4 → ∞: :∞ (y) = Dx L|x=F∞ (y) J (F∞ (y)) :
(364)
For sti6 dynamic systems, the limit (364) is qualitatively almost obvious: After some relaxation time 40 (for t ¿ t0 ), the motion T4 (x) is located in an j-neighborhood of F∞ (W ). Thus, for 440 , the natural projection : (Eqs. (362) and (363)) is de8ned by the vector 8eld attached to F∞ with any prede8ned accuracy. Rigorous proofs requires existence and uniqueness theorems, as well as homogeneous continuous dependence of solutions on initial conditions and right hand sides of equations. The method of natural projector is applied not only to dissipative systems but also (and even mostly) to conservative systems. One of the methods to study the natural projector is based on series expansion 13 in powers of 4. Various other approximation schemes like Pade approximation are possible too. The construction of natural projector was rediscovered in completely di6erent context by Chorin et al. [217]. They constructed the optimal prediction methods for estimation the solution of nonlinear time-dependent problems when that solution is too complex to be fully resolved or when data are missing. The initial conditions for the unresolved components of the solution are drawn from a probability distribution, and their e6ect on a small set of variables that are actually computed is evaluated via statistical projection. The formalism resembles the projection methods of irreversible statistical mechanics, supplemented by the systematic use of conditional expectations and methods of solution for the orthogonal dynamics equation, needed to evaluate a nonMarkovian memory term. The result of the computations is close to the best possible estimate that can be obtained given the partial data. Most of the methods of invariant manifold can be discussed as development of the Chapman– Enskog method. The idea is to construct the manifold of distribution functions, where the slow dynamics occurs. The change-over from solution of the Boltzmann equation to construction of an invariant manifold was a crucial idea of Enskog and Chapman. On the other hand, the method of natural projector gives development of ideas of the Hilbert method. This method was historically the 8rst in the solution of the Boltzmann equation. The Hilbert method is not very popular now, nevertheless, for some purposes it may be more convenient than the Chapman–Enskog method, for example, for a studying of stationary solutions [218]. In the method of natural projector we are looking for a solutions of kinetic equations with quasi-equilibrium initial state (and in Hilbert method we start from the local equilibrium too). The main new element in the method of natural projector with respect to the Hilbert method is construction of the macroscopic equation (363). In the 13
In the well known work of Lewis [216], this expansion was executed incorrectly (terms of di6erent orders were matched on the left and on the right hand sides of Eq. (363)). This created an obstacle in a development of the method. See more detailed discussion in the section Example 10.
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next example the solution for the matching condition (363) will be found in a form of Taylor expansion. Example 10: From reversible dynamics to Navier–Stokes and post-Navier–Stokes hydrodynamics by natural projector The starting point of our construction are microscopic equations of motion. A traditional example of the microscopic description is the Liouville equation for classical particles. However, we need to stress that the distinction between “micro” and “macro” is always context dependent. For example, Vlasov’s equation describes the dynamics of the one-particle distribution function. In one statement of the problem, this is a microscopic dynamics in comparison to the evolution of hydrodynamic moments of the distribution function. In a di6erent setting, this equation itself is a result of reducing the description from the microscopic Liouville equation. The problem of reducing the description includes a de8nition of the microscopic dynamics, and of the macroscopic variables of interest, for which equations of the reduced description must be found. The next step is the construction of the initial approximation. This is the well known quasi-equilibrium approximation, which is the solution to the variational problem, S → max, where S in the entropy, under given constraints. This solution assumes that the microscopic distribution functions depend on time only through their dependence on the macroscopic variables. Direct substitution of the quasi-equilibrium distribution function into the microscopic equation of motion gives the initial approximation to the macroscopic dynamics. All further corrections can be obtained from a more precise approximation of the microscopic as well as of the macroscopic trajectories within a given time interval 4 which is the parameter of our method. The method described here has several clear advantages: (i) It allows to derive complicated macroscopic equations, instead of writing them ad hoc. This fact is especially signi8cant for the description of complex =uids. The method gives explicit expressions for relevant variables with one unknown parameter (4). This parameter can be obtained from the experimental data. (ii) Another advantage of the method is its simplicity. For example, in the case where the microscopic dynamics is given by the Boltzmann equation, the approach avoids evaluation of Boltzmann collision integral. (iii) The most signi8cant advantage of this formalization is that it is applicable to nonlinear systems. Usually, in the classical approaches to reduced description, the microscopic equation of motion is linear. In that case, one can formally write the evolution operator in the exponential form. Obviously, this does not work for nonlinear systems, such as, for example, systems with mean 8eld interactions. The method which we are presenting here is based on mapping the expanded microscopic trajectory into the consistently expanded macroscopic trajectory. This does not require linearity. Moreover, the order-by-order recurrent construction can be, in principle, enhanced by restoring to other types of approximations, like Pad[e approximation, for example, but we do not consider these options here. In the present section we discuss in detail applications of the method of natural projector [16,17,21] to derivations of macroscopic equations in various cases, with and without mean 8eld interaction
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potentials, for various choices of macroscopic variables, and demonstrate how computations are performed in the higher orders of the expansion. The structure of the Example is as follows: In the next subsection, for the sake of completeness, we describe brie=y the formalization of Ehrenfest’s approach [16,17]. We stress the role of the quasi-equilibrium approximation as the starting point for the constructions to follow. We derive explicit expressions for the correction to the quasi-equilibrium dynamics, and conclude this section with the entropy production formula and its discussion. In Section 3, we begin the discussion of applications. We use the present formalism in order to derive hydrodynamic equations. Zeroth approximation of the scheme is the Euler equations of the compressible nonviscous =uid. The 8rst approximation leads to the system of Navier–Stokes equations. Moreover, the approach allows to obtain the next correction, so-called post-Navier–Stokes equations. The latter example is of particular interest. Indeed, it is well known that the post-Navier– Stokes equations as derived from the Boltzmann kinetic equation by the Chapman–Enskog method (Burnett and super-Burnett hydrodynamics) su6er from unphysical instability already in the linear approximation [53]. We demonstrate it by the explicit computation that the linearized higher-order hydrodynamic equations derived within our method are free from this drawback. General construction. Let us consider a microscopic dynamics given by an equation, f˙ = J (f) ;
(365)
where f(x; t) is a distribution function over the phase space x at time t, and where operator J (f) may be linear or nonlinear. We consider linear macroscopic variables Mk = @k (f), where operator @k maps f into Mk . The problem is to obtain closed macroscopic equations of motion, M˙k =Vk (M ). This is achieved in two steps: First, we construct an initial approximation to the macroscopic dynamics and, second, this approximation is further corrected on the basis of the coarse-gaining. The initial approximation is the quasi-equilibrium approximation, and it is based on the entropy maximum principle under 8xed constraints [153,125]: S(f) → max;
@(f) = M ;
(366)
where S is the entropy functional, which is assumed to be strictly concave, and M is the set of the macroscopic variables {M }, and @ is the set of the corresponding operators. If the solution to the problem (366) exists, it is unique thanks to the concavity of the entropy functionals. Solution to Eq. (366) is called the quasi-equilibrium state, and it will be denoted as f∗ (M ). The classical example is the local equilibrium of the ideal gas: f is the one-body distribution function, S is the Boltzmann entropy, @ are 8ve linear operators, @(f) = {1; C; v2 }f dC, with C the particle’s velocity; the corresponding f∗ (M ) is called the local Maxwell distribution function. If the microscopic dynamics is given by Eq. (365), then the quasi-equilibrium dynamics of the variables M reads: M˙ k = @k (J (f∗ (M ))) = V∗k :
(367)
The quasi-equilibrium approximation has important property, it conserves the type of the dynamics: If the entropy monotonically increases (or not decreases) due to Eq. (365), then the same is true for the quasi-equilibrium entropy, S ∗ (M ) = S(f∗ (M )), due to the quasi-equilibrium dynamics (367). That is, if 9S(f) ˙ 9S(f) f= J (f) ¿ 0 ; S˙ = 9f 9f
A.N. Gorban et al. / Physics Reports 396 (2004) 197 – 403
then S˙∗ =
9S ∗ 9S ∗ M˙ k = @k (J (f∗ (M ))) ¿ 0 : 9Mk 9Mk k
323
(368)
k
Summation in k always implies summation or integration over the set of labels of the macroscopic variables. Conservation of the type of dynamics by the quasi-equilibrium approximation is a simple yet a general and useful fact. If the entropy S is an integral of motion of Eq. (365) then S ∗ (M ) is the integral of motion for the quasi-equilibrium equation (367). Consequently, if we start with a system which conserves the entropy (for example, with the Liouville equation) then we end up with the quasi-equilibrium system which conserves the quasi-equilibrium entropy. For instance, if M is the one-body distribution function, and (365) is the (reversible) Liouville equation, then (367) is the Vlasov equation which is reversible, too. On the other hand, if the entropy was monotonically increasing on solutions to Eq. (365), then the quasi-equilibrium entropy also increases monotonically on solutions to the quasi-equilibrium dynamic equations (367). For instance, if Eq. (365) is the Boltzmann equation for the one-body distribution function, and M is a 8nite set of moments (chosen in such a way that the solution to problem (366) exists), then (367) are closed moment equations for M which increase the quasi-equilibrium entropy (this is the essence of a well known generalization of Grad’s moment method). Enhancement of quasi-equilibrium approximations for entropy-conserving dynamics The goal of the present subsection is to describe the simplest analytic implementation, the microscopic motion with periodic coarse-graining. The notion of coarse-graining was introduced by Ehrenfest’s in their seminal work [215]: The phase space is partitioned into cells, the coarse-grained variables are the amounts of the phase density inside the cells. Dynamics is described by the two processes, by the Liouville equation for f, and by periodic coarse-graining, replacement of f(x) in each cell by its average value in this cell. The coarse-graining operation means forgetting the microscopic details, or of the history. From the perspective of general quasi-equilibrium approximations, periodic coarse-graining amounts to the return of the true microscopic trajectory on the quasi-equilibrium manifold with the preservation of the macroscopic variables. The motion starts at the quasi-equilibrium state fi∗ . Then the true solution fi (t) of the microscopic equation (365) with the initial condition fi (0)=fi∗ is coarse-grained ∗ = f ∗ (@(f (4))). at a 8xed time t = 4, solution fi (4) is replaced by the quasi-equilibrium function fi+1 i This process is sketched in Fig. 15. From the features of the quasi-equilibrium approximation it follows that for the motion with periodic coarse-graining, the inequality is valid, ∗ S(fi∗ ) 6 S(fi+1 ) ;
(369)
the equality occurs if and only if the quasi-equilibrium is the invariant manifold of the dynamic system (365). Whenever the quasi-equilibrium is not the solution to Eq. (365), the strict inequality in (369) demonstrates the entropy increase. In other words, let us assume that the trajectory begins at the quasi-equilibrium manifold, then it takes o6 from this manifold according to the microscopic evolution equations. Then, after some time
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A.N. Gorban et al. / Physics Reports 396 (2004) 197 – 403 · f = J( f)
f
f∗
µ
µ
µ M · M = ϕ( M)
Fig. 15. Coarse-graining scheme. f is the space of microscopic variables, M is the space of the macroscopic variables, f∗ is the quasi-equilibrium manifold, @ is the mapping from the microscopic to the macroscopic space.
4, the trajectory is coarse-grained, that is the, state is brought back on the quasi-equilibrium manifold keeping the values of the macroscopic variables. The irreversibility is born in the latter process, and this construction clearly rules out quasi-equilibrium manifolds which are invariant with respect to the microscopic dynamics, as candidates for a coarse-graining. The coarse-graining indicates the way to derive equations for macroscopic variables from the condition that the macroscopic trajectory, M (t), which governs the motion of the quasi-equilibrium states, f∗ (M (t)), should match precisely the same points on the quasi-equilibrium manifold, f∗ (M (t + 4)), and this matching should be independent of both the initial time, t, and the initial condition M (t). The problem is then how to derive the continuous time macroscopic dynamics which would be consistent with this picture. The simplest realization suggested in Refs. [16,17] is based on using an expansion of both the microscopic and the macroscopic trajectories. Here we present this construction to the third order accuracy, in a general form, whereas only the second-order accurate construction has been discussed in [16,17]. Let us write down the solution to the microscopic equation (365), and approximate this solution by the polynomial of third order in 4. Introducing notation, J ∗ = J (f∗ (M (t))), we write, 42 9J ∗ ∗ 43 9J ∗ 9J ∗ ∗ 92 J ∗ ∗ ∗ ∗ ∗ f(t + 4) = f + 4J + J + J + + o(43 ) : J J (370) 2 9f 3! 9f 9f 9f2 Evaluation of the macroscopic variables on function (370) gives ∗ 9J ∗ 42 ∗ Mk (t + 4) = Mk + 4Vk + @k J 2 9f ∗ 2 ∗ 43 9J 9J ∗ ∗ 9J ∗ ∗ @k J + + @k + o(43 ) ; J J 3! 9f 9f 9f2
(371)
where V∗k = @k (J ∗ ) is the quasi-equilibrium macroscopic vector 8eld (the right hand side of Eq. (367)), and all the functions and derivatives are taken in the quasi-equilibrium state at time t.
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325
We shall now establish the macroscopic dynamic by matching the macroscopic and the microscopic dynamics. Speci8cally, the macroscopic dynamic equations (367) with the right hand side not yet de8ned, give the following third-order result: 42 9Vk Mk (t + 4) = Mk + 4Vk + Vj 2 j 9Mj 43 + 3! ij
9 2 Vk 9Vk 9Vi V i Vj + Vj 9Mi Mj 9Mi 9Mj
+ o(43 ) :
(372)
(1) (0) 2 (2) ∗ Expanding functions Vk into the series Vk =R(0) k +4Rk +4 Rk +· · ·, (Rk =V ), and requiring that 3 the microscopic and the macroscopic dynamics coincide to the order of 4 , we obtain the sequence of corrections for the right hand side of the equation for the macroscopic variables. Zeroth order is the quasi-equilibrium approximation to the macroscopic dynamics. The 8rst-order correction gives " ∗ # 9Vk ∗ 9J ∗ ∗ 1 (1) @k J : (373) − V Rk = 2 9f 9Mj j j
The next, second-order correction has the following explicit form: ∗ 2 ∗ 9J 9J ∗ ∗ 9J ∗ ∗ 1 9V∗k 9V∗i ∗ 1 (2) @k J Rk = + @k − J J V 3! 9f 9f 9f2 3! ij 9Mi 9Mj j 1 − 3! ij
92 V∗k V∗ V∗ 9Mi 9Mj i j
1 − 2 j
(1) 9V∗k (1) 9Rj R + V∗ 9Mj j 9Mj j
;
(374)
Further corrections are found by the same token. Eqs. (373)–(374) give explicit closed expressions for corrections to the quasi-equilibrium dynamics to the order of accuracy speci8ed above. They are used below in various speci8c examples. Entropy production. The most important consequence of the above construction is that the resulting continuous time macroscopic equations retain the dissipation property of the discrete time coarse-graining (369) on each order of approximation n ¿ 1. Let us 8rst consider the entropy production formula for the 8rst-order approximation. In order to shorten notations, it is convenient to introduce the quasi-equilibrium projection operator, 9f∗ P∗g = @k (g) : (375) 9Mk k
It has been demonstrated in [17] that the entropy production, 9S ∗ (0) S˙∗(1) = (R + 4R(1) k ) ; 9Mk k k
equals
2 ∗ 4 ∗ ∗ ∗ 9 S ˙ (1 − P ∗ )J ∗ : S (1) = − (1 − P )J 2 9f9f f∗
(376)
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Eq. (376) is nonnegative de8nite due to concavity of the entropy. Entropy production (376) is equal to zero only if the quasi-equilibrium approximation is the true solution to the microscopic dynamics, that is, if (1 − P ∗ )J ∗ ≡ 0. While quasi-equilibrium approximations which solve the Liouville equation are uninteresting objects (except, of course, for the equilibrium itself), vanishing of the entropy production in this case is a simple test of consistency of the theory. Note that the entropy production (376) is proportional to 4. Note also that projection operator does not appear in our consideration a priory, rather, it is the result of exploring the coarse-graining condition in the previous section. Though Eq. (376) natural, its existence is rather subtle. Indeed, Eq. (376) is a di6erence looks very ∗ ∗ of the two terms, k @k (J 9J =9f) (contribution of the second-order approximation to the micro (0) (0) scopic trajectory), and ik Ri 9Rk =9Mi (contribution of the derivative of the quasi-equilibrium vector 8eld). Each of these expressions separately gives a positive contribution to the entropy production, and Eq. (376) is the di6erence of the two positive de8nite expressions. In the higher order approximations, these subtractions are more involved, and explicit demonstration of the entropy production formulae becomes a formidable task. Yet, it is possible to demonstrate the increase-in-entropy without explicit computation, though at a price of smallness of 4. Indeed, let us denote S˙∗(n) the time derivative of the entropy on the nth order approximation. Then t+4 S˙∗(n) (s) ds = S ∗ (t + 4) − S ∗ (t) + O(4 n+1 ) ; t
where S ∗ (t + 4) and S ∗ (t) are true values of the entropy at the adjacent states of the H -curve. The di6erence S = S ∗ (t + 4) − S ∗ (t) is strictly positive for any 8xed 4, and, by Eq. (376), S ∼ 42 for small 4. Therefore, if 4 is small enough, the right hand side in the above expression is positive, and 4S˙∗(n) (
(n) ) ¿ 0
;
where t 6 (n) 6 t + 4. Finally, since S˙∗(n) (t) = S˙∗(n) (s) + O(4 n ) for any s on the segment [t; t + 4], we can replace S˙∗(n) ( (n) ) in the latter inequality by S˙∗(n) (t). The sense of this consideration is as follows: Since the entropy production formula (376) is valid in the leading order of the construction, the entropy production will not collapse in the higher orders at least if the coarse-graining time is small enough. More re8ned estimations can be obtained only from the explicit analysis of the higher-order corrections. Relation to the work of Lewis. Among various realizations of the coarse-graining procedures, the work of Lewis [216] appears to be most close to our approach. It is therefore pertinent to discuss the di6erences. Both methods are based on the coarse-graining condition, Mk (t + 4) = @k (T4 f∗ (M (t))) ;
(377)
where T4 is the formal solution operator of the microscopic dynamics. Above, we applied a consistent expansion of both, the left hand side and the right hand side of the coarse-graining condition (377), in terms of the coarse-graining time 4. In the work of Lewis [216], it was suggested, as a general way to exploring condition (377), to write the 8rst-order equation for M in the form of the di6erential pursuit, dMk (t) ≈ @k (T4 f∗ (M (t))) : Mk (t) + 4 (378) dt
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327
In other words, in the work of Lewis [216], the expansion to the 8rst order was considered on the left (macroscopic) side of Eq. (377), whereas the right hand side containing the microscopic trajectory T4 f∗ (M (t)) was not treated on the same footing. Clearly, expansion of the right hand side to 8rst order in 4 is the only equation which is common in both approaches, and this is the quasi-equilibrium dynamics. However, the di6erence occurs already in the next, second-order term (see Refs. [16,17] for details). Namely, the expansion to the second order of the right hand side of Lewis’ equation [216] results in a dissipative equation (in the case of the Liouville equation, for example) which remains dissipative even if the quasi-equilibrium approximation is the exact solution to the microscopic dynamics, that is, when microscopic trajectories once started on the quasi-equilibrium manifold belong to it in all the later times, and thus no dissipation can be born by any coarse-graining. On the other hand, our approach assumes a certain smoothness of trajectories so that application of the low-order expansion bears physical signi8cance. For example, while using lower-order truncations it is not possible to derive the Boltzmann equation because in that case the relevant quasi-equilibrium manifold (N -body distribution function is proportional to the product of one-body distributions, or uncorrelated states, see next section) is almost invariant during the long time (of the order of the mean free =ight of particles), while the trajectory steeply leaves this manifold during the short-time pair collision. It is clear that in such a case lower-order expansions of the microscopic trajectory do not lead to useful results. It has been clearly stated by Lewis [216], that the exploration of the condition (377) depends on the physical situation, and how one makes approximations. In fact, derivation of the Boltzmann equation given by Lewis on the basis of the condition (377) does not follow the di6erential pursuit approximation: As is well known, the expansion in terms of particle’s density of the solution to the BBGKY hierarchy is singular, and begins with the linear in time term. Assuming the quasi-equilibrium approximation for the N -body distribution function under 8xed one-body distribution function, and that collisions are well localized in space and time, one gets on the right hand side of Eq. (377), f(t + 4) = f(t) + n4JB (f(t)) + o(n) ; where n is particle’s density, f is the one-particle distribution function, and JB is the Boltzmann’s collision integral. Next, using the mean-value theorem on the left hand side of Eq. (377), the Boltzmann equation is derived (see also a recent elegant renormalization-group argument for this derivation [36]). We stress that our approach of matched expansion for exploring the coarse-graining condition (377) is, in fact, the exact (formal) statement that the unknown macroscopic dynamics which causes the shift of Mk on the left hand side of Eq. (377) can be reconstructed order-by-order to any degree of accuracy, whereas the low-order truncations may be useful for certain physical situations. A thorough study of the cases beyond the lower-order truncations is of great importance which is left for future work. Equations of hydrodynamics for simple Kuid. The method discussed above enables one to establish in a simple way the form of equations of the macroscopic dynamics to various degrees of approximation. In this section, the microscopic dynamics is given by the Liouville equation, similar to the previous case. However, we take another set of macroscopic variables: density, average velocity, and average temperature of the =uid. Under this condition the solution to problem (366) is the local Maxwell distribution. For the hydrodynamic equations, the zeroth (quasi-equilibrium) approximation
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is given by Euler’s equations of compressible nonviscous =uid. The next order approximation are the Navier–Stokes equations which have dissipative terms. Higher-order approximations to the hydrodynamic equations, when they are derived from the Boltzmann kinetic equation (so-called Burnett approximation), are subject to various diRculties, in particular, they exhibit an instability of sound waves at suRciently short wave length (see, e.g. [24] for a recent review). Here we demonstrate how model hydrodynamic equations, including postNavier–Stokes approximations, can be derived on the basis of coarse-graining idea, and investigate the linear stability of the obtained equations. We will 8nd that the resulting equations are stable. Two points need a clari8cation before we proceed further [17]. First, below we consider the simplest Liouville equation for the one-particle distribution, describing a free moving particle without interactions. The procedure of coarse-graining we use is an implementation of collisions leading to dissipation. If we had used the full interacting N -particle Liouville equation, the result would be di6erent, in the 8rst place, in the expression for the local equilibrium pressure. Whereas in the present case we have the ideal gas pressure, in the N -particle case the nonideal gas pressure would arise. Second, and more essential is that, to the order of the Navier–Stokes equations, the result of our method is identical to the lowest-order Chapman–Enskog method as applied to the Boltzmann equation with a single relaxation time model collision integral (the Bhatnagar–Gross–Krook model [89]). However, this happens only at this particular order of approximation, because already the next, post-Navier–Stokes approximation, is di6erent from the Burnett hydrodynamics as derived from the BGK model (the latter is linearly unstable). Derivation of the Navier–Stokes equations. Let us assume that reversible microscopic dynamics is given by the one-particle Liouville equation, 9f 9f = −vi ; 9t 9ri
(379)
where f =f(r; C; t) is the one-particle distribution function, and index i runs over spatial components {x; y; z}. Subject to appropriate boundary conditions which we assume, this equation conserves the Boltzmann entropy S = −kB f ln f dC dr. We introduce the following hydrodynamic moments as the macroscopic variables: M0 = f dC, Mi = vi f dC, M4 = v2 f dC. These variables are related to the more conventional density, average velocity and temperature, n; u; T as follows: M0 = n;
Mi = nui ;
n = M0 ;
ui = M0−1 Mi ;
3nkB T + nu2 ; m m T= (M4 − M0−1 Mi Mi ) : 3kB M0
M4 =
The quasi-equilibrium distribution function (local Maxwellian) reads: 3=2 m −m(v − u)2 : exp f0 = n 2,kB T 2kB T
(380)
(381)
Here and below, n, u, and T depend on r and t. Based on the microscopic dynamics (379), the set of macroscopic variables (380), and the quasi-equilibrium (381), we can derive the equations of the macroscopic motion.
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329
A speci8c feature of the present example is that the quasi-equilibrium equation for the density (the continuity equation), 9n 9nui ; =− 9t 9ri
(382)
should be excluded out of the further corrections. This rule should be applied generally: If a part of the chosen macroscopic variables (momentum =ux nu here) correspond to =uxes of other macroscopic variables, then the quasi-equilibrium equation for the latter is already exact, and has to be exempted of corrections. The quasi-equilibrium approximation for the rest of the macroscopic variables is derived in the usual way. In order to derive the equation for the velocity, we substitute the local Maxwellian into the one-particle Liouville equation, and act with the operator @k = vk dC on both the sides of Eq. (379). We have 9nuk uj 9 nkB T 9nuk =− − : 9t 9rk m 9rj Similarly, we derive the equation for the energy density, and the complete system of equations of the quasi-equilibrium approximation reads (Euler equations): 9n 9nui =− ; 9t 9ri 9nuk uj 9 nkB T 9nuk =− − ; 9t 9rk m 9rj 9 5kB T 9! 2 =− nui + u nui : 9t 9ri m
(383)
Now we are going to derive the next order approximation to the macroscopic dynamics (8rst order in the coarse-graining time 4). For the velocity equation we have 9Vnu 1 92 f0 k vk vi vj Rnuk = dC − Vj ; 2 9ri 9rj 9Mj j where Vj are the corresponding right hand sides of the Euler equations (383). In order to take derivatives with respect to macroscopic moments {M0 ; Mi ; M4 }, we need to rewrite Eqs. (383) in terms of these variables instead of {n; ui ; T }. After some computation, we obtain 9uj 1 9 nkB T 9uk 2 9un : (384) + − kj Rnuk = 2 9rj m 9rj 9rk 3 9rn For the energy we obtain 9V! 5 9 nkB2 T 9T 92 f0 1 2 v vi vj : dC − Vj = R! = 2 9ri 9rj 9Mj 2 9ri m2 9ri j
(385)
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Thus, we get the system of the Navier–Stokes equation in the following form: 9nui 9n =− ; 9t 9ri
9uj 9nuk uj 4 9 nkB T 9uk 2 9un 9nuk 9 nkB T + + − kj ; =− − 9t 9rk m 9rj 2 9rj m 9rj 9rk 3 9rn 9 5kB T 5 9 nkB2 T 9T 9! 2 =− nui + u nui + 4 : 9t 9ri m 2 9ri m2 9ri
(386)
We see that kinetic coeRcients (viscosity and heat conductivity) are proportional to the coarsegraining time 4. Note that they are identical with kinetic coeRcients as derived from the Bhatnagar– Gross–Krook model [89] in the 8rst approximation of the Chapman–Enskog method [51] (also, in particular, no bulk viscosity). Post-Navier–Stokes equations. Now we are going to obtain the second-order approximation to the hydrodynamic equations in the framework of the present approach. We will compare qualitatively the result with the Burnett approximation. The comparison concerns stability of the hydrodynamic modes near global equilibrium, which is violated for the Burnett approximation. Though the derivation is straightforward also in the general, nonlinear case, we shall consider only the linearized equations which is appropriate to our purpose here. Linearizing the local Maxwell distribution function, we obtain n mvn mv2 3 T 2 f = n0 e−mv =2kB T0 + un + − n0 kB T 0 2kB T0 2 T0 2 3 2 2 e−c ; c − = M0 + 2Mi ci + M4 − M0 3 2
m 2,kB T0
3=2
(387)
where we have introduced dimensionless variables: ci =vi =vT , vT = 2kB T0 =m is the thermal velocity, M0 = n=n0 , Mi = ui =vT , M4 = (3=2)(n=n0 + T=T0 ). Note that n, and T determine deviations of these variables from their equilibrium values, n0 ; and T0 . The linearized Navier–Stokes equations read 9Mi 9M0 =− ; 9t 9ri 1 9M4 4 9 9Mk =− + 9t 3 9rk 4 9rj
9Mj 9Mk 2 9Mn + − kj 9rj 3 9rn 9rk
5 9Mi 5 9 2 M4 9M4 =− +4 : 9t 2 9ri 2 9ri 9ri
; (388)
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331
Let us 8rst compute the post-Navier–Stokes correction to the velocity equation. In accordance with Eq. (374), the 8rst part of this term under linear approximation is 1 @k 3!
9J ∗ 9J ∗ ∗ J 9f 9f
1 − 3! ij
9V∗k 9V∗i ∗ V 9Mi 9Mj j
93 2 3 2 e− c d 3 c c2 − ci cj cn M0 + 2Mi ci + M4 − M0 9ri 9rj 9rn 3 2 3 9 2 M0 5 9 92 M 4 5 9 92 M 4 1 9 92 M 4 + + = − 108 9ri 9rs 9rs 6 9rk 4 9rs 9rs 9rs 9rs 108 9rk 9rs 9rs
=
1 6
=−
ck
1 9 92 M 0 13 9 92 M4 − : 8 9rk 9rs 9rs 108 9rk 9rs 9rs
(389)
The part of Eq. (374) proportional to the 8rst-order correction is 1 − 2 j
(1) 9V∗k (1) 9Rk Rj + V∗ 9Mj 9Mj j
=
5 9 92 M 4 1 9 92 M 4 + : 6 9rk 9rs 9rs 9 9rk 9rs 9rs
(390)
Combining together terms (389), and (390), we obtain R(2) Mk =
1 9 92 M 0 89 9 92 M4 + : 8 9rk 9rs 9rs 108 9rk 9rs 9rs
Similar calculation for the energy equation leads to the following result:
c2
− +
93 ci cj ck 9ri 9rj 9rk
M0 + 2Mi ci +
25 9 92 Mi 1 =− 72 9ri 9rs 9rs 6
2 M4 − M0 3
c2 −
21 9 92 Mi 25 9 92 Mi + 4 9ri 9rs 9rs 12 9ri 9rs 9rs
3 2
=−
2
e−c d 3 c
19 9 92 Mi : 36 9ri 9rs 9rs
The term proportional to the 8rst-order corrections gives 5 6
92 9Mi 9rs 9rs 9ri
25 + 4
92 9Mi 9rs 9rs 9ri
:
Thus, we obtain R(2) M4
59 = 9
92 9Mi 9rs 9rs 9ri
:
(391)
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Fig. 16. Attenuation rates of various modes of the post-Navier–Stokes equations as functions of the wave vector. Attenuation rate of the twice degenerated shear mode is curve 1. Attenuation rate of the two sound modes is curve 2. Attenuation rate of the di6usion mode is curve 3.
Finally, combining together all the terms, we obtain the following system of linearized hydrodynamic equations: 9Mi 9M0 =− ; 9t 9ri 9Mj 9Mk 1 9M4 4 9 9Mk 2 9Mn =− + + − kj 9t 3 9rk 4 9rj 9rj 9rk 3 9rn 1 9 92 M 0 89 9 92 M4 2 +4 ; + 8 9rk 9rs 9rs 108 9rk 9rs 9rs 2 5 9Mi 9 9M4 9Mi 5 9 2 M4 2 59 =− : (392) +4 +4 9t 2 9ri 2 9ri 9ri 9 9rs 9rs 9ri Now we are in a position to investigate the dispersion relation of this system. Substituting Mi = M˜ i exp(!t + i(k; r)) (i = 0; k; 4) into Eq. (392), we reduce the problem to 8nding the spectrum of the matrix: 0 −ikx −iky −ikz 0 2 2 kx ky 1 2 x kz −ikx k8 − 14 k 2 − 12 kx − 12 − k12 −ikx ( 13 + 89k ) 108 2 kx ky ky kz 1 2 −iky k 2 − 12 − 14 k 2 − 12 ky − 12 −iky ( 13 + 89k ) 8 108 : 2 ky kz 1 2 −ik k 2 x kz − k12 − 12 − 14 k 2 − 12 kz −ikz ( 13 + 89k ) z 8 108 0
−ikx ( 52 +
59k 2 ) 9
−iky ( 52 +
59k 2 ) 9
−ikz ( 52 +
59k 2 ) 9
− 52 k 2
This matrix has 8ve eigenvalues. The real parts of these eigenvalues responsible for the decay rate of the corresponding modes are shown in Fig. 16 as functions of the wave vector k. We see that all
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333
real parts of all the eigenvalues are nonpositive for any wave vector. In other words, this means that the present system is linearly stable. For the Burnett hydrodynamics as derived from the Boltzmann or from the single relaxation time Bhatnagar–Gross–Krook model, it is well known that the decay rate of the acoustic becomes positive after some value of the wave vector [53,24] which leads to the instability. While the method suggested here is clearly semi-phenomenological (coarse-graining time 4 remains unspeci8ed), the consistency of the expansion with the entropy requirements, and especially the latter result of the linearly stable post-Navier–Stokes correction strongly indicates that it might be more suited to establishing models of highly nonequilibrium hydrodynamics. Example 11: Natural projector for the McKean model In this section the =uctuation–dissipation formula recently derived by the method of natural projector [18] is illustrated by the explicit computation for McKean’s kinetic model [219]. It is demonstrated that the result is identical, on the one hand, to the sum of the Chapman–Enskog expansion, and, on the other hand, to the exact solution of the invariance equation. The equality between all the three results holds up to the crossover from the hydrodynamic to the kinetic domain. General scheme. Let us consider a microscopic dynamics (76) given by an equation for the distribution function f(x; t) over a con8guration space x: 9t f = J (f) ;
(393)
where operator J (f) may be linear or nonlinear. Let m(f) be a set of linear functionals whose values, M = m(f), represent the macroscopic variables, and also let f(M ; x) be a set of distribution functions satisfying the consistency condition, m(f(M )) = M :
(394)
The choice of the relevant distribution functions is the point of central importance which we discuss later on but for the time being we need only speci8cation (394). The starting point has been the following observation [16,17]: Given a 8nite time interval 4, it is possible to reconstruct uniquely the macroscopic dynamics from a single condition. For the sake of completeness, we shall formulate this condition here. Let us denote as M (t) the initial condition at the time t to the yet unknown equations of the macroscopic motion, and let us take f(M (t); x) for the initial condition of the microscopic equation (393) at the time t. Then the condition for the reconstruction of the macroscopic dynamics reads as follows: For every initial condition {M (t); t}, solutions to the macroscopic dynamic equations at the time t + 4 are equal to the values of the macroscopic variables on the solution to Eq. (393) with the initial condition {f(M (t); x); t}: M (t + 4) = m(T4 f(M (t))) ;
(395)
where T4 is the formal solution operator of the microscopic equation (393). The right hand side of Eq. (395) represents an operation on trajectories of the microscopic equation (393), introduced in a particular form by Ehrenfest’s [215] (the coarse-graining): The solution at the time t + 4 is replaced by the state on the manifold f(M ; x). Notice that the coarse-graining time 4 in Eq. (395) is 8nite, and we stress the importance of the required independence from the initial time t, and from the initial condition at t.
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The essence of the reconstruction of the macroscopic equations from the condition just formulated is in the following [16,17]: Seeking the macroscopic equations in the form, 9t M = R(M ; 4) ;
(396)
we proceed with Taylor expansion of the unknown functions R in terms of powers 4 n , where n = 0; 1; : : :, and require that each approximation, R(n) , of the order n, is such that resulting macroscopic solutions satisfy the condition (396) to the order 4 n+1 . This process of successive approximation is solvable. Thus, the unknown macroscopic equation (396) can be reconstructed to any given accuracy. Coming back to the problem of choosing the distribution function f(M ; x), we recall that many physically relevant cases of the microscopic dynamics (393) are characterized by existence of a concave functional S(f) (the entropy functional; discussions of S can be found in [151,152,125]). Traditionally, two cases are distinguished, the conservative [dS=dt ≡ 0 due to Eq. (393)], and the dissipative [dS=dt ¿ 0 due to Eq. (393), where equality sign corresponds to the stationary solution]. Approach (395) and (396) is applicable to both these situations. In both of these cases, among the possible sets of distribution functions f(M ; x), the distinguished role is played by the well-known quasi-equilibrium approximations, f∗ (M ; x), which are maximizers of the functional S(f) for 8xed M . We recall that, due to convexity of the functional S, if such a maximizer exists then it is unique. The special role of the quasi-equilibrium approximations is due to the well known fact that they preserve the type of dynamics: If dS=dt ¿ 0 due to Eq. (393), then dS ∗ =dt ¿ 0 due to the quasi-equilibrium dynamics, where S ∗ (M ) = S(f∗ (M )) is the quasi-equilibrium entropy, and where the quasi-equilibrium dynamics coincides with the zeroth order in the above construction, R(0) = m(J (f∗ (M ))). We notice it in passing that, since the well known work of Jaynes [153], the usefulness of quasi-equilibrium approximations is well understood in various versions of projection operator formalism for the conservative case [95,154–156], as well as for the dissipative dynamics [125,4– 6]. Relatively less studied remains the case of open or externally driven systems, where invariant quasi-equilibrium manifolds may become unstable [84]. The use of the quasi-equilibrium approximations for the above construction has been stressed in [16,17,20]. In particular, the strict increase in the quasi-equilibrium entropy has been demonstrated for the 8rst and higher order approximations [17]. Examples have been provided [17], focusing on the conservative case, and demonstrating that several well known dissipative macroscopic equations, such as the Navier–Stokes equation and the di6usion equation for the one-body distribution function, are derived as the lowest order approximations of this construction. The advantage of the approach [16,17] is the locality of construction, because only Taylor series expansion of the microscopic solution is involved. This is also its natural limitation. From the physical standpoint, 8nite and 8xed coarse-graining time 4 remains a phenomenological device which makes it possible to infer the form of the macroscopic equations by a noncomplicated computation rather than to derive a full form thereof. For instance, the form of the Navier–Stokes equations can be derived from the simplest model of free motion of particles, in which case the coarse-graining is a substitution for collisions. Going away from the limitations imposed by the 8nite coarse graining time [16,17] can be recognized as the major problem of a consistent formulation of the nonequilibrium statistical thermodynamics. Intuitively, this requires taking the limit 4 → ∞, allowing for all the relevant correlations to be developed by the microscopic dynamics, rather than to be cut o6 at the 8nite 4. Indeed, in the case of the dissipative dynamics, in particular, for the linearized Boltzmann equation, one typically expects an initial layer [87] which is completely cut o6 in the short-memory
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approximation, whereas those e6ects can be made small by taking 4 large enough. A way of doing this in the general nonlinear setting for entropy-conserving systems still requires further work at the time of this writing. Natural projector for linear systems. However, there is one important exception when the ‘4 → ∞ problem’ is readily solved [17,18]. This is the case where Eq. (393) is linear, 9t f = Lf ;
(397)
and where the quasi-equilibrium is a linear function of M . This is, in particular, the classical case of linear irreversible thermodynamics where one considers the linear macroscopic dynamics near the equilibrium, feq , Lfeq = 0. We assume, for simplicity of presentation, that the macroscopic variables M vanish at equilibrium, and are normalized in such a way that m(feq m† ) = 1, where † denotes transposition, and 1 is an appropriate identity operator. In this case, the linear dynamics of the macroscopic variables M has the form, 9t M = RM ;
(398)
where the linear operator R is determined by the coarse-graining condition (395) in the limit 4 → ∞: 1 ln[m(e4L feq m† )] : (399) 4→∞ 4 Formula (399) has been already brie=y mentioned in [17], and its relation to the Green–Kubo formula has been demonstrated in [18]. In our case, the Green–Kubo formula reads: ∞ RGK = m(0) ˙ m(t) ˙ ; (400) R = lim
0
where angular brackets denote equilibrium averaging, and where m˙ = L† m. The di6erence between formulae (399) and (400) stems from the fact that condition (395) does not use an a priori hypothesis of the separation of the macroscopic and the microscopic time scales. For the classical N -particle dynamics, Eq. (399) is a complicated expression, involving a logarithm of noncommuting operators. It is therefore very desirable to gain its understanding in simple model situations. Explicit example of the Kuctuation–dissipation formula. In this section we want to give explicit example of formula (399). In order to make our point, we consider here dissipative rather than conservative dynamics in the framework of the well known toy kinetic model introduced by McKean [219] for the purpose of testing various ideas in kinetic theory. In the dissipative case with a clear separation of time scales, existence of formula (399) is underpinned by the entropy growth in both the rapid and the slow parts of the dynamics. This physical idea underlies generically the extraction of the slow (hydrodynamic) component of motion through the concept of normal solutions to kinetic equations, as pioneered by Hilbert [52], and has been discussed by many authors, e.g. [87,157,158]. Case studies for linear kinetic equation help clarifying the concept of this extraction [159,160,219]. Therefore, since for the dissipative case there exist well established approaches to the problem of reducing the description, and which are exact in the present setting, it is very instructive to see their relation to formula (399). Speci8cally, we compare the result with the exact sum of the Chapman–Enskog expansion [51], and with the exact solution in the framework of the method of invariant manifold [4–6]. We demonstrate that both the three approaches, di6erent in their nature, give the same result as long as the hydrodynamic and the kinetic regimes are separated.
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The McKean model is the kinetic equation for the two-component vector function f (r; t) = (f+ (r; t); f− (r; t))† : −1 f+ + f− − f+ ; 9t f+ = −9r f+ + j 2 f+ + f− 9t f− = 9r f− + j−1 (401) − f− : 2 Eq. (401) describes the one-dimensional kinetics of particles with velocities +1 and −1 as a combination of the free =ight and a relaxation with the rate j−1 to the local equilibrium. Using the notation, (x; y), for the standard scalar product of the two-dimensional vectors, we introduce the 8elds, n(r; t) = (n; f ) [the local particle’s density, where n = (1; 1)], and j(r; t) = (j; f ) [the local momentum density, where j = (1; −1)]. Eq. (401) can be equivalently written in terms of the moments, 9t n = −9r j ; 9t j = −9r n − j−1 j :
(402)
The local equilibrium, n f ∗ (n) = n ; 2 is the conditional maximum of the entropy, S = − (f+ ln f+ + f− ln f− ) dr ;
(403)
under the constraint which 8xes the density, (n; f ∗ ) = n. The quasi-equilibrium manifold (403) is linear in our example, as well as is the kinetic equation. The problem of reducing the description for model (401) amounts to 8nding the closed equation for the density 8eld n(r; t). When the relaxation parameter j−1 is small enough (the relaxation dominance), then the 8rst Chapman–Enskog approximation to the momentum variable, j(r; t) ≈ −j9r n(r; t), amounts to the standard di6usion approximation. Let us consider now how formula (399), and other methods, extend this result. Because of the linearity of Eq. (401), and of the local equilibrium, it is natural to use the Fourier transform, hk = exp(ikr)h(r) dr. Eq. (401) is then written as 9 t fk = L k fk ; where
−ik −
(404) 1 2j
1 2j
(405) : 1 1 ik − 2j 2j Derivation of formula (399) in our example goes as follows: We seek the macroscopic dynamics of the form Lk =
9 t nk = R k nk ;
(406)
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where the function Rk is yet unknown. In the left hand side of Eq. (395) we have nk (t + 4) = e4Rk nk (t) :
(407)
In the right hand side of Eq. (395) we have 1 (408) (n; e4Lk f ∗ (nk (t))) = (n; e4Lk n)nk (t) : 2 After equating expressions (407) and (408), we require that the resulting equality holds in the limit 4 → ∞ independently of the initial data nk (t). Thus, we arrive at formula (399): 1 Rk = lim ln[(n; e4Lk n)] : (409) 4→∞ 4 Eq. (409) de8nes the macroscopic dynamics (406) within the present approach. Explicit evaluation of expression (409) is straightforward in the present model. Indeed, operator Lk has two eigenvalues, M± k , where & 1 1 ± Mk = − ± − k2 : (410) 2j 4j2 Let us denote as ek± two (arbitrary) eigenvectors of the matrix Lk , corresponding to the eigenvalues − − ± + + M± k . Vector n has a representation, n = 1k ek + 1k ek , where 1k are complex-valued coeRcients. With this, we obtain in Eq. (409), − + 1 Rk = lim ln[1k+ (n; ek+ )e4Mk + 1k− (n; ek− )e4Mk ] : (411) 4→∞ 4 − For k 6 kc , where kc2 = 4j, we have M+ k ¿ Mk . Therefore, Rk = M + k
for k ¡ kc :
(412)
As was expected, formula (399) in our case results in the exact hydrodynamic branch of the spectrum of the kinetic equation (401). The standard di6usion approximation is recovered from Eq. (412) as the 8rst nonvanishing approximation in terms of the (k=kc )2 . At k = kc , the crossover from the extended hydrodynamic to the kinetic regime takes place, and − Re M+ k = Re Mk . However, we may still extend the function Rk for k ¿ kc on the basis of formula (409): Rk = Re M+ k
for k ¿ kc :
(413)
Notice that the function Rk as given by Eqs. (412) and (413) is continuous but nonanalytic at the crossover. Comparison with the Chapman–Enskog method and solution of invariance equation Let us now compare this result with the Chapman–Enskog method. Since the exact Chapman–Enskog solution for the systems like Eq. (403) has been recently discussed in detail elsewhere [23,24,162–165], we shall be brief here. Following the Chapman–Enskog method, we seek the momentum variable j in terms of an expansion, ∞ j CE = jn+1 j (n) : (414) n=0
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The Chapman–Enskog coeRcients, j (n) , are found from the recurrence equations, n− 1 (n−1−m) 9(m) ; j (n) = − t j
(415)
m=0
where the Chapman–Enskog operators 9(m) are de8ned by their action on the density n: t (m) : 9(m) t n = −9r j
(416)
The recurrence equations (414)–(416), become well de8ned as soon as the aforementioned zero-order approximation j (0) is speci8ed, j (0) = −9r n :
(417)
From Eqs. (415)–(417), it follows that the Chapman–Enskog coeRcients j (n) have the following structure: j (n) = bn 9r2n+1 n ;
(418)
where coeRcients bn are found from the recurrence equation, bn =
n− 1
bn − 1 − m bm ;
b0 = −1 :
(419)
m=0
Notice that coeRcients (419) are real-valued, by the sense of the Chapman–Enskog procedure. The Fourier image of the Chapman–Enskog solution for the momentum variable has the form, jkCE = ikBkCE nk ;
(420)
where BkCE
=
∞
bn (−jk 2 )n :
(421)
n=0
Equation for the function B (421) is easily found upon multiplying Eq. (419) by (−k 2 )n , and summing in n from zero to in8nity: jk 2 Bk2 + Bk + 1 = 0 :
(422)
Solution to the latter equation which respects condition (417), and which constitutes the exact Chapman–Enskog solution (421) is " −2 + k Mk ; k ¡ kc ; CE Bk = (423) none; k ¿ kc : Thus, the exact Chapman–Enskog solution derives the macroscopic equation for the density as follows: 9t nk = −ikjkCE = RCE k nk ; where
" RCE k
=
M+ k;
k ¡ kc ;
none;
k ¿ kc :
(424)
(425)
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The Chapman–Enskog solution does not extends beyond the crossover at kc . This happens because the full Chapman–Enskog solution appears as a continuation the di6usion approximation, whereas formula (409) is not based on such an extension a priori. Finally, let us discuss brie=y the comparison with the solution within the method of invariant manifold [4–6]. Speci8cally, the momentum variable jkinv = ikBkinv nk is required to be invariant of both the microscopic and the macroscopic dynamics, that is, the time derivative of jkinv due to the macroscopic subsystem, 9jkinv 9t nk = ikBkinv (−ik)[ikBkinv ] ; 9nk
(426)
should be equal to the derivative of jkinv due to the microscopic subsystem, 9t jkinv = −iknk − j−1 ikBkinv nk ;
(427)
and that the equality between Eqs. (426) and (427) should hold independently of the speci8c value of the macroscopic variable nk . This amounts to a condition for the unknown function Bkinv , which is essentially the same as Eq. (422), and it is straightforward to show that the same selection procedure of the hydrodynamic root as above in the Chapman–Enskog case results in Eq. (425). In conclusion, in this Example we have given the explicit illustration for formula (399). The example considered above demonstrates that formula (399) gives the exact macroscopic evolution equation, which is identical to the sum of the Chapman–Enskog expansion, as well as to the invariance principle. This identity holds up to the point where the hydrodynamics and the kinetics cease to be separated. Whereas the Chapman–Enskog solution does not extend beyond the crossover point, formula (399) demonstrates a nonanalytic extension. The example considered adds to the con8dence of the correctness of the approach suggested in [16–19]. 11. Slow invariant manifold for a closed system has been found. What next? Suppose that the slow invariant manifold is found for a dissipative system. What have we constructed it for? First of all, for solving the Cauchy problem, in order to separate motions. This means that the Cauchy problem is divided in the following two subproblems: • Reconstruct the “fast” motion from the initial conditions to the slow invariant manifold (the initial layer problem). • Solve the Cauchy problem for the “slow” motions on the manifold. Thus, solving the Cauchy problem becomes easier (and in some complicated cases it just becomes possible). Let us stress here that for any suRciently reliable solution of the Cauchy problem one must solve not only the reduced Cauchy problem for the slow motion, but also the initial layer problem for fast motions. While solving the latter problem it was found to be surprisingly e6ective to use piece-wise linear approximations with smoothing or even without it [14,15]. This method was used for the Boltzman equation, for chemical kinetics equations, and for the Pauli equation.
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There exists a di6erent way to model the initial layer in kinetics problems: it is the route of model equations. For example, the Bhatnagar, Gross, Krook (BGK) equation [89] is the simplest model for the Boltzmann equation. It describes relaxation into a small neighborhood of the local Maxwell distribution. There are many types and hierarchies of the model equations [89,87,90,10,129]. The principal idea of any model equation is to replace the fast processes by a simple relaxation term. As a rule, it has a form d x=dt = · · · − (x − xsl (x))=4, where xsl (x) is a point of the approximate slow manifold. Such form is used in the BGK-equation, or in the quasi-equilibrium models [90]. It also can take a gradient form, like in the gradient models [10,129]. These simpli8cations not only allows to study the fast motions separately but it also allows to zoom in the details of the interaction of fast and slow motions in the vicinity of the slow manifold. What concerns solving the Cauchy problem for the “slow” motions, this is the basic problem of the hydrodynamics, of the gas dynamics (if the initial “big” systems describes kinetics of a gas or a =uid), etc. Here invariant manifold methods provide equations for a further study. However, even a preliminary consideration of the practical aspects of these studies shows a de8nite shortcoming. In practice, obtained equations are exploited not only for “closed” systems. The initial equations (76) describe a dissipative system that approaches the equilibrium. The equations of slow motion describe dissipative system too. Then these equations are supplied with various forces and Kows, and after that they describe systems with more or less complex dynamics. Because of this, there is a di6erent answer to our question, what have we constructed the invariant manifold for? First of all, in order to construct models of open system dynamics in the neighborhood of the slow manifold. Various approaches to this modeling are described in the following subsections. 11.1. Slow dynamics in open systems. Zero-order approximation and the thermodynamic projector Let the initial dissipative system (76) be “spoiled” by an additional term (“external vector 8eld” Jex (x; t)): dx = J (x) + Jex (x; t); dt
x⊂U :
(428)
For this new system the entropy does not increase everywhere. In the new system (428) di6erent dynamic e6ects are possible, such as a nonuniqueness of stationary states, auto-oscillations, etc. The “inertial manifold” e6ect is well-known: solutions of (428) approach some relatively low-dimensional manifold on which all the nontrivial dynamics takes place [234,235,136]. This “inertial manifold” can have a 8nite dimension even for in8nite-dimensional systems, for example, for the “reaction+ di6usion” systems [237]. In the theory of nonlinear control of partial di6erential equation systems a strategy based on approximate inertial manifolds [238] is proposed to facilitate the construction of 8nite-dimensional systems of ordinary di6erential equations (ODE), whose solutions can be arbitrarily close to the solutions of the in8nite-dimensional system [240]. It is natural to expect that the inertial manifold of system (428) is located somewhere close to the slow manifold of the initial dissipative system (76). This hypothesis has the following basis. Suppose
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that the vector 8eld Jex (x; t) is suRciently small. Let’s introduce, for example, a small parameter ! ¿ 0, and consider ! Jex (x; t) instead of Jex (x; t). Let’s assume that for system (76) a separation of motions into “slow” and “fast” takes place. In this case, there exists such interval of positive ! that ! Jex (x; t) is comparable to J only in a small neighborhood of the given slow motion manifold of system (76). Outside this neighborhood, ! Jex (x; t) is negligibly small in comparison with J and only negligibly in=uences the motion (for this statement to be true, it is important that system (76) is dissipative and every solution comes in 8nite time to a small neighborhood of the given slow manifold). Precisely this perspective on system (428) allows to exploit slow invariant manifolds constructed for the dissipative system (76) as the ansatz and the zero-order approximation in a construction of the inertial manifold of the open system (428). In the zero-order approximation, the right part of Eq. (428) is simply projected onto the tangent space of the slow manifold. The choice of the projector is determined by the motion separation which was described above: fast motion is taken from the dissipative system (76). A projector which is suitable for all dissipative systems with given entropy function is unique. It is constructed in the following way (detailed consideration of this is given above in the sections “Entropic projector without a priori parametrization” and in Ref. [177]). Let a point x ∈ U be de8ned and some vector space T , on which one needs to construct a projection (T is the tangent space to the slow manifold at the point x). We introduce the entropic scalar product | x : a | bx = −(a; Dx2 S(b)) :
(429)
Let us consider T0 that is a subspace of T and which is annulled by the di6erential S at the point x. T0 = {a ∈ T | Dx S(a) = 0} :
(430)
Suppose 14 that T0 = T . Let eg ∈ T , eg ⊥ T0 with respect to the entropic scalar product | x , and Dx S(eg ) = 1. These conditions de8ne vector eg uniquely. The projector onto T is de8ned by the formula P(J ) = P0 (J ) + eg Dx S(J )
(431)
where P0 is the orthogonal projector onto T0 with respect to the entropic scalar product | x . For example, if T a 8nite-dimensional space, then projector (431) is constructed in the following way. Let e1 ; : : : ; en be a basis in T , and for de8niteness, Dx S(e1 ) = 0. 1) Let us construct a system of vectors bi = ei+1 − i e1 ;
(i = 1; : : : ; n − 1) ;
(432)
where i = Dx S(ei+1 )=Dx S(e1 ), and hence Dx S(bi ) = 0. Thus, {bi }1n−1 is a basis in T0 . 2) Let us orthogonalize {bi }1n−1 with respect to the entropic scalar product | x (76). We thus derived an orthonormal with respect to | x basis {gi }1n−1 in T0 . 3) We 8nd eg ∈ T from the conditions: eg | gi x = 0; (i = 1; : : : ; n − 1); 14
Dx S(eg ) = 1
(433)
If T0 = T , then the thermodynamic projector is the orthogonal projector on T with respect to the entropic scalar product |x .
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and, 8nally we get P(J ) =
n− 1
gi gi | J x + eg Dx S(J ) :
(434)
i=1
If Dx S(T ) = 0, then the projector P is simply the orthogonal projector with respect to the | x scalar product. This is possible if x is the global maximum of entropy point (equilibrium). Then P(J ) =
n
gi gi |J x ; gi |gj = ij :
(435)
i=1
Remark. In applications, Eq. (76) often has additional linear balance constraints such as numbers of particles, momentum, energy, etc. Solving the closed dissipative system (76) we simply choose balance values and consider the dynamics of (76) on the corresponding aRne balance subspace. For driven system (428) the balances can be violated. Because of this, for the open system (428) the natural balance subspace includes the balance subspace of (76) with di6erent balance values. For every set of balance values there is a corresponding equilibrium. Slow invariant manifold of the dissipative systems that is applied to the description of the driven systems (428) is usually the union of slow manifolds for all possible balance values. The equilibrium of the dissipative closed system corresponds to the entropy maximum given the balance values are 1xed. In the phase space of the driven system (428) the entropy gradient in the equilibrium points of system (76) is not necessarily equal to zero. In particular, for the Boltzmann entropy in the local 8nite-dimensional case one gets the thermodynamic projector in the following form: S = − f(v)(ln(f(v)) − 1) dv ; Df S(J ) = −
J (v)ln f(v) dv ;
(v)’(v) dv ; f(v) n− 1 gi (v)J (v) dv − eg (v) J (v)ln f(v) dv ; P(J ) = gi (v) f(v) i=1 | ’f = −(
; Df2 S(’))
=
where gi (v) and eg (v) are constructed accordingly to the scheme described above, gi (v)gj (v) dv = ij ; f(v)
(436)
(437)
gi (v)ln f(v) dv = 0 ;
(438)
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gi (v)eg (v) dv = 0 ;
(439)
eg (v)ln f(v) dv = 1 :
(440)
If for all g ∈ T we have g(v)ln f(v) dv = 0, then the projector P is de8ned as the orthogonal projector with respect to the | f scalar product. 11.2. Slow dynamics in open systems. First-order approximation Thermodynamic projector (431) de8nes a “slow and fast motions” duality: if T is the tangent space of the slow motion manifold then T = im P, and ker P is the plane of fast motions. Let us denote by Px the projector at a point x of a given slow manifold. The vector 8eld Jex (x; t) can be decomposed in two components: Jex (x; t) = Px Jex (x; t) + (1 − Px )Jex (x; t) :
(441)
Let us denote Jex s = Px Jex , Jex f = (1 − Px )Jex . The slow component Jex s gives a correction to the motion along the slow manifold. This is a zero-order approximation. The “fast” component shifts the slow manifold in the fast motions plane. This shift changes Px Jex accordingly. Consideration of this e6ect gives a 8rst-order approximation. In order to 8nd it, let us rewrite the invariance equation taking Jex into account: (1 − Px )(J (x + x) + ! Jex (x; t)) = 0 Px x = 0 :
(442)
The 8rst iteration of the Newton method subject to incomplete linearization gives: (1 − Px )(Dx J (x) + ! Jex (x; t)) = 0 Px x = 0 :
(443)
(1 − Px )Dx J (1 − Px )J (x) = −! Jex (x; t) :
(444)
Thus, we have derived a linear equation in the space ker P. The operator (1 − P)Dx J (1 − P) is de8ned in this space. Utilization of the self-adjoint linearization instead of the traditional linearization Dx J operator (see “Decomposition of motions, nonuniqueness of selection: : :” section) considerably simpli8es solving and studying Eq. (444). It is necessary to take into account here that the projector P is a sum of the orthogonal projector with respect to the | x scalar product and a projector of rank one. Assume that the 8rst-order approximation equation (444) has been solved and the following function has been found: 1 x(x; ! Jex f ) = −[(1 − Px )Dx J (1 − Px )]−1 ! Jex f ; where Dx J is either the di6erential of J or symmetrized di6erential of J (309).
(445)
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Let x be a point on the initial slow manifold. At the point x + x(x; ! Jex f ) the right hand side of Eq. (428) in the 8rst-order approximation is given by J (x) + ! Jex (x; t) + Dx J (x(x; ! Jex f )) :
(446)
Due to the 8rst-order approximation (446), the motion of a point projection onto the manifold is given by the following equation: dx = Px (J (x) + ! Jex (x; t) + Dx J (x(x; ! Jex f (x; t)))) : dt
(447)
Note that, in Eq. (447), the vector 8eld J (x) enters only in the form of projection, Px J (x). For the invariant slow manifold it holds Px J (x) = J (x), but actually we always deal with approximately invariant manifolds, hence, it is necessary to use the projection Px J instead of J in (447). Remark. The notion “projection of a point onto the manifold” needs to be speci8ed. For every point x of the slow invariant manifold M there are de8ned both the thermodynamic projector Px (431) and the fast motions plane ker Px . Let us de8ne a projector L of some neighborhood of M onto M in the following way: L(z) = x
if Px (z − x) = 0 :
(448)
Qualitatively, it means that z, after all fast motions took place, comes into a small neighborhood of x. Operation (431) is de8ned uniquely in some small neighborhood of the manifold M . A derivation of slow motions equations requires not only an assumption that ! Jex is small but it must be slow as well: d=dt(! Jex ) must be small too. One can get the further approximations for slow motions of system (428), taking into account the time derivatives of Jex . This is an alternative to the usage of the projection operators methods [154]. This is considered in a more detail in example 12 for a particularly interesting driven system of dilute polymeric solutions. A short scheme description is given in the next subsection. 11.3. Beyond the 1rst-order approximation: higher-order dynamical corrections, stability loss and invariant manifold explosion Let us pose formally the invariance problem for the driven system (428) in the neighborhood of the slow manifold M of the initial (dissipative) system. Let for a given neighborhood of M an operator L (448) be de8ned. One needs to de8ne the function x(x; : : :) = x(x; Jex ; J˙ex ; JD ex ; : : :), x ∈ M , with the following properties: Px (x(x; : : :)) = 0 ; J (x + x(x; : : :)) + Jex (x + x(x; : : :); t) = x˙sl + Dx x(x; : : :)x˙sl +
∞ n=0
(n+1) DJex(n) x(x; : : :)Jex ; (449)
(n) = d n Jex =dt n , DJex(n) x(x; : : :) is a partial where x˙sl = Px (J (x + x(x; : : :)) + Jex (x + x(x; : : :); t)), Jex (n) (n) ; : : :) with respect to the variable Jex . One can di6erential of the function x(x; Jex ; J˙ex ; JD ex ; : : : ; Jex
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rewrite Eqs. (449) in the following form: (1 − Px − Dx x(x; : : :))(J (x + x(x; : : :)) + Jex (x + x(x; : : :); t)) =
∞ n=0
(n+1) DJex(n) x(x; : : :)Jex :
(450)
For solving Eq. (450) one can use an iteration method and take into account smallness consideration. The series in the right hand side of Eq. (450) can be rewritten as RHS =
k −1 n=0
(n+1) !n+1 DJex(n) x(x; : : :)Jex
(451)
at the kth iteration, considering expansion terms only to order less than k. The 8rst iteration equation was solved in the previous subsection. On second iteration one gets the following equation: (1 − Px − Dx 1 x(x; Jex ))(J (x + 1 x(x; Jex )) + Dz J (z)|z=x+1 x(x;Jex ) · (2 x − 1 x(x; Jex )) + Jex ) ˙ : = DJex 1 x(x; Jex )Jex
(452)
This is a linear equation with respect to 2 x. The solution 2 x(x; Jex ; J˙ex ) depends linearly on ˙ J ex , but nonlinearly on Jex . Let us remind that the 8rst iteration equation solution depends linearly on Jex . In all these iteration equations the 8eld Jex and it’s derivatives are included in the formulas as if they were functions of time t only. Indeed, for any solution x(t) of Eqs. (428) Jex (x; t) can be substituted for Jex (x(t); t). The function x(t) will be a solution of system (428) in which Jex (x; t) is substituted for Jex (t) in this way. However, in order to obtain the macroscopic equations (447) one must return to Jex (x; t). For the 8rst iteration such return is quite simple as one can see from (446). There Jex (x; t) is calculated in (k) points of the initial slow manifold. For general case, suppose that x = x(x; Jex ; J˙ex ; : : : ; Jex ) has been found. The motion equations for x (447) have the following form: dx (453) = Px (J (x + x) + Jex (x + x; t)) : dt In these equations the shift x must be a function of x and t (or a function of x; t; 1, where 1 are external 8elds, see example 12, but from the point of view of this consideration dependence on the external 8elds is not essential). One calculates the shift x(x; t) using the following equation: (k) Jex = Jex (x + x(x; Jex ; J˙ex ; : : : ; Jex ); t) :
(454)
It can be solved, for example, by the iterative method, taking Jex 0 = Jex (x; t): (k) Jex(n+1) = Jex (x + x(x; Jex(n) ; J˙ex(n) ; : : : ; Jex(n) ); t) :
(455)
We hope that using Jex in Eqs. (454) and (455) both as a variable and as a symbol of unknown function Jex (x; t) will not lead to a confusion. In all the constructions introduced above it was assumed that x is suRciently small and the driven system (428) will not deviate too far from the slow invariant manifold of the initial system. However,
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a stability loss is possible: solutions of Eq. (428) can deviate arbitrarily far given some perturbations level. The invariant manifold can loose it’s stability. Qualitatively, this e6ect of invariant manifold explosion can be represented as follows. Suppose that Jex includes the parameter !: one has ! Jex in Eq. (428). When ! is small, system motions are located in a small neighborhood of the initial manifold. This neighborhood grows monotonically and continuously with increase of !, but after some !0 a sudden change happens (“explosion”) and the neighborhood, in which the motion takes place, becomes signi8cantly wider at ! ¿ !0 than at ! ¡ !0 . The stability loss is not necessarily connected with the invariance loss. In example 13 it is shown how the invariant manifold (which is at the same time the quasi-equilibrium manifold in the example) can loose it’s stability. This “explosion” of the invariant manifold leads to essential physical consequences (see example 13). 11.4. Lyapunov norms, 1nite-dimensional asymptotic and volume contraction In a general case, it is impossible to prove the existence of a global Lyapunov function on the basis of local data. We can only verify or falsify the hypothesis about a given function, is it a global Lyapunov function, or is it not. On the other hand, there exists a more strict stability property which can be veri8ed or falsi8ed (in principle) on the base of local data analysis. This is a Lyapunov norm existence. A norm • is the Lyapunov norm for system (428), if for any two solutions x(1) (t); x(2) (t); t ¿ 0; the function x(1) (t) − x(2) (t) is nonincreasing in time. Linear operator A is dissipative with respect to a norm • , if exp(At) (t ¿ 0) is a semigroup of contractions: exp(At)x 6 x for any x and t ¿ 0. The family of linear operators {A1 }1∈K is simultaneously dissipative, if all operators A1 are dissipative with respect to some norm • (it should be stressed that there exists one norm for all A1 ; 1 ∈ K). The mathematical theory of simultaneously dissipative operators for 8nite-dimensional spaces was developed in Refs. [220–224]. Let system (428) be de8ned in a convex set U ⊂ E, and Ax be Jacobi operator at the point x: Ax = Dx (J (x) + Jex (x)). This system has a Lyapunov norm, if the family of operators {Ax }x∈U is simultaneously dissipative. If one can choose such ! ¿ 0 that for all Ax , t ¿ 0, any vector z, and this Lyapunov norm exp(Ax t)z 6 exp(−!t) z , then for any two solutions x(1) (t); x(2) (t); t ¿ 0 of Eqs. (428) x(1) (t) − x(2) (t) 6 exp(−!t) x(1) (0) − x(2) (0) . The simplest class of nonlinear kinetic (open) systems with Lyapunov norms was described in the paper [225]. These are reaction systems without interactions of various substances. The stoichiometric equation of each elementary reaction has a form "rj Aj ; (456) 1ri Ai → j
where r enumerates reactions, 1ri ; "rj are nonnegative stoichiometric coeRcients (usually they are integer), Ai are symbols of substances. In the right hand side of Eq. (456) there is one initial reagent, though 1ri ¿ 1 is possible (there may be several copies of Ai , for example 3A → 2B + C). 1 Kinetic equations for reaction system (456) have a Lyapunov norm [225]. This is l norm with weights: x = i wi |xi |, wi ¿ 0. There exists no quadratic Lyapunov norm for reaction systems without interaction of various substances.
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Existence of the Lyapunov norm is a very strong restriction on nonlinear systems, and such systems are not wide spread in applications. But if we go from distance contraction to contraction of k-dimensional volumes (k = 2; 3; : : : ; ) [231], the situation changes. There exist many kinetic systems with a monotonous contraction of k-dimensional volumes for suRciently big k (see, for example, [234–237]). Let x(t); t ¿ 0 be a solution of Eq. (428). Let us write a 8rst approximation equation for small deviations from x(t): dYx = Ax(t) Yx : (457) dt This is linear system with coeRcients depending on t. Let us study how system (457) changes k-dimensional volumes. A k-dimensional parallelepiped with edges x(1) ; x(2) ; : : : ; x(k) is an element of the kth exterior power: x(1) ∧ x(2) ∧ · · · ∧ x(k) ∈ E ∧ E ∧ · · · ∧ E k
(this is an antisymmetric tensor). A norm in the kth exterior power of the space E is a measure of k-dimensional volumes (one of the possible measures). Dynamics of parallelepipeds induced by system (457) can be described by equations d (Yx(1) ∧ Yx(2) ∧ · · · ∧ Yx(k) ) dt = (Ax(t) Yx(1) ) ∧ Yx(2) ∧ · · · ∧ Yx(k) + Yx(1) ∧ (Ax(t) Yx(2) ) ∧ · · · ∧ Yx(k) + · · · ∧k + Yx(1) ∧ Yx(2) ∧ · · · ∧ (Ax(t) Yx(k) ) = ADx(t) (Yx(1) ∧ Yx(2) ∧ · · · ∧ Yx(k) ) :
(458)
D ∧k are operators of induced action of Ax(t) on the kth exterior power of E. Again, a Here Ax(t) decreasing of Yx(1) ∧ Yx(2) ∧ · · · ∧ Yx(k) in time is equivalent to dissipativity of all operators D ∧k Ax(t) ; t ¿ 0 in the norm • . Existence of such norm for all AxD∧k (x ∈ U ) is equivalent to decreasing of volumes of all parallelepipeds due to 8rst approximation system (457) for any solution x(t) of Eqs. (428). If one can choose such ! ¿ 0 that for all Ax (x ∈ U ), any vector z ∈ E ∧ E ∧ · · · ∧ E, and this norm exp(AxD∧k t)z 6 exp(−!t) z , then the volumes of parallelepipeds decrease exponentially as exp(−!t). For such systems we can estimate the HausdorF dimension of the attractor [226] (under some additional technical conditions about solutions boundedness): it cannot exceed k. It is necessary to stress here that this estimation of the Hausdor6 dimension does not solve the problem of construction of the invariant manifold containing this attractor, and one needs special technique and additional restriction on the system to obtain this manifold (see [235,242,238,243]). The simplest way for construction slow invariant manifold is possible for systems with a dominance of linear part in highest dimensions. Let an in8nite-dimensional system have a form: u˙ + Au = R(u), where A is self-adjoint, and has discrete spectrum i → ∞ with suRciently big gaps between
i , and R(u) is continuous. One can build the slow manifold as the graph over a root space of A. Let the basis consists of eigenvectors of A. In this basis u˙ i = − i ui + Ri (u), and it seems very plausible that for some k and suRciently big i functions ui (t) exponentially fast tend to ui (u1 (t); : : : uk (t)), if Ri (u) are bounded and continuous in a suitable sense. Di6erent variants of rigorous theorems about systems with such a dominance of linear part in highest dimensions linear may be found in literature (see, for example, the textbook [76]). Even if
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all the suRcient conditions hold, it remains the problem of eRcient computing of these manifold, and di6erent ways for calculation are proposed: from Euler method for manifold correction [13] to di6erent algorithms of discretisations [3,81,239]. The simplest conditions of simultaneous dissipativity for the family of operators {Ax } can be created in a following way: let us take a norm • . If all operators Ax are dissipative with respect to this norm, then the family Ax is (evidently) simultaneously dissipative in this norm. So, we can verify or falsify a hypothesis about simultaneous dissipativity for a given norm. Simplest examples give us quadratic and l1 norms. For quadratic norm associated with a scalar product | dissipativity of operator A is equivalent to nonpositivity of all points of spectrum A + A+ , where A+ is the adjoint to A operator with respect to scalar product |. For l1 norm with weights x = i wi |xi |; wi ¿ 0. The condition of operator A dissipativity for this norm is the weighted diagonal predominance for columns of the A matrix A = (aij ): aii ¡ 0; wi |aii | ¿ wj |aji | : j; j =i
For exponential contraction it is necessary and suRcient that some gap exists in the dissipativity inequalities: + + for quadratic norm (A + A+ ) ¡ ! ¡ 0, where (A + A ) is the spectrum of A + A ; 1 for l norm with weights aii ¡ 0; wi |aii | ¿ j; j=i wj |aji | + !, ! ¿ 0. The suRcient conditions of simultaneous dissipativity can have another form (not only the form of dissipativity checking with respect to a given norm) [221–224], but the problem of necessary and suRcient conditions in general case is open. The dissipativity conditions for operators ADx ∧k of induced action of Ax on the kth exterior power of E have the similar form, for example, if we know the spectrum of A + A+ , then it is easy to 8nd D ∧k D ∧k + the spectrum of Ax(t) + (Ax(t) ) : each eigenvalue of this operator is a sum of k distinct eigenvalues + D ∧k D ∧k + of A + A ; the Ax(t) + (Ax(t) ) spectrum is a closure of set of sums of k distinct points A + A+ spectrum. A basis the kth exterior power of E can be constructed from the basis {ei } of E: it is {ei1 i2 :::ik } = {ei1 ∧ ei2 ∧ · · · ∧ eik };
i1 ¡ i2 ¡ · · · ¡ ik :
For l1 norm with weights in the kth exterior power of E the set of weights is {wi1 i2 :::ik ¿ 0; i1 ¡ i2 ¡ · · · ¡ ik }. The norm of a vector z is z = wi1 i2 ···ik |zi1 i2 :::ik | : i1 ¡i2 ¡···¡ik
The dissipativity conditions for operators AD∧k of induced action of A in l1 norm with weights have the form ai1 i1 + ai2 i2 + · · · + aik ik ¡ 0 ; wi1 i2 :::ik |ai1 i1 + ai2 i2 + · · · + aik ik | ¿
k
l=1 j; j =i1 ; i2 ;:::; ik
wil;1 ij2 :::ik |ajil |
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for any i1 ¡ i2 ¡ · · · ¡ ik ;
(459)
wil;1 ij2 :::ik
= wI , multiindex I consists of indices ip (p = l), and j. where For in8nite-dimensional systems the problem of volume contraction and Lyapunov norms for exterior powers of E consists of three parts: geometrical part concerning the choice of norm for simultaneous dissipativity of operator families, topological part concerning topological nonequivalence of constructed norms, and estimation of the bounded set containing compact attractor. The diRcult problem may concern the appropriate a priori estimations of the bounded convex positively invariant set V ⊂ U where the compact attractor is situated. It may be crucial to solve the problem of simultaneous dissipativity for the most narrow family of operators {Ax ; x ∈ V } (and their induced action on the kth exterior power of E). The estimation of attractor dimension based on Lyapunov norms in the exterior powers is rather rough. This is a local estimation. More exact estimations are based on global Lyapunov exponents (Lyapunov or Kaplan–Yorke dimension [228,229]). There are many di6erent measures of dimension [227,230], and many e6orts are applied to create good estimations for di6erent dimensions [241]. Estimations of attractor dimension was given for di6erent systems: from the Navier–Stokes hydrodynamic [233] to climate dynamics [232]. The introduction and review of many results is given in the book [235]. But local estimations remain the main tools for estimation of attractors dimension, because global estimations for complex systems are much more complicated and often unattainable because of computation complexity. Example 12: The universal limit in dynamics of dilute polymeric solutions The method of invariant manifold is developed for a derivation of reduced description in kinetic equations of dilute polymeric solutions. It is demonstrated that this reduced description becomes universal in the limit of small Deborah and Weissenberg numbers, and it is represented by the (revised) Oldroyd 8 constants constitutive equation for the polymeric stress tensor. CoeRcients of this constitutive equation are expressed in terms of the microscopic parameters. A systematic procedure of corrections to the revised Oldroyd 8 constants equations is developed. Results are tested with simple =ows. Kinetic equations arising in the theory of polymer dynamics constitute a wide class of microscopic models of complex =uids. Same as in any branch of kinetic theory, the problem of reduced description becomes actual as soon as the kinetic equation is established. However, in spite of an enormous amount of work in the 8eld of polymer dynamics [113–115,249,259], this problem remains less studied as compared to other classical kinetic equations. It is the purpose of this section to suggest a systematic approach to the problem of reduced description for kinetic models of polymeric =uids. First, we would like to specify our motivation by comparing the problem of the reduced description for that case with a similar problem in the familiar case of the rare8ed gas obeying the classical Boltzmann kinetic equation [87,51]. The problem of reduced description begins with establishing a set of slow variables. For the Boltzmann equation, this set is represented by 8ve hydrodynamic 8elds (density, momentum and energy) which are low-order moments of the distribution function, and which are conserved quantities of the dissipation process due to particle’s collisions. The reduced description is a closed system of
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equations for these 8elds. One starts with the manifold of local equilibrium distribution functions (local Maxwellians), and 8nds a correction by the Chapman–Enskog method [51]. The resulting reduced description (the Navier–Stokes hydrodynamic equations) is universal in the sense that the form of equations does not depend on details of particle’s interaction whereas the latter shows up explicitly only in the transport coeRcients (viscosity, temperature conductivity, etc.). Coming back to the complex =uids, we shall consider the simplest case of dilute polymer solutions represented by dumbbell models studied below. Two obstacles preclude an application of the traditional techniques. First, the question which variables should be regarded as slow is at least less evident because the dissipative dynamics in the dumbbell models has no nontrivial conservation laws compared to the Boltzmann case. Consequently, a priori, there are no distinguished manifolds of distribution functions like the local equilibria which can be regarded as a starting point. Second, while the Boltzmann kinetic equation provides a self-contained description, the dumbbell kinetic equations are coupled to the hydrodynamic equations. This coupling manifests itself as an external =ux in the kinetic equation. The well-known distinguished macroscopic variable associated with the dumbbell kinetic equations is the polymeric stress tensor [113,259]. This variable is not the conserved quantity but nevertheless it should be treated as a relevant slow variable because it actually contributes to the macroscopic (hydrodynamic) equations. Equations for the stress tensor are known as the constitutive equations, and the problem of reduced description for the dumbbell models consists in deriving such equations from the kinetic equation. Our approach is based on the method of invariant manifold [6], modi8ed for systems coupled with external 8elds. This method suggests constructing invariant sets (or manifolds) of distribution functions that represent the asymptotic states of slow evolution of the kinetic system. In the case of dumbbell models, the reduced description is produced by equations which constitute stress-strain relations, and two physical requirements are met by our approach: The 8rst is the principle of frame-indiFerence with respect to any time-dependent reference frame. This principle requires that the resulting equations for the stresses contain only frame-indi6erent quantities. For example, the frame-dependent vorticity tensor should not show up in these equations unless being presented in frame-indi6erent combinations with another tensors. The second principle is the thermodynamic stability: In the absence of the =ow, the constitutive model should be purely dissipative, in other words, it should describe the relaxation of stresses to their equilibrium values. The physical picture addressed below takes into account two assumptions: (i) In the absence of the =ow, deviations from the equilibrium are small. Then the invariant manifold is represented by eigenvectors corresponding to the slowest relaxation modes. (ii) When the external =ow is taken into account, it is assumed to cause a small deformation of the invariant manifolds of the purely dissipative dynamics. Two characteristic parameters are necessary to describe this deformation. The 8rst is the characteristic time variation of the external 8eld. The second is the characteristic intensity of the external 8eld. For dumbbell models, the 8rst parameter is associated with the conventional Deborah number while the second one is usually called the Weissenberg number. An iteration approach which involves these parameters is developed. Two main results of the analysis are as follows: First, the lowest-order constitutive equations with respect to the characteristic parameters mentioned above has the form of the revised phenomenological Oldroyd 8 constants model. This result is interpreted as the macroscopic limit of the microscopic dumbbell dynamics whenever the rate of the strain is low, and the Deborah number is small. This
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limit is valid generically, in the absence or in the presence of the hydrodynamic interaction, and for the arbitrary nonlinear elastic force. The phenomenological constants of the Oldroyd model are expressed in a closed form in terms of the microscopic parameters of the model. The universality of this limit is similar to that of the Navier–Stokes equations which are the macroscopic limit of the Boltzmann equation at small Knudsen numbers for arbitrary hard-core molecular interactions. The test calculation for the nonlinear FENE force demonstrates a good quantitative agreement of the constitutive equations with solutions to the microscopic kinetic equation within the domain of their validity. The second result is a regular procedure of 8nding corrections to the zero-order model. These corrections extend the model into the domain of higher rates of the strain, and to =ows which alternate faster in time. Same as in the zero-order approximation, the higher-order corrections are linear in the stresses, while their dependence on the gradients of the =ow velocity and its time derivatives becomes highly nonlinear. The section is organized as follows: For the sake of completeness, we present the nonlinear dumbbell kinetic models in the next subsection, “The problem of reduced description in polymer dynamics”. In the section, “The method of invariant manifold for weakly driven systems”, we describe in details our approach to the derivation of macroscopic equations for an abstract kinetic equation coupled to external 8elds. This derivation is applied to the dumbbell models in the section, “Constitutive equations”. The zero-order constitutive equation is derived and discussed in detail in this section, as well as the structure of the 8rst correction. Tests of the zero-order constitutive equation for simple =ow problems are given in the section, “Tests on the FENE dumbbell model”. The problem of reduced description in polymer dynamics Elastic dumbbell models. The elastic dumbbell model is the simplest microscopic model of polymer solutions [113]. The dumbbell model re=ects the two features of real-world macromolecules to be orientable and stretchable by a =owing solvent. The polymeric solution is represented by a set of identical elastic dumbbells placed in an isothermal incompressible liquid. In this example we adopt notations used in kinetic theory of polymer dynamics [113]. Let Q be the connector vector between the beads of a dumbbell, and :(x; Q; t) be the con8guration distribution function which depends on the location in the space x at time t. We assume that dumbbells are distributed uniformly, and consider the normalization, :(x; Q; t) dQ =1. The Brownian motion of beads in the physical space causes a di6usion in the phase space described by the Fokker–Planck equation (FPE) [113]: D: 2kB T 9 9 F 9 · k · Q: + ·D· :+ =− : : (460) Dt 9Q S 9Q 9Q kB T Here D=Dt = 9=9t + C · ∇ is the material derivative, ∇ is the spatial gradient, k(x; t) = (∇C)† is the gradient of the velocity of the solvent C, † denotes transposition of tensors, D is the dimensionless di6usion matrix, kB is the Boltzmann constant, T is the temperature, S is the dimensional coeRcient characterizing a friction exerted by beads moving through solvent media (friction coeRcient [113,114]), and F = 9V=9Q is the elastic spring force de8ned by the potential V. We consider forces of the form F = Hf(Q2 )Q, where f(Q2 ) is a dimensionless function ofthe variable Q2 = Q · Q, and H is the dimensional constant. Incompressibility of solvent implies i kii = 0.
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Let us introduce a time dimensional constant S ;
r = 4H which coincides with a characteristic relaxation time of dumbbell con8guration in the case when the force F is linear: f(Q2 ) = 1. It proves convenient to rewrite the FPE (460) in the dimensionless form 9 ' ' 9 9 D: ' =− · k · Q: + ·D· : + F: : (461) ' ' ' D't 9Q 9Q 9Q ' = (H=kB T )1=2 Q, D=D't = 9=9't + C · ∇, ] 't = t= r is the Various dimensionless quantities used are: Q † ' ] ] dimensionless time, ∇ = r ∇ is the reduced space gradient, and k = k r = (∇C) is the dimensionless ' and F ' are tensor of the gradients of the velocity. In the sequel, only dimensionless quantities Q used, and we keep notations Q and F for them for the sake of simplicity. The quantity of interest is the stress tensor introduced by Kramers [113]: & = −Js ˙ + nkB T (1 − FQ) ;
(462) ˙ = k + k†
is the rate-of-strain tensor, n is the concentration where Js is the viscosity of the solvent, of polymer molecules, and the angle brackets stand for the averaging with the distribution function :: • ≡ •:(Q) dQ. The tensor &p = nkB T (1 − FQ)
(463)
gives a contribution to stresses caused by the presence of polymer molecules. The stress tensor is required in order to write down a closed system of hydrodynamic equations: DC (464) = −%−1 ∇p − ∇ · &[:] : Dt Here p is the pressure, and % = %s + %p is the mass density of the solution where %s is the solvent, and %p is the polymeric contributions. Several models of the elastic force are known in the literature. The Hookean law is relevant to small perturbations of the equilibrium con8guration of the macromolecule: F =Q :
(465)
In that case, the di6erential equation for & is easily derived from the kinetic equation, and is the well known Oldroyd-B constitutive model [113]. The second model, the FENE force law [250], was derived as an approximation to the inverse Langevin force law [113] for a more realistic description of the elongation of a polymeric molecule in a solvent: Q F= : (466) 1 − Q 2 =Q02 This force law takes into account the nonlinear sti6ness and the 8nite extendibility of dumbbells, where Q0 is the maximal extendibility. The features of the di6usion matrix are important for both the microscopic and the macroscopic behavior. The isotropic di6usion is represented by the simplest di6usion matrix DI = 12 1 :
(467)
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Here 1 is the unit matrix. When the hydrodynamic interaction between the beads is taken into account, this results in an anisotropic contribution to the di6usion matrix (467). The original form of this contribution is the Oseen–Burgers tensor DH [251,252]: QQ 1 1+ 2 D = DI − TDH ; DH = ; (468) Q Q where
T=
H kB T
1=2
S : 16,Js
Several modi8cations of the Oseen–Burgers tensor can be found in the literature (the Rotne–Prager– Yamakawa tensor [253,254]), but here we consider only the classical version. Properties of the Fokker–Planck operator. Let us review some of the properties of the Fokker– Planck operator J in the right hand side of Eq. (461) relevant to what will follow. This operator can be written as J = Jd + Jh , and it represents two processes. The 8rst term, Jd , is the dissipative part, 9 9 ·D· +F : (469) Jd = 9Q 9Q This part is responsible for the di6usion and friction which a6ect internal con8gurations of dumbbells, and it drives the system to the unique equilibrium state, :eq = c−1 exp(−V(Q2 )) ; where c = exp(−V) dQ is the normalization constant. The second part, Jh , describes the hydrodynamic drag of the beads in the =owing solvent: 9 ' Jh = − ·k·Q : (470) 9Q The dissipative nature of the operator Jd is re=ected by its spectrum. We assume that this spectrum consists of real-valued nonpositive eigenvalues, and that the zero eigenvalue is not degenerated. In the sequel, the following scalar product will be useful: −1 gh dQ : g; hs = :eq The operator Jd is symmetric and nonpositive de8nite in this scalar product: Jd g; hs = g; Jd hs Since
Jd g; gs = −
and
Jd g; gs 6 0 :
(471)
−1 :eq (9g=9Q) · :eq D · (9g=9Q) dQ ;
the above inequality is valid if the di6usion matrix D is positive semide8nite. This happens if D=DI (467) but is not generally valid in the presence of the hydrodynamic interaction (468). Let us split the operator Jd according to the splitting of the di6usion matrix D: Jd = JdI − TJdH
where JdI; H = 9=9Q · DI; H · (9=9Q + F) :
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Both the operators JdI and JdH have nondegenerated eigenvalue 0 which corresponds to their common eigenfunction :eq : JdI; H :eq = 0, while the rest of the spectrum of both operators belongs to the nonpositive real semi-axis. Then the spectrum of the operator Jd = JdI − TJdH remains nonpositive for suRciently small values of the parameter T. The spectral properties of both operators JdI; H depend only on the choice of the spring force F. Thus, in the sequel we assume that the hydrodynamic interaction parameter T is suRciently small so that the thermodynamic stability property (471) holds. We note that the scalar product •; •s coincides with the second di6erential D2 S|:eq of an entropy functional S[:]: •; •s = −D2 S|:eq [•; •], where the entropy has the form ( ) : : S[:] = − : ln dQ = − ln : (472) :eq :eq The entropy S grows in the course of dissipation: DS[Jd :] ¿ 0 : This inequality similar to inequality (471) is satis8ed for suRciently small T. Symmetry and nonpositiveness of operator Jd in the scalar product de8ned by the second di6erential of the entropy is a common property of linear dissipative systems. Statement of the problem. Given the kinetic equation (460), we aim at deriving di6erential equations for the stress tensor & (462). The latter includes the moments FQ = FQ: dQ. In general, when the di6usion matrix is nonisotropic and/or the spring force is nonlinear, closed equations for these moments are not available, and approximations are required. With this, any derivation should be consistent with the three requirements: (i) Dissipativity or thermodynamic stability: the macroscopic dynamics should be dissipative in the absence of the =ow. (ii) Slowness: the macroscopic equations should represent slow degrees of freedom of the kinetic equation. (iii) Material frame indiFerence: the form of equations for the stresses should be invariant with respect to the Eucluidian, time dependent transformations of the reference frame [113,255]. While these three requirements should be met by any approximate derivation, the validity of our approach will be restricted by two additional assumptions: (a) Let us denote 1 the inertial time of the =ow, which we de8ne via characteristic value of the gradient of the =ow velocity: 1 = |∇C|−1 , and 2 the characteristic time of the variation of the =ow velocity. We assume that the characteristic relaxation time of the molecular con8guration r is small as compared to both the characteristic times 1 and 2 : r 1
and
r 2
:
(473)
(b) In the absence of the =ow, the initial deviation of the distribution function from the equilibrium is small so that the linear approximation is valid. While assumption (b) is merely of a technical nature, and it is intended to simplify the treatment of the dissipative part of the Fokker–Planck operator (469) for elastic forces of a complicated form, assumption (a) is crucial for taking into account the =ow in an adequate way. We have assumed
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that the two parameters characterizing the composed system ‘relaxing polymer con8guration+=owing solvent’ should be small: These two parameters are: !1 =
r = 1;
!2 =
r= 2
:
(474)
The characteristic relaxation time of the polymeric con8guration is de8ned via the coeRcient r : r = c r , where c is some positive dimensionless constant which is estimated by the absolute value of the lowest nonzero eigenvalue of the operator Jd . The 8rst parameter !1 is usually termed the Weissenberg number while the second one !2 is the Deborah number (cf. Ref. [256, Section 7-2]). The method of invariant manifold for weakly driven systems The Newton iteration scheme. In this section we introduce an extension of the method of invariant manifold [6] onto systems coupled with external 8elds. We consider a class of dynamic systems of the form d: = Jd : + Jex (1): ; dt
(475)
where Jd is a linear operator representing the dissipative part of the dynamic vector 8eld, while Jex (1) is a linear operator which represents an external =ux and depends on a set of external 8elds 1 = {11 ; : : : ; 1k }. Parameters 1 are either known functions of the time, 1 = 1(t), or they obey a set of equations, d1 = ;(:; 1): dt
(476)
Without any restriction, parameters 1 are adjusted in such a way that Jex (1=0) ≡ 0. Kinetic equation (461) has form (475), and general results of this section will be applied to the dumbbell models below in a straightforward way. We assume that the vector 8eld Jd : has the same dissipative properties as the Fokker–Planck operator (469). Namely there exists the globally convex entropy function S which obeys: DS[Jd :] ¿ 0, and the operator Jd is symmetric and nonpositive in the scalar product •; •s de8ned by the second di6erential of the entropy: g; hs = −D2 S[g; h]. Thus, the vector 8eld Jd : drives the system irreversibly to the unique equilibrium state :eq . We consider a set of n real-valued functionals, Mi∗ [:] (macroscopic variables), in the phase space F of system (475). A macroscopic description is obtained once we have derived a closed set of equations for the variables Mi∗ . Our approach is based on constructing a relevant invariant manifold in phase space F. This manifold is thought as a 8nite-parametric set of solutions :(M ) to Eqs. (475) which depends on time implicitly via the n variables Mi [:]. The latter may di6er from the macroscopic variables Mi∗ . For systems with external =uxes (475), we assume that the invariant manifold depends also on the parameters 1, and on their time derivatives taken to arbitrary order: :(M; A), where A={1; 1(1) ; : : :} is the set of time derivatives 1(k) = d k 1=dt k . It is convenient to consider time derivatives of 1 as independent parameters. This assumption is important because then we do not need an explicit form of the Eqs. (476) in the course of construction of the invariant manifold.
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By a de8nition, the dynamic invariance postulates the equality of the “macroscopic” and the “microscopic” time derivatives: J:(M; A) =
n 9:(M; A) dMi i=1
9Mi
dt
+
∞ k 9:(M; A) n=0 j=1
91j(n)
1j(n+1) ;
(477)
where J = Jd + Jex (1). The time derivatives of the macroscopic variables, dMi =dt, are calculated as follows: dMi = DMi [J:(M; A)] ; (478) dt where DMi stands for di6erentials of the functionals Mi . Let us introduce the projector operator associated with the parameterization of the manifold :(M; A) by the values of the functionals Mi [:]: n 9:(M; A) DMi [ • ] : (479) PM = 9M i i=1 It projects vector 8elds from the phase space F onto tangent bundle T:(M; A) of the manifold :(M; A). Then Eq. (477) is rewritten as the invariance equation: (1 − PM )J:(M; A) =
∞ k 9:
91j(n)
n=0 j=1
1j(n+1) ;
(480)
which has the invariant manifolds as its solutions. Furthermore, we assume the following: (i) The external =ux Jex (1): is small in comparison to the dissipative part Jd :, i.e. with respect to some norm we require: |Jex (1):||Jd :|. This allows us to introduce a small parameter !1 , and to replace the operator Jex with !1 Jex in Eq. (475). Parameter !1 is proportional to the characteristic value of the external variables 1. (ii) The characteristic time 1 of the variation of the external 8elds 1 is large in comparison to the characteristic relaxation time r , and the second small parameter is !2 = r = 1 1. The parameter !2 does not enter the vector 8eld J explicitly but it shows up in the invariance equation. Indeed, with a substitution, 1(i) → !2i 1(i) , the invariance equation (477) is rewritten in a form which incorporates both the parameters !1 and !2 : (1 − PM ){Jd + !1 Jex }: = !2
k 9: (i) j=1 91j
i
1j(i+1) :
(481)
We develop a modi8ed Newton scheme for solution of this equation. Let us assume that we have some initial approximation to desired manifold :(0) . We seek the correction of the form :(1) = :(0) + :1 . Substituting this expression into Eq. (481), we derive (1 −
(0) PM ){Jd
= − (1 −
+ !1 Jex }:1 − !2
k 9:1 i
(0) PM )J:(0)
+ !2
(i) j=1 91j
k 9:(0) i
j=1
91j(i)
1j(i+1) 1j(i+1) :
(482)
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(0) Here PM is a projector onto tangent bundle of the manifold :(0) . Further, we neglect two terms in the left hand side of this equation, which are multiplied by parameters !1 and !2 , regarding them small in comparison to the 8rst term. In the result we arrive at the equation, (0) (0) (1 − PM )Jd :1 = −(1 − PM )J:(0) + !2
k 9:(0) (i) j=1 91j
i
1j(i+1) :
(483)
For (n + 1)th iteration we obtain (n) (0) )Jd :n+1 = −(1 − PM )J:(n) + !2 (1 − PM
k 9:(n) i
(i) j=1 91j
1j(i+1) ;
(484)
(n) where :(n) = ni=0 :i is the approximation of nth order and PM is the projector onto its tangent bundle. It should be noted that deriving Eqs. (483) and (484) we have not varied the projector PM with (n) respect to yet unknown term :n+1 , i.e. we have kept PM = PM and have neglected the contribution from the term :n+1 . The motivation for this action comes from the original paper [6], where it was shown that such modi8cation generates iteration schemes properly converging to slow invariant manifold. In order to gain the solvability of Eq. (484) an additional condition is required: (n) PM :n+1 = 0 :
(485)
This condition is suRcient to provide the existence of the solution to linear system (484), while the additional restriction onto the choice of the projector is required in order to guarantee the uniqueness of the solution. This condition is (n) (n) ker [(1 − PM )Jd ] ∩ ker PM =0 :
(486)
Here ker denotes a null space of the corresponding operator. How this condition can be met is discussed in the next subsection. It is natural to begin the iteration procedure (484) starting from the invariant manifold of the nondriven system. In other words, we choose the initial approximation :(0) as the solution of the invariance equation (481) corresponding to !1 = 0 and !2 = 0: (0) (1 − PM )Jd :(0) = 0 :
(487)
We shall return to the question how to construct solutions to this equation in the subsection “Linear zero-order equations”. The above recurrent equations (484), (485) present the simpli8ed Newton method for the solution of invariance equation (481), which involves the small parameters. A similar procedure for the Grad equations of the Boltzmann kinetic theory was used recently in the Ref. [9]. When these parameters are not small, one should proceed directly with Eqs. (482). Above, we have focused our attention on how to organize the iterations to construct invariant manifolds of weakly driven systems. The only question we have not yet answered is how to choose projectors in iterative equations in a consistent way. In the next subsection we discuss the problem of derivation of the reduced dynamics and its relation to the problem of the choice of projector.
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Projector and reduced dynamics. Below we suggest the projector which is equally applicable for constructing invariant manifolds by the iteration method (484), (485) and for generating macroscopic equations based on given manifold. Let us discuss the problem of constructing closed equations for macroparameters. Having some approximation to the invariant manifold, we nevertheless deal with a noninvariant manifold and we * is found the face the problem how to construct the dynamics on it. If the n-dimensional manifold : * as follows [6]: macroscopic dynamics is induced by any projector P onto the tangent bundle of : dMi∗ * : (488) = DMi∗ |:˜ [PJ :] dt To specify the projector we involve the two above mentioned principles: dissipativity and slowness. The dissipativity is required to have the unique and stable equilibrium solution for macroscopic equations, when the external 8elds are absent (1 = 0). The slowness condition requires the induced vector 8eld PJ: to match the slow modes of the original vector 8eld J:. * ) by the parameters Mi [:]. This paLet us consider the parameterization of the manifold :(M rameterization generates associated projector P = PM by Eq. (479). This leads us to look for the admissible parameterization of this manifold, where by admissibility we understand the concordance with the dissipativity and the slowness requirements. We solve the problem of the admissible parameterization in the following way. Let us de8ne the functionals Mi i = 1; : : : ; n by the set of the lowest eigenvectors ’i of the operator Jd : * = ’i ; : * s ; Mi [:] where Jd ’i = i ’i . The lowest eigenvectors ’1 ; : : : ; ’n are taken as a join of basises in the eigenspaces of the eigenvalues with smallest absolute values: 0 ¡ | 1 | 6 | 2 | 6 · · · 6 | n |. For simplicity we shall work with the orthonormal set of eigenvectors: ’i ; ’j s = ij with ij the Kronecker symbol. Since the function :eq is the eigenvector of the zero eigenvalue we have: Mi [:eq ] = ’i ; :eq s = 0. Then the associated projector PM , written as n * 9: ’i ; •s ; (489) PM = 9Mi i=1 will generate the equations in terms of the parameters Mi as follows: * s = ’i J : * s : dMi =dt = ’i PM J : Their explicit form is dMi + * )s ; = i Mi + Jex (1)gi ; :(M (490) dt + is the adjoint to operator Jex with respect to the scalar product •; •s . where the Jex Apparently, in the absence of forcing (1 ≡ 0) the macroscopic equations dMi =dt = i Mi are thermodynamically stable. They represent the dynamics of slowest eigenmodes of equations d:=dt = Jd :. Thus, projector (489) complies with the above stated requirements of dissipativity and slowness in the absence external =ux. To rewrite the macroscopic equations (490) in terms of the required set of macroparameters, ∗ Mi [:] = m∗i ; :s , we use formula (488) which is equivalent to the change of variables
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359
* )s in Eqs. (490). Indeed, this is seen from the relation: {M } → {M ∗ (M )}, Mi∗ = m∗i ; :(M * = DMi∗ |:˜ [PM J :]
9M ∗ i
9Mj
j
* : DMj |:˜ [J :]
We have constructed the dynamics with the help of the projector PM associated with the lowest eigenvectors of the operator Jd . It is directly veri8ed that such projector (489) ful8lls condition * For this reason it is natural to use the projector (489) for (485) for arbitrary manifold :(n) = :. both procedures, constructing the invariant manifold, and deriving the macroscopic equations. We have to note that the above described approach to de8ning the dynamics via the projector is di6erent from the concept of “thermodynamic parameterization” proposed in Refs. [6,5]. The latter was applicable for arbitrary dissipative systems including nonlinear ones, whereas the present derivations are applied solely for linear systems. Linear zero-order equations. In this section we focus our attention on the solution of the zero-order invariance equation (487). We seek the linear invariant manifold of the form :(0) (a) = :eq +
n
ai m i ;
(491)
i=1
where ai are coordinates on this manifold. This manifold can be considered as an expansion of the relevant slow manifold near the equilibrium state. This limits the domain of validity of manifolds (491) because they are not generally positively de8nite. This remark indicates that nonlinear invariant manifolds should be considered for large deviations from the equilibrium but this goes beyond the scope of this Example. The linear n-dimensional manifold representing the slow motion for the linear dissipative system (475) is associated with n slowest eigenmodes. This manifold should be built up as the linear hull of the eigenvectors ’i of the operator Jd , corresponding to the lower part of its spectrum. Thus we choose mi = ’i . Dynamic equations for the macroscopic variables M ∗ are derived in two steps. First, following the subsection, “Projector and reduced dynamics”, we parameterize the linear manifold :(0) with the values of the moments Mi [:] = ’i ; :s . We obtain that the parameterization of manifold (491) is given by ai = Mi , or: :(0) (M ) = :eq +
n
Mi ’i ;
i=1
Then the reduced dynamics in terms of variables Mi reads dMi + + Jex ’i ; ’j s Mj + Jex ’i ; :eq s ; = i Mi + dt j where i = ’i ; Jd ’i s are eigenvalues which correspond to eigenfunctions ’i .
(492)
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Second, we switch from the variables Mi to the variables Mi∗ (M ) = m∗i ; :(0) (M )s in Eq. (492). Resulting equations for the variables M ∗ are also linear: dMi∗ −1 + = (B )ij Mjk Bkl YMl∗ + (B−1 )ij Jex ’j ; ’k s YMk∗ dt jkl
+
jk
+ (B−1 )ij Jex ’j ; :eq s :
(493)
j
∗ is the deviation of the variable M ∗ from its equilibrium value M ∗ , and Here YMi∗ = Mi∗ − Meq i |i eq |i ∗ Bij = mi ; ’j and Mij = i ij .
Auxiliary formulas: 1. Approximations to eigenfunctions of the Fokker–Planck operator In this subsection we discuss the question how to 8nd the lowest eigenvectors :eq m0 (Q2 ) and ◦
:eq1 (Q2 )QQ of the operator Jd (469) in the classes of functions having a form: w0 (Q) and ◦
w1 (Q)QQ. The results presented in this subsection were used in the subsections: “Constitutive equations” and “Tests on the FENE dumbbell model”. It is directly veri8ed that Jd w0 = G0h w0 ; ◦
◦
Jd w1 QQ = (G1h w1 )QQ ; where the operators G0h and G1h are given by G0h = G0 − TH0 ;
G1h = G1 − TH1 :
(494)
The operators G0; 1 and H0; 1 act in the space of isotropic functions (i.e. dependent only on Q = (Q · Q)1=2 ) as follows: 2 9 1 92 9 + ; (495) − fQ G0 = 2 9Q2 9Q Q 9Q 6 9 1 92 9 + − 2f ; (496) − fQ G1 = 2 9Q2 9Q Q 9Q 2 9 2 9 2 9 ; (497) H0 = − fQ + Q 9Q2 9Q Q 9Q 2 5 9 1 9 2 9 + − 2f + 2 : (498) − fQ H1 = Q 9Q2 9Q Q 9Q Q The following two properties of the operators G0;h 1 are important for our analysis: Let us de8ne two scalar products •; •0 and •; •1 : y; x0 = xye ; y; x1 = xyQ4 e :
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361
Here •e is the equilibrium average as de8ned in (511). Then we state that for suRciently small T the operators G0h and G1h are symmetric and nonpositive in the scalar products •; •0 and •; •1 , respectively. Thus for obtaining the desired eigenvectors of the operator Jd we need to 8nd the eigenfunctions m0 and m1 related to the lowest nonzero eigenvalues of the operators G0;h 1 . Since we regard the parameter T small it is convenient, 8rst, to 8nd lowest eigenfunctions g0; 1 of the operators G0; 1 and, then, to use standard perturbation technique in order to obtain m0; 1 . For the perturbation of the 8rst order one 8nds [262]: m0 = g0 + Th0 ;
h0 = −g0
g0 H0 G0 g0 0 − G 0 H 0 g0 ; g0 ; g0 0
m1 = g1 + Th1 ;
h1 = −g1
g1 H1 G1 g1 1 − G 1 H 1 g1 : g1 ; g1 1
(499)
For the rest of this subsection we describe one recurrent procedure for obtaining the functions m0 and m1 in a constructive way. Let us solve this problem by minimizing the functionals '0; 1 : m0; 1 ; G0;h 1 m0; 1 0; 1 → min ; M0; 1 [m0; 1 ] = − m0; 1 ; m0; 1 0; 1
(500)
by means of the gradient descent method. Let us denote e0; 1 the eigenfunctions of the zero eigenvalues of the operators G0;h 1 . Their explicit values are e0 = 1 and e1 = 0. Let the initial approximations m(0) 0; 1 to the lowest eigenfunctions m0; 1 (0) be chosen so that m0; 1 ; e0; 1 0; 1 = 0. We de8ne the variation derivative M0; 1 =m0; 1 and look for the correction in the form (0) (0) m(1) 0; 1 = m0; 1 + m0; 1 ;
m(0) 0; 1 = 1
M0; 1 ; m0; 1
where scalar parameter 1 ¡ 0 is found from the condition: 9M0; 1 [m(1) 0; 1 (1)] =0 : 91 In the explicit form the result reads: (0) (0) m(0) 0; 1 = 10; 1 G0; 1 ;
where G(0) 0; 1 =
0;(0)1
=
2 (0) m(0) 0; 1 ; m0; 1 0; 1
(0) (0) h (m(0) 0; 1 0; 1 − G0; 1 m0; 1 ) ;
(0) h m(0) 0; 1 ; G0; 1 m0; 1 0; 1 (0) m(0) 0; 1 ; m0; 1 0; 1
;
(501)
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+ , (0) , m(0) 0; 1 ; m0; 1 0; 1 (0) 10; 1 = q0; 1 − -q0;2 1 + (0) (0) ; G0; 1 ; G0; 1 0; 1 (0) (0) h m0; 1 ; G0;h 1 m(0) G(0) 1 0; 1 0; 1 0; 1 ; G0; 1 G0; 1 0; 1 q0; 1 = : − (0) (0) (0) G(0) m(0) G(0) 0; 1 ; G0; 1 0; 1 0; 1 ; m0; 1 0; 1 0; 1 ; G0; 1 0; 1
(502)
Having the new correction m(1) 0; 1 we can repeat the procedure and eventually generate the recurrence scheme. Since by the construction all iterative approximations m0;(n)1 remain orthogonal to zero eigenfunctions e0; 1 : m0;(n)1 ; e0; 1 0; 1 = 0 we avoid the convergence of this recurrence procedure to the eigenfunctions e0; 1 . The quantities 0;(n)1 : 0;(n)1 =
G0;(n)1 ; G0;(n)1 0; 1
m0;(n)1 ; m0;(n)1 0; 1
can serve as relative error parameters for controlling the convergence of the iteration procedure (501). Auxiliary formulas: 2. Integral relations Let 8 be a sphere in R3 with the center at the origin of the coordinate system or be the entire space R3 . For any function s(x2 ), where x2 = x · x, x ∈ R3 , and any square 3 × 3 matrices A, B, C independent of x the following integral relations are valid: 2 ◦ ◦ 2 ◦ 4 s(x )xx(xx : A) dx = A sx dx ; 15 8 8 ◦ 4 ◦ ◦ ◦ (A · B +B · A) sx6 dx ; s(x2 )xx(xx : A)(xx : B) dx = 105 8 8 ◦ ◦ ◦ 2 ◦ s(x )xx(xx : A)(xx : B)(xx : C ) dx 8
◦ ◦ 4 ◦ = {A(B : C ) + B(A : C ) + C(A : B)} 315
8
sx8 dx :
Microscopic derivation of constitutive equations Iteration scheme. In this section we apply the above developed formalism to the elastic dumbbell ' model (461). External 8eld variables 1 are the components of the tensor k. Since we aim at constructing a closed description for the stress tensor & (462) with the six independent components, the relevant manifold in our problem should be six-dimensional. Moreover, ' k '(i) = Di k=Dt i . we allow a dependence of the manifold on the material derivatives of the tensor k: ∗ (1) ' ' Let : (M; K) K = {k; k ; : : :} be the desired manifold parameterized by the six variables Mi '(l) ) of the tensors k '(l) . Small i = 1; : : : ; 6 and the independent components (maximum eight for each k
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363
parameters !1 and !2 , introduced in the section: “The problem of reduced description in polymer dynamics”, are established by Eq. (474). Then we de8ne the invariance equation: ∞ 9: '(i+1) h (1 − PM )(Jd + !1 Jh ): = !2 ; (503) '(i) lm i=0 lm 9hlm where PM =(9:=9Mi )DMi [•] is the projector associated with chosen parameterization and summation ' indexes l; m run only eight independent components of tensor k. Following the further procedure we straightforwardly obtain the recurrent equations: 9:(n) (i+1) (n) (n) ' k )Jd :n+1 = −(1 − PM )[Jd + !1 Jh ]:(n) + !2 ; (504) (1 − PM (i) lm ' i lm 9klm (n) PM :n+1 = 0 ; (505) n where :n+1 is the correction to the manifold :(n) = i=0 :i . The zero-order manifold is found as the relevant solution to equation: (0) (1 − PM )Jd :(0) = 0 :
(506)
We construct zero-order manifold :(0) in the subsection, “Zero-order constitutive equation”. * K) The dynamics in general form. Let us assume that some approximation to invariant manifold :(a; is found (here a = {a1 ; : : : ; a6 } are some coordinates on this manifold). The next step is constructing the macroscopic dynamic equations. In order to comply with dissipativity and slowness by means of the recipe from the previous section we need to 8nd six lowest eigenvectors of the operator Jd . We shall always assume in a sequel that the hydrodynamic interaction parameter T is small enough that the dissipativity of Jd (471) is not violated. ◦
Let us consider two classes of functions: C1 = {w0 (Q2 )} and C2 = {w1 (Q2 )QQ}, where w0; 1 are functions of Q2 and the notation ◦ indicates traceless parts of tensor or matrix, e.g. for the dyad QQ: ◦ 1 (QQ)ij = Qi Qj − ij Q2 : 3 Since the sets C1 and C2 are invariant with respect to operator Jd , i.e. Jd C1 ⊂ C1 and Jd C2 ⊂ C2 , ◦
and densities FQ = fQQ + (1=3)1fQ2 of the moments comprising the stress tensor &p (463) belong to the space C1 + C2 , we shall seek the desired eigenvectors in the classes C1 and C2 . Namely, we intend to 8nd one lowest isotropic eigenvector :eq m0 (Q2 ) of eigenvalue − 0 ( 0 ¿ 0) and 8ve ◦
nonisotropic eigenvectors mij = :eq m1 (Q2 )(QQ)ij of another eigenvalue − 1 ( 1 ¿ 0). The method of derivation and analytic evaluation of these eigenvalues are discussed in the subsection “Auxiliary formulas, 1”. For a while we assume that these eigenvectors are known. * by the values of the functionals: In the next step we parameterize given manifold : * dQ ; * s = m0 : M0 = :eq m0 ; : ◦
◦
* s= M = :eq m1 QQ; :
◦
* dQ : m1 QQ:
(507)
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* 0 ; M; K) is obtained, the dynamic equations are found as Once a desired parameterization :(M ◦ DM0 + 0 M0 = (ˆ˙ : QQ)m0 ; D't ◦ ◦ ◦ ◦ ◦ 1 1 ˆ ˙ 1 Q2 + QQ(ˆ˙ : QQ)m1 ; (508) M[1] + 1 M = − 1ˆ˙ : M − m 3 3 * i.e. • = •: * dQ, m = dm0; 1 (Q2 )=d(Q2 ) and where all averages are calculated with the df :, 0; 1 subscript [1] represents the upper convective derivative of tensor:
'[1] =
D' '·'+'·k '† } : − {k D't
The parameters 0; 1 , which are absolute values of eigenvalues of operator Jd , are calculated by formulas (for de8nition of operators G1 and G2 see subsection “Auxiliary formulas, 1”):
0 = −
m0 G0 m0 e ¿0 ; m0 m0 e
(509)
1 = −
Q4 m1 G1 m1 e ¿0 ; m1 m1 Q4 e
(510)
where we have introduced the notation of the equilibrium average: ye = :eq y dQ :
(511)
Equations on components of the polymeric stress tensor &p (463) are constructed as a change of ◦ * makes this operation straightforward: variables {M0 ; M} → &p . The use of the projector P ◦ D&p * 0 (&p ; K); M(&p ; K); K) dQ : * :(M = −nkB T FQ PJ (512) D't ◦
* is associated with the parameterization by the variables M0 and M: Here, the projector P 9: * * ◦ * = 9: :eq m0 ; •s + P :eq m1 (QQ)kl ; •s : ◦ 9M0 9Mkl
(513)
kl
We note that sometimes it is easier to make transition to the variables &p after solving Eqs. (508) rather than to construct explicitly and solve equations in terms of &p . It allows to avoid inverting ◦
the functions &p (M0 ; M) and to deal with simpler equations. Zero-order constitutive equation. In this subsection we derive the closure based on the zero-order manifold :(0) found as appropriate solution to Eq. (506). Following the approach described in subsection, “Linear zero-order equations”, we construct such a solution as the linear expansion near ◦ the equilibrium state :eq (491). After parameterization by the values of the variables M0 and M
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365
◦
associated with the eigenvectors :eq m0 and :eq m1 QQ we 8nd: ◦ 15 ◦ m1 m0 :(0) = :eq 1 + M0 : + M : QQ m0 m0 e 2 m1 m1 Q4 e Then with the help of projector (513): ◦ ◦ m1 m0 15 (0) PM = :eq m0 ; •e + QQ : m1 QQ; •e m0 m0 e 2 m1 m1 Q4 e by formula (512) we obtain D tr &p ◦ ˆ˙ ; + 0 tr &p = a0 (&p : ) D't &p[1] + 0 &p = b0 [&p · ˆ˙ + ˆ˙ · &p ] − ◦
◦
◦
◦
where the constants bi , a0 are fm0 Q2 e m0 m1 Q4 m1 e a0 = ; fm0 Q4 e m20 e
1 ◦ ˆ 1(&p : ) ˙ + (b1 tr &p − b2 nkB T )ˆ˙ ; 3
(514)
(515)
(516)
2 m1 m2 Q6 e ; 7 m21 Q4 e m0 m2 Q4 e 1 fm1 Q4 e m0 m1 Q2 e b1 = 2 ; +5 15 fm0 Q2 e m1 m1 Q4 e m21 Q4 e
b0 =
b2 =
1 fm1 Q4 e {2m2 Q4 e + 5m1 Q2 e } : 15 m1 m1 Q4 e
(517)
We remind that m0; 1 = 9m0; 1 =9(Q2 ). These formulas were obtained using the auxiliary results from subsection “Auxiliary formulas, 2”. Revised Oldroyd 8 constant constitutive equation for the stress. It is remarkable that being rewritten in terms of the full stresses & = −Js ˙ + &p the dynamic system (516) takes a form: ˙ + c5 (tr &)˙ + 1(c6 & : ˙ + c8 tr &) & + c1 &[1] + c3 {˙ · & + & · } = − J{˙ + c2 ˙[1] + c4 ˙ · ˙ + c7 (˙ : )1} ˙ ;
(518)
where the parameters J, ci are given by the following relationships: J = r Js @; c1 = r = 1 ;
@ = 1 + nkB T 1 b2 =Js ; c2 = r =(@ 1 ) ;
c3 = −b0 r = 0 ;
c4 = −2b0 r =(@ 1 ) ;
c5 =
r (2b0 − 3b1 − 1); 3 1
c7 =
r (2b0 + 1 − a0 );
1 @
c6 = c8 =
r (2b0 + 1 − a0 ) ;
1
1 ( 0 = 1 − 1) : 3
(519)
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In the last two formulas we returned to the original dimensional quantities: time t and gradient of velocity tensor k=∇C, and at the same time we kept the old notations for the dimensional convective derivative '[1] = D'=Dt − k · ' − ' · k† . Note that all parameters (519) are related to the entropic spring law f due to Eq. (517). Thus, the constitutive relation for the stress & (518) is fully derived from the microscopic kinetic model. If the constant c8 were equal to zero, then the form of Eq. (518) would be recognized as the Oldroyd 8 constant model [257], proposed by Oldroyd about 40 years ago on a phenomenological basis. Nonzero c8 indicates a presence of di6erence between r = 0 and r = 1 which are relaxation ◦ times of trace tr & and traceless components & of the stress tensor &. Higher-order constitutive equations. In this subsection we discuss the properties of corrections to (0) the zero-order model (518). Let PM (515) be the projector onto the zero-order manifold :(0) (514). The invariance equation (504) for the 8rst-order correction :(1) = :(0) + :1 takes a form (0) L:1 = −(1 − PM )(Jd + Jh ):(0) ; (0) :1 = 0 ; PM
(520)
(0) (0) where L = (1 − PM )Jd (1 − PM ) is the symmetric operator. If the manifold :(0) is parameterized ◦ ◦ ◦ by the functionals M0 = g0 :(0) dQ and M = m1 QQ:(0) dQ, where :eq m0 and :eq QQm1 are lowest eigenvectors of Jd , then the general form of the solution is given by . ◦ ◦ ◦ ◦ :1 = :eq z0 M0 (˙ : QQ) + z1 (M : QQ)(˙ : QQ)
◦ 1 + z2 {˙ · M + M · } ˙ : QQ + z3 ˙ : M + ˙ : QQ 2 ◦
◦
◦
◦
:
(521)
The terms z0 –z3 are the functions of Q2 found as the solutions to some linear di6erential equations. ◦ We observe two features of the new manifold: 8rst, it remains linear in variables M0 and M and second it contains the dependence on the rate of strain tensor . ˙ As the consequence, the transition to variables & is given by the linear relations: ◦
◦
◦ ◦ ◦ ◦ &p − = r0 M + r1 M0 ˙ + r2 {˙ · M + M · } ˙ + r3 ˙ · ˙ ; nkB T ◦ tr &p = p0 M0 + p1 ˙ : M ; (522) nkB T where ri and pi are some constants. Finally the equations in terms of & should be also linear. Analysis shows that the 8rst-order correction to the modi8ed Oldroyd 8 constants model (518) will be transformed into the equations of the following general structure:
−
& + c1 &[1] + {(1 · & · (2 + (†2 · & · (†1 } + (3 (tr &) + (4 ((5 : &) = −J0 (6 ;
(523)
where (1 –(6 are tensors dependent on the rate-of-strain tensor ˙ and its 8rst convective derivative ˙[1] , constant c1 is the same as in Eq. (519) and J0 is a positive constant. Because the explicit form of the tensors (i is quite extensive we do not present them in this section. Instead we give several general remarks about the structure of the 8rst- and higher-order
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corrections: (1) Since manifold (521) does not depend on the vorticity tensor ! = k − k† the latter enters Eqs. (523) only via convective derivatives of & and . ˙ This is suRcient to acquire the frame indi6erence feature, since all the tensorial quantities in dynamic equations are indi6erent in any time dependent reference frame [256]. (2) When k = 0 the 8rst order equations (523) as well as equations for any order reduce to linear relaxation dynamics of slow modes: ◦
D & 1 ◦ + &=0 ; Dt
r D tr & 0 + tr & = 0 ; Dt
r which is obviously concordant with the dissipativity and the slowness requirements. (3) In all higher-order corrections one will be always left with linear manifolds if the projector ◦ associated with functionals M0 [:] and M[:] is used in every step. It follows that the resulting constitutive equations will always take a linear form (523), where all tensors i depend on higher order convective derivatives of ˙ (the highest possible order is limited by the order of the correction). Similarly to the 8rst and zero orders the frame indi6erence is guaranteed if the manifold does not depend on the vorticity tensor unless the latter is incorporated in any frame invariant time derivatives. It is reasonable to eliminate the dependence on vorticity (if any) at the stage of constructing the solution to iteration equations (504). (4) When the force F is linear F = Q our approach is proven to be also correct since it leads the Oldroyd-B model (Eq. (518) with ci = 0 for i = 3; : : : ; 8). This follows from the fact that the spectrum of the corresponding operator Jd is more degenerated, in particular 0 = 1 = 1 and the corresponding lowest eigenvectors comprise a simple dyad :eq QQ. Tests on the FENE dumbbell model In this section we specify the choice of the force law as the FENE springs (466) and present results of test calculations for the revised Oldroyd 8 constants (516) equations on the examples of two simple viscometric =ows. We introduce the extensibility parameter of FENE dumbbell model b: 2 ' 2 = H Q0 : b=Q (524) 0 kB T It was estimated [113] that b is proportional to the length of polymeric molecule and has a meaningful variation interval 50–1000. The limit b → ∞ corresponds to the Hookean case and therefore to the Oldroyd-B constitutive relation. In our test calculations we will compare our results with the Brownian dynamic (BD) simulation data made on FENE dumbbell equations [258], and also with one popular approximation to the FENE model known as FENE-P (FENE-Peterelin) model [259,113,260]. The latter is obtained by self-consistent approximation to FENE force: 1 F= Q : (525) 1 − Q 2 =b
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Table 4 Values of constants to the revised Oldroyd 8 constants model computed on the base of the FENE dumbbells model b
0
1
b0
b1
b2
a0
20 50 100 200 ∞
1.498 1.198 1.099 1.050 1
1.329 1.135 1.068 1.035 1
−0.0742 −0.0326 −0.0179 0.000053 0
0.221 0.279 0.303 0.328 1/3
1.019 1.024 1.015 1.0097 1
0.927 0.982 0.990 1.014 1
Table 5 Corrections due to hydrodynamic interaction to the constants of the revised Oldroyd 8 constants model based on FENE force b
0
1
b0
b1
b2
a0
20 50 100
−0.076 −0.0618 −0.0574
−0.101 −0.109 −0.111
0.257 −0.365 −1.020
−0.080 0.0885 0.109
−0.0487 −0.0205 −0.020
−0.0664 −0.0691 −0.0603
This force law like Hookean case allows for the exact moment closure leading to nonlinear constitutive equations [113,260]. Speci8cally we will use the modi8ed variant of FENE-P model, which matches the dynamics of original FENE in near equilibrium region better than the classical variant. This modi8cation is achieved by a slight modi8cation of Kramers de8nition of the stress tensor: &p = nkB T (1 − b)1 − FQ :
(526)
The case = 0 gives the classical de8nition of FENE-P, while more thorough estimation [249,260] is = (b(b + 2))−1 . Constants. The speci8c feature of the FENE model is that the length of dumbbells Q can vary only in a bounded domain of R3 , namely inside a sphere Sb = {Q2 6 b}. The sphere Sb de8nes the domain of integration for averages •e = Sb :eq • dQ, where the equilibrium distribution reads :eq = c−1 (1 − Q2 =b)b=2 , c = Sb (1 − Q2 =b)b=2 dQ. In order to 8nd constants for the zero-order model (516) we do the following: First we analytically ◦
compute the lowest eigenfunctions of operator Jd : g1 (Q2 )QQ and g0 (Q2 ) without account of the hydrodynamic interaction (T = 0). The functions g0 and g1 are computed by a procedure presented in subsection “Auxiliary formulas, 1” with the help of the symbolic manipulation software Maple V.3 [261]. Then we calculate the perturbations terms h0; 1 by formulas (499) introducing the account of hydrodynamic interaction. Table 4 presents the constants 0; 1 , ai , bi (510), (517) of the zero-order model (516) without inclusion of hydrodynamic interaction T = 0 for several values of extensibility parameter b. The relative error 0; 1 (see subsection “Auxiliary formulas, 1”) of approximation for these calculations did not exceed the value 0:02. Table 5 shows the linear correction terms for constants from Table 4 which account a hydrodynamic interaction e6ect: 0;h 1 = 0; 1 (1 + T( 0; 1 )), ahi =ai (1+T(ai )), bhi =bi (1+T(bi )). The latter are calculated by substituting the perturbed functions
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369
m0; 1 = g0; 1 + Th0; 1 into (510) and (517), and expanding them up to 8rst-order in T. One can observe, since T ¿ 0, the e6ect of hydrodynamic interaction results in the reduction of the relaxation times. Dynamic problems. The rest of this section concerns the computations for two particular =ows. The shear =ow is de8ned by 0 1 0 k(t) = >(t) ˙ (527) 0 0 0 ; 0 0 0 where >(t) ˙ is the shear rate, and the elongation =ow corresponds to the choice: 1 0 0 0 k(t) = !(t) ˙ 0 −1=2 ; 0 0 −1=2
(528)
where !(t) ˙ is the elongation rate. In test computations we will look at the so-called viscometric material functions de8ned through the components of the polymeric part of the stress tensor &p . Namely, for shear =ow they are the shear viscosity J, the 8rst and the second normal stress coeRcients 1 , 2 , and for elongation =ow the only function is the elongation viscosity J. ] In dimensionless form they are written as 'J =
&p; 12 J − Js ; =− nkB T r >nk ] BT
'1 = '2 = #=
1
nkB T r2 2
nkB T r2
(529)
=
&p; 22 − &p; 11 ; >] 2 nkB T
(530)
=
&p; 33 − &p; 22 ; >] 2 nkB T
(531)
J] − 3Js &p; 22 − &p; 11 ; = nkB T r !nk ] BT
(532)
where >] = >
˙ r and !] = !
˙ r are dimensionless shear and elongation rates. Characteristic values of latter parameters >] and !] allow to estimate the parameter !1 (474). For all =ows considered below the second =ow parameter (Deborah number) !2 is equal to zero. Let us consider the steady state values of viscometric functions in steady shear and elongation =ows: >˙ = const, !˙ = const. For the shear =ow the steady values of these functions are found from Eqs. (516) as follows: 'J = b2 =( 1 − c>] 2 );
'1 = 2'J= 1 ;
'2 = 2b0'J= 1 ;
where c = 2=3(2b20 + 2b0 − 1)= 1 + 2b1 a0 = 0 . Estimations for the constants (see Table 1) shows that c 6 0 for all values of b (case c=0 corresponds to b=∞), thus all three functions are monotonically decreasing in absolute value with increase of quantity >, ] besides the case when b = ∞. Although
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b=50 1
ν^
0.8 0.6 0.4 0.2
100
_
γ
102
b=50 2
^
Ψ1
1.5
1
0.5
100
_
102
γ
Fig. 17. Dimensionless shear viscosity 'J and 8rst normal stress coeRcient '1 vs. shear rate: (——) revised Oldroyd 8 constants model; (· · · · · ·) FENE-P model; (◦ ◦ ◦) BD simulations on the FENE model; (− · − · −) Hookean dumbbell model.
they qualitatively correctly predict the shear thinning for large shear rates due to power law, but the exponent −2 of power dependence in the limit of large >] from the values −0:66 for parameter 'J and −1:33 for '1 observed in Brownian dynamic simulations [258]. It is explained by the fact that slopes of shear thining lie out of the applicability domain of our model. A comparison with BD simulations and modi8ed FENE-P model is depicted in Fig. 17. The predictions for the second normal stress coeRcient indicate one more di6erence between revised Oldroyd 8 constant equation and FENE-P model. FENE-P model shows identically zero values for '2 in any shear =ow, either steady or time dependent, while the model (516), as well as BD simulations (see Fig. 9 in Ref. [258]) predict small, but nonvanishing values for this quantity. Namely, due to model (516) in shear =ows the following relation '2 = b0 '1 is always valid, with proportionality coeRcient b0 small and mostly negative, what leads to small and mostly negative values of '2 .
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371
b=50 120 100
ϑ
80 60 40 20
10
-1
10
0
_
ε
10
1
10
2
Fig. 18. Dimensionless elongation viscosity vs. elongation rate: (——) revised Oldroyd 8 constants model, (· · · · · ·) FENE-P model, (◦ ◦ ◦) BD simulations on the FENE model; (− · − · −) Hookean dumbbell model. Table 6 Singular values of elongation rate b !] ∗
20 0.864
50 0.632
100 0.566
120 0.555
In the elongation =ow the steady state value to # is found as 3b2 #= : 5
1 − 6 (2b0 + 1)!] − 7b1 a0 !]2 = 0 The denominator has one root on positive semi-axis 1=2 5 0 (2b0 + 1) 5 0 (2b0 + 1) 2
1 0 !]∗ = − + + ; 84b1 a0 84b1 a0 7b1 a0
200 0.520
∞ 0.5
(533)
(534)
which de8nes a singularity point for the dependence #(!). ] The BD simulation experiments [258] on the FENE dumbbell models shows that there is no divergence of elongation viscosity for all values of elongation rate (see Fig. 18). For Hookean springs !]∗ = 1=2 while in our model (516) the singularity point shifts to higher values with respect to decreasing values of b as it is demonstrated in Table 6. Fig. 19 gives an example of dynamic behavior for elongation viscosity in the instant start-up of the elongational =ow. Namely it shows the evolution of initially vanishing polymeric stresses after instant jump of elongation rate at the time moment t = 0 from the value !] = 0 to the value !] = 0:3. It is possible to conclude that the revised Oldroyd 8 constants model (516) with estimations given by (517) for small and moderate rates of strain up to !1 = r ||=(2
˙ 1 ) ∼ 0:5 yields a good approximation to original FENE dynamics. The quality of the approximation in this interval is the same or better than the one of the nonlinear FENE-P model.
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A.N. Gorban et al. / Physics Reports 396 (2004) 197 – 403 _
γ=1 1
b=100 b=50
0.8
ν^
b=20 0.6 0.4 0.2 0 −2 10
10
−1
10
^
0
10
1
t
_
γ=1 2
b=100 b=50
^
ψ1
1.5
b=20
1
0.5
0 10
−1
10
0
10
1
^
t
Fig. 19. Time evolution of elongation viscosity after inception of the elongation =ow with elongation rate !] = 0:3: (——) revised Oldroyd 8 constants model, (· · · · · ·) FENE-P model, (− − −) BD simulations on FENE model; (− · − · −) Hookean dumbbell model.
The main results of this Example are as follows: (i) We have developed a systematic method of constructing constitutive equations from the kinetic dumbbell models for the polymeric solutions. The method is free from a priori assumptions about the form of the spring force and is consistent with basic physical requirements: frame invariance and dissipativity of internal motions of =uid. The method extends so-called the method of invariant manifold onto equations coupled with external 8elds. Two characteristic parameters of =uid =ow were distinguished in order to account for the e6ect of the presence of external 8elds. The iterative Newton scheme for obtaining a slow invariant manifold of the system driven by the =ow with relatively low values of both characteristic parameters was developed.
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(ii) We demonstrated that the revised phenomenological Oldroyd 8 constants constitutive equations represent the slow dynamics of microscopic elastic dumbbell model with any nonlinear spring force in the limit when the rate of strain and frequency of time variation of the =ow are suRciently small and microscopic states at initial time of evolution are taken not far from the equilibrium. (iii) The corrections to the zero-order manifold lead generally to linear in stresses equations but with highly nonlinear dependence on the rate of strain tensor and its convective derivatives. (iv) The zero-order constitutive equation is compared to the direct Brownian dynamics simulation for FENE dumbbell model as well as to predictions of FENE-P model. This comparison shows that the zero-order constitutive equation gives the correct predictions in the domain of its validity, but does not exclude qualitative discrepancy occurring out of this domain, particularly in elongation =ows. This discrepancy calls for a further development, in particular, the use of nonlinear manifolds for derivation of zero-order model. The reason is in the necessity to provide concordance with the requirement of the positivity of distribution function. It may lead to nonlinear constitutive equation on any order of correction. These issues are currently under consideration and results will be reported separately. Example 13: Explosion of invariant manifold, limits of macroscopic description for polymer molecules, molecular individualism, and multimodal distributions Derivation of macroscopic equations from the simplest dumbbell models is revisited [84]. It is demonstrated that the onset of the macroscopic description is sensitive to the =ows. For the FENE-P model, small deviations from the Gaussian solution undergo a slow relaxation before the macroscopic description sets on. Some consequences of these observations are discussed. A new class of closures is discussed, the kinetic multipeak polyhedra. Distributions of this type are expected in kinetic models with multidimensional instability as universally, as the Gaussian distribution appears for stable systems. The number of possible relatively stable states of a nonequilibrium system grows as 2m , and the number of macroscopic parameters is in order mn, where n is the dimension of con8guration space, and m is the number of independent unstable directions in this space. The elaborated class of closures and equations pretends to describe the e6ects of “molecular individualism”. Dumbbell models and the problem of the classical Gaussian solution stability We shall consider the simplest case of dilute polymer solutions represented by dumbbell models. The dumbbell model re=ects the two features of real-world macromolecules to be orientable and stretchable by a =owing solvent [113]. Let us consider the simplest one-dimensional kinetic equation for the con8guration distribution function :(q; t), where q is the reduced vector connecting the beads of the dumbbell. This equation is slightly di6erent from the usual Fokker–Planck equation. It is nonlinear, because of the dependence of potential energy U on the moment M2 [:] = q2 :(q) dq. This dependence allows us to get the exact quasi-equilibrium equations on M2 , but this equations are not solving the problem: this
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quasi-equilibrium manifold may become unstable when the =ow is present [84]. Here is this model: 9t : = −9q {1(t)q:} +
1 2 9: : 2 q
(535)
Here 1 (536) f(M2 (t)) ; 2 T(t) is the given time-independent velocity gradient, t is the reduced time, and the function −fq is the reduced 2 spring force. Function f may depend on the second moment of the distribution function M2 = q :(q; t) dq. In particular, the case f ≡ 1 corresponds to the linear Hookean spring, while f = [1 − M2 (t)=b]−1 corresponds to the self-consistent 8nite extension nonlinear elastic spring (the FENE-P model, 8rst introduced in [260]). The second moment M2 occurs in the FENE-P force f as the result of the pre-averaging approximation to the original FENE model (with nonlinear spring force f = [1 − q2 =b]−1 ). The parameter b changes the characteristics of the force law from Hookean at small extensions to a con8ning force for q2 → b. Parameter b is roughly equal to the number of monomer units represented by the dumbbell and should therefore be a large number. In the limit b → ∞, the Hookean spring is recovered. Recently, it has been demonstrated that FENE-P model appears as 8rst approximation within a systematic self-con8dent expansion of nonlinear forces [16]. Eq. (535) describes an ensemble of noninteracting dumbbells subject to a pseudo-elongational =ow with 8xed kinematics. As is well known, the Gaussian distribution function, 1 q2 :G (M2 ) = √ ; (537) exp − 2M2 2,M2 1(t) = T(t) −
solves Eq. (535) provided the second moment M2 satis8es dM2 = 1 + 21(t)M2 : (538) dt Solution (537) and (538) is the valid macroscopic description if all other solutions of Eq. (535) are rapidly attracted to the family of Gaussian distributions (537). In other words [6], the special solution (537) and (538) is the macroscopic description if Eq. (537) is the stable invariant manifold of the kinetic equation (535). If not, then the Gaussian solution is just a member of the family of solutions, and Eq. (538) has no meaning of the macroscopic equation. Thus, the complete answer to the question of validity of Eq. (538) as the macroscopic equation requires a study of dynamics in the neighborhood of manifold (537). Because of the simplicity of model (535), this is possible to a satisfactory level even for M2 -dependent spring forces. Dynamics of the moments and explosion of the Gaussian manifold In the paper [84] it was shown, that there is a possibility of “explosion” of the Gaussian manifold: with the small initial deviation from it, the solutions of Eq. (535) are very fast going far from, and then slowly come back to the stationary point which is located on the Gaussian manifold. The distribution function : is stretched fast, but loses the Gaussian form, and after that the Gaussian form recovers slowly with the new value of M2 . Let us describe brie=y the results of [84].
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375
Let M2n = q2n : dq denote the even moments (odd moments vanish by symmetry). We consider G G deviations @2n = M2n − M2n , where M2n = q2n :G dq are moments of the Gaussian distribution function (537). Let :(q; t0 ) be the initial condition to Eq. (535) at time t = t0 . Introducing functions, t p2n (t; t0 ) = exp 4n (539) 1(t ) dt ; t0
where t ¿ t0 , and 2n ¿ 4, the exact time evolution of the deviations @2n for 2n ¿ 4 reads @4 (t) = p4 (t; t0 )@4 (t0 ) ; and
(540)
t −1 @2n (t) = @2n (t0 ) + 2n(4n − 1) @2n−2 (t )p2n (t ; t0 ) dt p2n (t; t0 ) ; t0
(541)
for 2n ¿ 6. Eqs. (539)–(541) describe evolution near the Gaussian solution for arbitrary initial condition :(q; t0 ). Notice that explicit evaluation of the integral in Eq. (539) requires solution to the moment equation (538) which is not available in the analytical form for the FENE-P model. It is straightforward to conclude that any solution with a nonGaussian initial condition converges to the Gaussian solution asymptotically as t → ∞ if t lim 1(t ) dt ¡ 0 : (542) t →∞
t0
However, even if this asymptotic condition is met, deviations from the Gaussian solution may survive for considerable t 1nite times. For example, if for some 8nite time T , the integral in Eq. (539) is estimated as t0 1(t ) dt ¿ 1(t − t0 ), 1 ¿ 0, t 6 T , then the Gaussian solution becomes exponentially unstable during this time interval. If this is the case, the moment equation (538) cannot be regarded as the macroscopic equation. Let us consider speci8c examples. For the Hookean spring (f ≡ 1) under a constant elongation (T = const), the Gaussian solution is exponentially stable for T ¡ 0:5, and it becomes exponentially unstable for T ¿ 0:5. The exponential instability in this case is accompanied by the well known breakdown of the solution to Eq. (538) due to in8nite stretching of the dumbbell. The situation is much more interesting for the FENE-P model because this nonlinear spring force does not allow the in8nite stretching of the dumbbell [274,275]. Eqs. (538) and (540) were integrated by the 5th order Runge–Kutta method with adaptive time step. The FENE-P parameter b was set equal to 50. The initial condition was :(q; 0)=C(1−q2 =b)b=2 , where C is the normalization (the equilibrium of the FENE model, notoriously close to the FENE-P equilibrium [258]). For this initial condition, in particular, @4 (0) = −6b2 =[(b + 3)2 (b + 5)] which is about 4% of the value of M4 in the Gaussian equilibrium for b = 50. In Fig. 20 we demonstrate deviation @4 (t) as a function of time for several values of the =ow. Function M2 (t) is also given for comparison. For small enough T we 8nd an adiabatic regime, that is @4 relaxes exponentially to zero. For stronger =ows, we observe an initial fast runaway from the invariant manifold with |@4 | growing over three orders of magnitude compared to its initial value. After the maximum deviation has been reached, @4 relaxes to zero. This relaxation is exponential as soon as the solution to Eq. (538) approaches the steady state. However, the time constant for this exponential relaxation |1∞ | is very small. Speci8cally, for large T, 1 1∞ = lim 1(t) = − + O(T−1 ) : (543) t →∞ 2b
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A.N. Gorban et al. / Physics Reports 396 (2004) 197 – 403 0.8
X
0.6
0.4
0.2
Y
0.0
−0.2
0
5
10
15
20
t
Fig. 20. Deviations of reduced moments from the Gaussian solution as a function of reduced time t in pseudo-elongation =ow for the FENE-P model. Upper part: Reduced second moment X = M2 =b. Lower part: Reduced deviation of fourth moment from Gaussian solution Y = −@41=2 =b. Solid: T = 2, dash-dot: T = 1, dash: T = 0:75, long dash: T = 0:5. (The 8gure from the paper [84], computed by P. Ilg.)
Thus, the steady state solution is unique and Gaussian but the stronger is the =ow, the larger is the initial runaway from the Gaussian solution, while the return to it thereafter becomes =ow-independent. Our observation demonstrates that, though the stability condition (542) is met, signi1cant deviations from the Gaussian solution persist over the times when the solution of Eq. (538) is already reasonably close to the stationary state. If we accept the usually quoted physically reasonable minimal value of parameter b of the order 20 then the minimal relaxation time is of order 40 in the reduced time units of Fig. 20. We should also stress that the two limits, T → ∞ and b → ∞, are not commutative, thus it is not surprising that estimation (543) does not reduce to the above mentioned Hookean result as b → ∞. Finally, peculiarities of convergence to the Gaussian solution are even furthered if we consider more complicated (in particular, oscillating) =ows T(t). Further numerical experiments are presented in [85]. The statistics of FENE-P solutions with random strains was studied recently by J.-L. Thi6eault [263]. Two-peak approximation for polymer stretching in Kow and explosion of the Gaussian manifold In accordance with [264] the ansatz for : can be suggested in the following form: :An ({ ; &}; q) =
1 √
2 2,
2
2
2
2
(e−(q+&) =2 + e−(q−&) =2 ) :
(544)
Natural inner coordinates on this manifold are and &. Note, that now 2 = M2 . The value 2 is a dispersion of one of the Gaussian summands in (544), M2 (:An ({ ; &}; q)) = 2 + &2 :
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377
To build the thermodynamic projector on manifold (544), the thermodynamic Lyapunov function is necessary. It is necessary to emphasize, that Eqs. (535) are nonlinear. For such equations, the arbitrarity in the choice of the thermodynamic Lyapunov function is much smaller than for the linear Fokker–Planck equation. Nevertheless, such a function exists. It is the free energy F = U (M2 [:]) − TS[:] ; where
(545)
S[:] = −
:(ln : − 1) dq ;
U (M2 [:]) is the potential energy in the mean 8eld approximation, T is the temperature (further we assume that T = 1). Note, that Kullback-form entropy [118] Sk = − : ln(:=:∗ ) also has the form Sk = −F=T : :∗ = exp(−U ) ; Sk [:] = −U −
: ln : dq :
If U (M2 [:]) in the mean 8eld approximation is the convex function of M2 , then the free energy (545) is the convex functional too. For the FENE-P model U = −ln[1 − M2 =b]. In accordance to the thermodynamics the vector I of =ow of : must be proportional to the gradient of the corresponding chemical potential @: I = −B(:)∇q @ ;
(546)
where @ = F=:, B ¿ 0. From Eq. (545) it follows, that dU (M2 ) 2 · q + ln : dM2 dU · q + : −1 ∇q : : I = −B(:) 2 dM2
@=
(547)
If we suppose here B = D=2:, then we get 1 dU · q: + ∇q : I = −D dM2 2 9: dU (M2 ) D = divq I = D 9q (q:) + 92 q: ; 9t dM2 2
(548)
When D = 1 this equations coincide with (535) in the absence of the =ow: due to Eq. (548) dF=dt 6 0.
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Let us construct the thermodynamic projector with the help of the thermodynamic Lyapunov function F (545). Corresponding entropic scalar product at the point : has the form d 2 U f(q)g(q) 2 2 dq : (549) f|g: = · q f(q) dq · q g(q) dq + 2 :(q) dM2 M2 =M2 [:] During the investigation of ansatz (544) the scalar product (549), constructed for the corresponding point of the Gaussian manifold with M2 = 2 , will be used. It will let us to investigate the neighborhood of the Gaussian manifold (and to get all the results in the analytical form): √ q2 =2 2 d 2 U 2 2 f|g 2 = · q f(q) dq · q g(q) dq + 2, e f(q)g(q) dq : (550) dM22 M2 = 2 Also we will need to know the functional DF at the point of Gaussian manifold: dU (M2 ) 1 DF 2 (f) = q2 f(q) dq ; − dM2 M2 = 2 2 2 (with the condition f(q) dq = 0). The point dU (M2 ) 1 = 2 ; dM2 2 2
(551)
M2 =
corresponds to the equilibrium. The tangent space to manifold (544) is spanned by the vectors f =
9:An ; 9( 2 )
f& =
9:An ; 9(&2 )
2 2 2 2 −(q+&)2 =2 2 (q + &) − −(q−&)2 =2 2 (q − &) − f = ; e +e 2 2 4 3 2, 1 2 2 q + & 2 2 (q − &) √ + e−(q−&) =2 : f& = −e−(q+&) =2 4 2 & 2, 1 √
(552)
The Gaussian entropy (free energy) production in the directions f and f& (551) has a very simple form dU (M2 ) 1 DF 2 (f& ) = DF 2 (f ) = − : (553) dM2 M2 = 2 2 2 The linear subspace ker DF 2 in lin{f ; f& } is spanned by the vector f& − f . Let us have the given vector 8eld d:=dt = J (:) at the point :({ ; &}). We need to build the projection of J onto the tangent space T ; & at the point :({ ; &}): P ;th & (J ) = ’ f + ’& f& :
(554)
This equation means, that the equations for 2 and &2 will have the form d 2 = ’ ; dt
d&2 = ’& : dt
(555)
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Projection (’ ; ’& ) can be found from the following two equations: ’ + ’& = q2 J (:)(q) dq ; ’ f + ’& f& |f − f& 2 = J (:)|f − f& 2 ;
(556)
where f|g 2 = J (:)|f − f& 2 , (549). First equation of (556) means, that the time derivative dM2 =dt is the same for the initial and the reduced equations. Due to the formula for the dissipation of the free energy (551), this equality is equivalent to the persistence of the dissipation in the neighborhood of the Gaussian manifold. Indeed, in according to (551) dF=dt=A( 2 ) q2 J (:)(q) dq= A( 2 ) dM2 =dt, where A( 2 ) does not depend of J . On the other hand, time derivative of M2 due to projected equation (555) is ’ + ’& , because M2 = 2 + &2 . The second equation in (556) means, that J is projected orthogonally on ker DS ∩ T ; & . Let us use the orthogonality with respect to the entropic scalar product (550). The solution of Eqs. (556) has the form d 2 J |f − f& 2 + M2 (J )(f& |f& 2 − f |f& 2 ) ; = ’ = dt f − f& |f − f& 2 d&2 −J |f − f& 2 + M2 (J )(f |f 2 − f |f& 2 ) ; (557) = ’& = dt f − f& |f − f& 2 where J = J (:), M2 (J ) = q2 J (:) dq. It is easy to check, that formulas (557) are indeed de8ning the projector: if f (or f& ) is substituted there instead of the function J , then we will get ’ = 1; ’& = 0 (or ’ = 0; ’& = 1, respectively). Let us substitute the right part of the initial kinetic equations (535), calculated at the point :(q) = :({ ; &}; q) (see Eq. (544)) in Eq. (557) instead of J . We will get the closed system of equations on 2 ; &2 in the neighborhood of the Gaussian manifold. This system describes the dynamics of the distribution function :. The distribution function is represented as the half-sum of two Gaussian distributions with the averages of distribution ±& and mean-square deviations . All integrals in the right hand part of (557) are possible to calculate analytically. Basis (f ; f& ) is convenient to use everywhere, except the points in the Gaussian manifold, & = 0, because if &, then 2 & f − f& = O 2 → 0 : Let us analyze the stability of the Gaussian manifold to the “dissociation” of the Gaussian peak in two peaks (544). To do this, it is necessary to 8nd 8rst nonzero term in the Taylor expansion in &2 of the right hand side of the second equation in system (557). The denominator has the order of &4 , the numerator has, as it is easy to see, the order not less, than &6 (because the Gaussian manifold is invariant with respect to the initial system). With the accuracy up to &4 : 4 1 d&2 & &2 (558) = 21 2 + o 4 ; 2 dt
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Fig. 21. Phase trajectories for the two-peak approximation, FENE-P model. The vertical axis (& = 0) corresponds to the Gaussian manifold. The triangle with 1(M2 ) ¿ 0 is the domain of exponential instability.
Fig. 22. Phase trajectories for the two-peak approximation, FENE model: (a) A stable equilibrium on the vertical axis, one stable peak; (b) A stable equilibrium with & ¿ 0, stable two-peak con8guration.
where
dU (M2 ) : 1=T− dM2 M2 = 2
So, if 1 ¿ 0, then &2 grows exponentially (& ∼ e1t ) and the Gaussian manifold is unstable; if 1 ¡ 0, then &2 decreases exponentially and the Gaussian manifold is stable. Near the vertical axis d 2 =dt = 1 + 21 2 . 15 The form of the phase trajectories is shown qualitative on Fig. 21. Note that this result completely agrees with Eq. (540). For the real equation FPE (for example, with the FENE potential) the motion in presence of the =ow can be represented as the motion in the e6ective potential well U˜ (q) = U (q) − 12 Tq2 . Di6erent variants of the phase portrait for the FENE potential are present on Fig. 22. Instability and dissociation of the unimodal distribution functions (“peaks”) for the FPE is the general e6ect when the =ow is present. 15
Pavel Gorban calculated the projector (557) analytically without Taylor expansion and with the same, but exact result: d&2 =dt = 21&2 d 2 =dt = 1 + 21 2 .
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˜ ˜ The instability occurs when the matrix 92 U =9qi 9qj starts to have negative eigenvalues (U is the 1 ˜ e6ective potential energy, U (q) = U (q) − 2 i; j Ti; j qi qj ). Polymodal polyhedron and molecular individualism The discovery of the molecular individualism for dilute polymers in the =ow [265] was the challenge to theory from the very beginning. “Our data should serve as a guide in developing improved microscopic theories for polymer dynamics”... was the concluding sentence of the paper [265]. P. de Gennes invented the term “molecular individualism” [266]. He stressed that in this case the usual averaging procedures are not applicable. At the highest strain rates distinct conformation shapes with di6erent dynamics were observed [265]. Further works for shear =ow demonstrated not only shape di6erences, but di6erent large temporal =uctuations [267]. Equation for the molecules in a =ow are known. These are the Fokker–Planck equations with external force. The theory of the molecular individualism is hidden inside these equations. Following the logic of model reduction we should solve two problems: to construct the slow manifold, and to project the equation on this manifold. The second problem is solved: the thermodynamic projector is necessary for this projection. How to solve the 8rst problem? We can 8nd a hint in previous subsections. The Gaussian distributions form the invariant manifold for the FENE-P model of polymer dynamics, but, as it was discovered in [84], this manifold can become unstable in the presence of a =ow. We propose to model this instability as dissociation of the Gaussian peak into two peaks. This dissociation describes appearance of an unstable direction in the con8guration space. In the classical FENE-P model of polymer dynamics a polymer molecule is represented by one coordinate: the stretching of molecule (the connector vector between the beads). There exists a simple mean 8eld generalized models for multidimensional con8guration spaces of molecules. In these models dynamics of distribution functions is described by the Fokker–Planck equation in a quadratic potential well. The matrix of coeRcients of this quadratic potential depends on the matrix of the second order moments of the distribution function. The Gaussian distributions form the invariant manifold for these models, and the 8rst dissociation of the Gaussian peak after appearance of the unstable direction in the con8guration space has the same nature and description, as for the one-dimensional models of molecules considered below. At the highest strain there can appear new unstable directions, and corresponding dissociations of Gaussian peaks form a cascade of dissociation. For m unstable directions we get the Gaussian parallelepiped: The distribution function is represented as a sum of 2m Gaussian peaks located in the vertixes of parallelepiped: m m 1 1 √ X −1 q + ; (559) exp − !i &i ; q + !i &i :(q) = 2 2m (2,)n=2 det X i=1 i=1 !i =±1;(i=1;:::; m)
where n is dimension of con8guration space, 2&i is the vector of the ith edge of the parallelepiped, X is the one peak covariance matrix (in this model X is the same for all peaks). The macroscopic variables for this model are: 1. The covariance matrix X for one peak; 2. The set of vectors &i (or the parallelepiped edges).
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The stationary polymodal distribution for the Fokker–Planck equation corresponds to the persistence of several local minima of the function U˜ (q). The multidimensional case is di6erent from one-dimensional because it has the huge amount of possible con8gurations. An attempt to describe this picture quantitative meet the following obstacle: we do not know the potential U , on the other hand, the e6ect of molecular individualism [265–267] seems to be universal in its essence, without dependence of the qualitative picture on details of interactions. We should 8nd a mechanism that is as general, as the e6ect. The simplest dumbbell model which we have discussed in previous subsection does not explain the e6ect, but it gives us a hint: the =ow can violate the stability of unimodal distribution. If we assume that the whole picture is hidden insight a multidimensional Fokker–Planck equation for a large molecule in a =ow, then we can use this hint in such a way: when the =ow strain grows there appears a sequence of bifurcations, and for each of them a new unstable direction arises. For qualitative description of such a picture we can apply a language of normal forms [268], but with some modi8cation. The bifurcation in dimension one with appearance of two point of minima from one point has the simplest polynomial representation: U (q; 1) = q4 + 1q2 . If 1 ¿ 0, then this potential has one minimum, if 1 ¡ 0, then there are two points of minima. The normal form of degenerated singularity is U (q) = q4 . Such polynomial forms as q4 + 1q2 are very simple, but they have inconvenient asymptotic at q → ∞. For our goals it is more appropriate to use logarithms of convex combinations of Gaussian distributions instead of polynomials. It is the same class of jets near the bifurcation, but with given quadratic asymptotic q → ∞. If one needs another class of asymptotic, it is possible just to change the choice of the basic peak. All normal forms of the critical form of functions, and families of versal deformations are well investigated and known [268]. Let us represent the deformation of the probability distribution under the strain in multidimensional case as a cascade of peak dissociation. The number of peaks will duplicate on the each step. The possible cascade of peaks dissociation is presented qualitatively on Fig. 23. The important property of this qualitative picture is the linear complexity of dynamical description with exponential complexity of geometrical picture. Let m be the number of bifurcation steps in the cascade. Then • For description of parallelepiped it is suRcient to describe m edges; • There are 2m−1 geometrically di6erent conformations associated with 2m vertex of parallelepiped (central symmetry halved this number). Another important property is the threshold nature of each dissociation: It appears in points of stability loss for new directions, in these points the dimension of unstable direction increases. Positions of peaks correspond to parallelepiped vertices. Di6erent vertices in con8guration space present di6erent geometric forms. So, it seems plausible 16 that observed di6erent forms (“dumbbells”, “half-dumbbells”, “kinked”, “folded” and other, not classi8ed forms) correspond to
16
We cannot prove it now, and it is necessary to determine the status of proposed qualitative picture: it is much more general than a speci8c model, it is the mechanism which acts in a wide class of models. The cascade of instabilities can appear and, no doubt, it appears for the Fokker–Planck equation for a large molecule in a =ow. But it is not proven yet that the e6ects observed in well-known experiments have exactly this mechanism. This proof requires quantitative veri8cation of a speci8c model. And now we talk not about a proven, but about the plausible mechanism which typically appears for systems with instabilities.
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Fig. 23. Cartoon representing the steps of molecular individualism. Black dots are vertices of Gaussian parallelepiped. Zero, one, and four-dimensional polyhedrons are drawn. The three-dimensional polyhedron used to draw the four-dimensional object is also presented. Each new dimension of the polyhedron adds as soon as the corresponding bifurcation occurs. Quasi-stable polymeric conformations are associated with each vertex. First bifurcation pertinent to the instability of a dumbbell model in elongational =ow is described in the text.
these vertices of parallelepiped. Each vertex is a metastable state of a molecule and has its own basin of attraction. A molecule goes to the vertex which depends strongly on details of initial conditions. The simplest multidimensional dynamic model is the Fokker–Planck equation with quadratic mean 8eld potential. This is direct generalization of the FENE-P model: the quadratic potential U (q) depends on the tensor of second moments M2 =qi qj (here the angle brackets denote the averaging). This dependence should provide the 8nite extensibility. This may be, for example, a simple matrix generalization of the FENE-P energy: Kij qi qj ; K = K0 + V(M2 =b); U (q) = tr(KM2 =b) U (q) = ij
where b is a constant (the limit of extensibility), K0 is a constant matrix, M2 is the matrix of second moments, and V is a positive analytical monotone increasing function of one variable on the interval (0; 1), V(x) → ∞ for x → 1 (for example, V(x) = −ln(1 − x)=x, or V(x) = (1 − x)−1 ). For quadratic multidimensional mean 8eld models persists the qualitative picture of Fig. 21: there is nonstationary molecular individualism for stationary “molecular collectivism”. The stationary
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distribution is the Gaussian distribution, and on the way to this stationary point there exists an unstable region, where the distribution dissociates onto 2m peaks (m is the number of unstable degrees of freedom). Dispersion of individual peak in unstable region increases too. This e6ect can deform the observed situation: If some of the peaks have signi8cant intersection, then these peaks join into new extended classes of observed molecules. The stochastic walk of molecules between connected peaks can be observed as “large nonperiodical =uctuations”. This walk can be unexpected fast, because it can be e6ectively a motion in a low-dimensional space, for example, in one-dimensional space (in a neighborhood of a part of one-dimensional skeleton of the polyhedron). We discussed the important example of ansatz: the multipeak models. Two examples of these type of models demonstrated high eRciency during decades: the Tamm–Mott–Smith bimodal ansatz for shock waves, and the Langer–Bar-on–Miller [269–271] approximation for spinodal decomposition. The multimodal polyhedron appears every time as an appropriate approximation for distribution functions for systems with instabilities. We create such an approximation for the Fokker–Planck equation for polymer molecules in a =ow. Distributions of this type are expected to appear in each kinetic model with multidimensional instability as universally, as Gaussian distribution appears for stable systems. This statement needs a clari8cation: everybody knows that the Gaussian distribution is stable with respect to convolutions, and the appearance of this distribution is supported by central limit theorem. Gaussian polyhedra form a stable class: convolution of two Gaussian polyhedra is a Gaussian polyhedron, convolution of a Gaussian polyhedron with a Gaussian distribution is a Gaussian polyhedron with the same number of vertices. On the other hand, a Gaussian distribution in a potential well appears as an exponent of a quadratic form which represents the simplest stable potential (a normal form of a nondegenerated critical point). Families of Gaussian parallelepipeds appear as versal deformations with given asymptotic for systems with cascade of simplest bifurcations. The usual point of view is: The shape of the polymers in a =ow is either a coiled ball, or a stretched ellipsoid, and the Fokker–Planck equation describes the stretching from the ball to the ellipsoid. It is not the whole truth, even for the FENE-P equation, as it was shown in Refs. [84,264]. The Fokker–Planck equation describes the shape of a probability cloud in the space of conformations. In the =ow with increasing strain this shape changes from the ball to the ellipsoid, but, after some thresholds, this ellipsoid transforms into a multimodal distribution which can be modeled as the peak parallelepiped. The peaks describe the 8nite number of possible molecule conformations. The number of this distinct conformations grows for a parallelepiped as 2m with the number m of independent unstable direction. Each vertex has its own basin of attraction. A molecule goes to the vertex which depends strongly on details of initial conditions. These models pretend to be the kinetic basis for the theory of molecular individualism. The detailed computations will be presented in following works, but some of the qualitative features of the models are in agreement with some of qualitative features of the picture observed in experiment [265–267]: e6ect has the threshold character, di6erent observed conformations depend signi8cantly on the initial conformation and orientation. Some general questions remain open: • Of course, appearance of 2m peaks in the Gaussian parallelepiped is possible, but some of these peaks can join in following dynamics, hence the 8rst question is: what is the typical number of signi8cantly di6erent peaks for a m-dimensional instability?
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• How can we decide what scenario is more realistic from the experimental point of view: the proposed universal kinetic mechanism, or the scenario with long living metastable states (for example, the relaxation of knoted molecules in the =ow can give an other picture than the relaxation of unknoted molecules)? • The analysis of random walk of molecules from peak to peak should be done, and results of this analysis should be compared with observed large =uctuations. The systematic discussion of the di6erence between the Gaussian elipsoid (and its generalizations) and the Gaussian multipeak polyhedron (and its generalizations) seems to be necessary. This polyhedron appears generically as the e6ective ansatz for kinetic systems with instabilities. 12. Accuracy estimation and postprocessing in invariant manifolds construction Assume that for the dynamical system (76) the approximate invariant manifold has been constructed and the slow motion equations have been derived: d xsl = Pxsl (J (xsl )); xsl ∈ M ; (560) dt where Pxsl is the corresponding projector onto the tangent space Txsl of M . Suppose that we have solved system (560) and have obtained xsl (t). Let’s consider the following two questions: • How well this solution approximates the true solution x(t) given the same initial conditions? • Is it possible to use the solution xsl (t) for it’s re8nement? These two questions are interconnected. The 8rst question states the problem of the accuracy estimation. The second one states the problem of postprocessing [244–246,277]. The simplest (“naive”) estimation is given by the “invariance defect”: Gxsl = (1 − Pxsl )J (xsl ) ;
(561)
which can be compared with J (xsl ). For example, this estimation is given by j = Gxsl = J (xsl ) using some appropriate norm. Probably, the most comprehensive answer to this question can be given by solving the following equation: d(x) (562) = Gxsl (t) + Dx J (x)|xsl (t) x : dt This linear equation describes the dynamics of the deviation x(t) = x(t) − xsl (t) using the linear approximation. The solution with zero initial condition x(0) = 0 allows to estimate the robustness of xsl , as well as the error value. Substituting xsl (t) for xsl (t) + x(t) gives the required solution re8nement. This dynamical postprocessing [246] allows to re8ne the solution substantially and to estimate its accuracy and robustness. However, the price for this is solving Eq. (562) with variable coeRcients. Thus, this dynamical postprocessing can be addressed by a whole hierarchy of simpli8cations, both dynamical and static. Let us mention some of them, starting from the dynamical ones.
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(1) Freezing the coeJcients. In Eq. (562) the linear operator Dx J (x)|xsl (t) is replaced by it’s value in some distinguished point x∗ (for example, in the equilibrium) or it is frozen somehow else. As a result, one gets the equation with constant coeRcients and the explicit integration formula: t x(t) = exp(D∗ (t − 4))Gxsl (4) d4 ; (563) 0
where D∗ is the “frozen” operator and x(0) = 0. Another important way of freezing is substituting (562) for some model equation, i.e. substituting Dx J (x) for −1=4∗ , where 4∗ is the relaxation time. In this case the formula for x(t) has very simple form: t ∗ x(t) = e(4−t)=4 Gxsl (4) d4 : (564) 0
(2) One-dimensional Galerkin-type approximation. Another “scalar” approximation is given by projecting (562) on G(t) = Gxsl (t) . Using the ansatz x(t) = (t) · G(t) ;
(565)
substituting it into Eq. (562) after orthogonal projection on G(t) we obtain ˙ G|DG − G|G d(t) ; =1+ dt G|G
(566)
where | is an appropriate scalar product, which can depend on the point xsl (for example, the entropic scalar product), D = Dx J (x)|xsl (t) or the self-adjoint linearizarion of this operator, or some approximation of it, G˙ = dG(t)=dt. A “hybrid” between Eqs. (566) and (562) has the simplest form (but it is more diRcult for computation than Eq. (566)): G|DG d(x) = G(t) + x : dt G|G
(567)
Here one uses the normalized matrix element G|DG=G|G instead of the linear operator D = Dx J (x)|xsl (t) . Both Eqs. (566) and (567) can be solved explicitly: t t (t) = d4 exp k( ) d ; (568) 0
4
x(t) =
0
t
G(4) d4 exp
4
t
k1 ( ) d
;
(569)
˙ where k(t) = G|DG − G|G=G|G, k1 (t) = G|DG=G|G. The projection of Gxsl (t) on the slow motion is equal to zero, hence, for the post-processing analysis of the slow motion, the one-dimensional model (566) should be supplemented by one more
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iteration in order to 8nd the 8rst nonvanishing term in xsl (t): d(xsl (t)) = (t)Pxsl (t) (Dx J (x)|xsl (4) )(G(t)) ; dt t xsl (t) = (4)Pxsl (4) (Dx J (x)|xsl (4) )(G(4)) d4 : 0
387
(570)
where (t) is the solution of (566). (3) For a static post-processing one uses stationary points of dynamical equations (562) or their simpli8ed versions (563), (566). Instead of (562) one gets: Dx J (x)|xsl (t) x = −Gxsl (t)
(571)
with one additional condition Pxsl x=0. This is exactly the iteration equation of the Newton’s method in solving the invariance equation. A clari8cation is in order here. Static post-processing (571) as well as other post-processing formulas should not be confused with the Newton method and other for correcting the approximately invariant manifold. Here, only a single trajectory xsl (t) on the manifold is corrected, not the whole manifold. The corresponding stationary problems for the model equations and for the projections of (562) on G are evident. We only mention that in the projection on G one gets a step of the relaxation method for the invariant manifold construction. In Example 14 it will be demonstrated how one can use function G(xsl (t)) in the accuracy estimation of macroscopic equations on example of polymer solution dynamics. Example 14: Defect of invariance estimation and switching from the microscopic simulations to macroscopic equations A method which recognizes the onset and breakdown of the macroscopic description in microscopic simulations was developed in [16,276,201]. The method is based on the invariance of the macroscopic dynamics relative to the microscopic dynamics, and it is demonstrated for a model of dilute polymeric solutions where it decides switching between Direct Brownian Dynamics simulations and integration of constitutive equations. Invariance principle and micro–macro computations. Derivation of reduced (macroscopic) dynamics from the microscopic dynamics is the dominant theme of nonequilibrium statistical mechanics. At the present time, this very old theme demonstrates new facets in view of a massive use of simulation techniques on various levels of description. A two-side bene8t of this use is expected: on the one hand, simulations provide data on molecular systems which can be used to test various theoretical constructions about the transition from micro to macro description. On the other hand, while the microscopic simulations in many cases are based on limit theorems [such as, for example, the central limit theorem underlying the Direct Brownian Dynamics simulations (BD)] they are extremely time-consuming in any real situation, and a timely recognition of the onset of a macroscopic description may considerably reduce computational e6orts. In this section, we aim at developing a “device” which is able to recognize the onset and the breakdown of a macroscopic description in the course of microscopic computations. Let us 8rst present the main ideas of the construction in an abstract setting. We assume that the microscopic description is set up in terms of microscopic variables S. In the examples considered
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below, microscopic variables are distribution functions over the con8guration space of polymers. ˙ The microscopic dynamics of variables S is given by the microscopic time derivative dS=dt = S(S). We also assume that the set of macroscopic variables M is chosen. Typically, the macroscopic variables are some lower-order moments if the microscopic variables are distribution functions. The reduced (macroscopic) description assumes (a) The dependence S(M ), and (b) The macroscopic ˙ (M ). We do not discuss here in any detail the way one gets the dependence dynamics dM =dt = M S(M ), however, we should remark that, typically, it is based on some (explicit or implicit) idea about decomposition of motions into slow and fast, with M as slow variables. With this, such tools as maximum entropy principle, quasi-stationarity, cumulant expansion etc. become available for constructing the dependence S(M ). Let us compare the microscopic time derivative of the function S(M ) with its macroscopic time derivative due to the macroscopic dynamics: G(M ) =
9S(M ) ˙ ˙ · M (M ) − S(S(M )) : 9M
(572)
If the defect of invariance G(M ) (572) is equal to zero on the set of admissible values of the macroscopic variables M , it is said that the reduced description S(M ) is invariant. Then the function S(M ) represents the invariant manifold in the space of microscopic variables. The invariant manifold is relevant if it is stable. Exact invariant manifolds are known in a very few cases (for example, the exact hydrodynamic description in the kinetic Lorentz gas model [159], in Grad’s systems [23,24], and one more example will be mentioned below). Corrections to the approximate reduced description through minimization of the defect of invariance is a part of the so-called method of invariant manifolds [6]. We here consider a di6erent application of the invariance principle for the purpose mentioned above. The time dependence of the macroscopic variables can be obtained in two di6erent ways: First, if the solution of the microscopic dynamics at time t with initial data at t0 is St; t0 , then evaluation of the macroscopic variables on this solution gives Mt;micro t0 . On the other hand, solving dynamic equations of the reduced description with initial data at t0 gives Mt;macro t0 . Let G be a value of defect of invariance with respect to some norm, and j ¿ 0 is a 8xed tolerance level. Then, if at the time t the following inequality is valid, G(Mt;micro t0 ) ¡ j ;
(573)
this indicates that the accuracy provided by the reduced description is not worse than the true microscopic dynamics (the macroscopic description sets on). On the other hand, if G(Mt;macro t0 ) ¿ j ;
(574)
then the accuracy of the reduced description is insuRcient (the reduced description breaks down), and we must use the microscopic dynamics. Thus, evaluating the defect of invariance (572) on the current solution to macroscopic equations, and checking the inequality (574), we are able to answer the question whether we can trust the solution without looking at the microscopic solution. If the tolerance level is not exceeded then we can safely integrate the macroscopic equation. We now proceed to a speci8c example of this approach. We consider a well-known class of microscopic models of dilute polymeric solutions.
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Application to dynamics of dilute polymer solution A well-known problem of the nonNewtonian =uids is the problem of establishing constitutive equations on the basis of microscopic kinetic equations. We here consider a model introduced by Lielens et al. [272]: 1 1 2 ˙ f(q; t) = −9q T(t)qf − f9q U (q ) + 92q f : (575) 2 2 With the potential U (x) = −(b=2) ln(1 − x=b) Eq. (575) becomes the one-dimensional version of the FENE dumbbell model which is used to describe the elongational behavior of dilute polymer solutions. The reduced description seeks a closed time evolution equation for the stress 4 = q9q U (q2 ) − 1. Due to its nonpolynomial character, the stress 4 for the FENE potential depends on all moments of f. We have shown in [273] how can be approximated systematically by a set n such potentials j of polynomial potentials Un (x) = j=1 1=2jcj x of degree n with coeRcients cj depending on the even moments Mj = q2j of f up to order n, with n = 1; 2; : : :, formally converging to the original potential as n tends to in8nity.In this approximation, the stress 4 becomes a function of the 8rst n even moments of f, 4(M ) = nj=1 cj Mj − 1, where the set of macroscopic variables is denoted by M = {M1 ; : : : ; Mn }. The 8rst two potentials approximating the FENE potential are: U1 (q2 ) = U (M1 )q2 ;
(576)
1 1 (577) U2 (q2 ) = (q4 − 2M1 q2 )U (M1 ) + (M2 − M12 )q2 U (M1 ) ; 2 2 where U , U and U denote the 8rst, second and third derivative of the potential U, respectively. The potential U1 corresponds to the well-known FENE-P model. The kinetic equation (575) with the potential U2 (577) will be termed the FENE-P+1 model below. Direct Brownian Dynamics simulation (BD) of the kinetic equation (575) with the potential U2 for the =ow situations studied in [272] demonstrates that it is a reasonable approximation to the true FENE dynamics whereas the corresponding moment chain is of a simpler structure. In [16] this was shown for a periodic =ow, while Fig. 24 shows results for the =ow " 100t(1 − t)e−4t ; 0 6 t 6 1 ; T(t) = (578) 0; else : The quality of the approximation indeed increases with the order of the polynomial. For any potential Un , the invariance equation can be studied directly in terms of the full set of the moments, which is equivalent to studying the distribution functions. The kinetic equation (575) can be rewritten equivalently in terms of moment equations, M˙ k = Fk (M1 ; : : : ; Mk+n−1 ) ; Fk = 2kT(t)Mk + k(2k − 1)Mk −1 − k
n j=1
cj Mk+j−1 :
(579)
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A.N. Gorban et al. / Physics Reports 396 (2004) 197 – 403 500 FENE FENE-P FENE-P+1
400
stress
300
200
100
0 0
1
2
3
4
5
time
Fig. 24. Stress 4 versus time from direct Brownian dynamics simulation: symbols—FENE, dashed line—FENE-P, solid line—FENE-P+1.
We seek functions Mkmacro (M ), k = n + 1; : : : which are form-invariant under the dynamics: n 9M macro (M ) k
j=1
9Mj
Fj (M ) = Fk (M1 ; : : : ; Mn ; Mn+1 (M ); : : : ; Mn+k (M )) :
(580)
This set of invariance equations states the following: The time derivative of the form Mkmacro (M ) when computed due to the closed equation for M (the 8rst contribution on the left hand side of Eq. (580), or the ‘macroscopic’ time derivative) equals the time derivative of Mk as computed by true moment equation with the same form Mk (M ) (the second contribution, or the ‘microscopic’ time derivative), and this equality should hold whatsoever values of the moments M are. Eqs. (580) in case n = 1 (FENE-P) are solvable exactly with the result Mkmacro = ak M1k
with ak = (2k − 1)ak −1 ; a0 = 1 :
This dependence corresponds to the Gaussian solution in terms of the distribution functions. As expected, the invariance principle give just the same result as the usual method of solving the FENE-P model. Let us brie=y discuss the potential U2 , considering a simple closure approximation Mkmacro (M1 ; M2 ) = ak M1k + bk M2 M1k −2 ;
(581)
where ak = 1 − k(k − 1)=2 and bk = k(k − 1)=2. The function M3macro closes the moment equations for the two independent moments M1 and M2 . Note, that M3macro di6ers from the corresponding moment M3 of the actual distribution function by the neglect of the 6th cumulant. The defect of invariance
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4
3
variance
2
1 t* 0 0 -1
1
2
3
4
5
time
-2
-3
Fig. 25. Defect of invariance G3 =b3 , Eq. (582), versus time extracted from BD simulation (the FENE-P+1 model) for the =ow situation of Eq. (578).
of this approximation is a set of functions Gk where 9M3macro 9M3macro G3 (M1 ; M2 ) = F1 + F2 − F 3 ; (582) 9M1 9M2 and analogously for k ¿ 3. In the sequel, we make all conclusions based on the defect of invariance G3 (582). It is instructive to plot the defect of invariance G3 versus time, assuming the functions M1 and M2 are extracted from the BD simulation (see Fig. 25). We observe that the defect of invariance is a nonmonotonic function of the time, and that there are three pronounced domains: From t0 = 0–t1 the defect of invariance is almost zero which means that the ansatz is reasonable. In the intermediate domain, the defect of invariance jumps to high values (so the quality of approximation is poor). However, after some time t = t ∗ , the defect of invariance again becomes negligible, and remains so for later times. Such behavior is typical of so-called “kinetic layer”. Instead of attempting to improve the closure, the invariance principle can be used directly to switch from the BD simulation to the solution of the macroscopic equation without loosing the accuracy to a given tolerance. Indeed, the defect of invariance is a function of M1 and M2 , and it can be easily evaluated both on the data from the solution to the macroscopic equation, and the BD data. If the defect of invariance exceeds some given tolerance on the macroscopic solution this signals to switch to the BD integration. On the other hand, if the defect of invariance becomes less than the tolerance level on the BD data signals that the BD simulation is not necessary anymore, and one can continue with the integration of the macroscopic equations. This reduces the necessity of using BD simulations only to get through the kinetic layers. A realization of this hybrid approach is demonstrated in Fig. 26: for the same =ow we have used the BD dynamics only for the 8rst period of the =ow while integrated the macroscopic equations in all the later times. The quality of the result is comparable to
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Fig. 26. Switching from the BD simulations to macroscopic equations after the defect of invariance has reached the given tolerance level (the FENE-P+1 model): symbols—the BD simulation, solid line—the BD simulation from time t = 0 up to time t ∗ , dashed line—integration of the macroscopic dynamics with initial data from BD simulation at time t = t ∗ . For comparison, the dot–dashed line gives the result for the integration of the macroscopic dynamics with equilibrium conditions from t = 0. Inset: Transient dynamics at the switching from BD to macroscopic dynamics on a 8ner time scale.
the BD simulation whereas the total integration time is much shorter. The transient dynamics at the point of switching from the BD scheme to the integration of the macroscopic equations (shown in the inset in Fig. 26) deserves a special comment: The initial conditions at t ∗ are taken from the BD data. Therefore, we cannot expect that at the time t ∗ the solution is already on the invariant manifold, rather, at best, close to it. Transient dynamics therefore signals the stability of the invariant manifold we expect: Even though the macroscopic solution starts not on this manifold, it nevertheless attracts to it. The transient dynamics becomes progressively less pronounced if the switching is done at later times. The stability of the invariant manifold in case of the FENE-P model is studied in detail in [84]. The present approach of combined microscopic and macroscopic simulations can be realized on the level of moment closures (which then needs reconstruction of the distribution function from the moments at the switching from macroscopic integration to BD procedures), or for parametric sets of distribution functions if they are available [272]. It can be used for a rigorous construction of domain decomposition methods in various kinetic problems. 13. Conclusion To construct slow invariant manifolds is useful. E6ective model reduction becomes impossible without them for complex kinetic systems. Why to reduce description in the times of supercomputers? First, in order to gain understanding. In the process of reducing the description one is often able to extract the essential, and the mechanisms of the processes under study become more transparent. Second, if one is given the detailed description of the system, then one should be able also to solve the initial-value problem for this system. But what should one do in the case where the system
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is representing just a small part of the huge number of interacting systems? For example, a complex chemical reaction system may represent only a point in a three-dimensional =ow. Third, without reducing the kinetic model, it is impossible to construct this model. This statement seems paradoxal only at the 8rst glance: How come, the model is 8rst simpli8ed, and is constructed only after the simpli8cation is done? However, in practice, the typical for a mathematician statement of the problem, (Let the system of di6erential equations be given, then ...) is rather rarely applicable for detailed kinetics. Quite on the contrary, the thermodynamic data (energies, enthalpies, entropies, chemical potentials etc) for suRciently rare8ed systems are quite reliable. Final identi8cation of the model is always done on the basis of comparison with the experiment and with a help of 8tting. For this purpose, it is extremely important to reduce the dimension of the system, and to reduce the number of tunable parameters. And, 8nally, for every supercomputer there exist too complicated problems. Model reduction makes these problems less complicated and sometimes gives us the possibility to solve them. It is useful to apply thermodynamics and the quasi-equilibrium concept while seeking slow invariant manifolds Though the open systems are important for many applications, however, it is useful to begin their study and model reduction with the analysis of closed (sub)systems. Then the thermodynamics equips these systems with Lyapunov functions (entropy, free energy, free enthalpy, depending on the context). These Lyapunov functions are usually known much better than the right hand sides of kinetic equations (in particular, this is the case in reaction kinetics). Using this Lyapunov function, one constructs the initial approximation to the slow manifold, that is, the quasi-equilibrium manifold, and also one constructs the thermodynamic projector. The thermodynamic projector is the unique operator which transforms the arbitrary vector 8eld equipped with the given Lyapunov function into a vector 8eld with the same Lyapunov function (and also this happens on any manifold which is not tangent to the level of the Lyapunov function). The quasi-chemical approximation is an extremely rich toolbox for assembling equations. It enables to construct and study wide classes of evolution equations equipped with prescribed Lyapunov functions, with Onsager reciprocity relations and like. Slow invariant manifolds of thermodynamically closed systems are useful for constructing slow invariant manifolds of corresponding open systems. The necessary technic is developed. Postprocessing of the invariant manifold construction is important both for estimation of the accuracy and for the accuracy improvement. The main result of this work can be formulated as follows: It is possible indeed to construct invariant manifolds. The problem of constructing invariant manifolds can be formulated as the invariance equation, subject to additional conditions of slowness (stability). The Newton method with incomplete linearization, relaxation methods, the method of natural projector, and the method of invariant grids enables educated approximations to the slow invariant manifolds. Studies on invariant manifolds were initiated by Lyapunov [144] and Poincare [145] (see [146]). Essential stages of the development of these ideas in the XX century are re=ected in the books [146,278,136,235]. It becomes more and more evident at the present time that the constructive methods of invariant manifold are useful on a host of subjects, from applied hydrodynamics [279] to physical and chemical kinetics.
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Acknowledgements First of all, we are grateful to our coauthors: M.S. S. Ansumali (ZDurich), Prof. V.I. Bykov (Krasnoyarsk), Prof. M. Deville (Lausanne), Dr. G. Dukek (Ulm), Dr. P. Ilg (ZDurich-Berlin), Prof. D T.F. Nonnenmacher (Ulm), Prof. H.C. Ottinger (ZDurich), M.S. P.A. Gorban (Krasnoyarsk-OmskZDurich), M.S. A. Ricksen (ZDurich), Prof. S. Succi (Roma), Dr. L.L. Tatarinova (Krasnoyarsk-ZDurich), Prof. G.S. Yablonskii (Novosibirsk-Saint-Louis), Dr. V.B. Zmievskii (Krasnoyarsk-LausanneMontreal) for years of collaboration, stimulating discussion and support. We thank Prof. M. Grmela (Montreal) for detailed and encouraging discussion of the geometrical foundations of nonequilibrium thermodynamics. Prof. M. Shubin (Moscow-Boston) explained us some important chapters of the pseudodi6erential operators theory. Finally, it is our pleasure to thank Prof. Misha Gromov (IHES, Bures-sur-Yvette) for encouragement and the spirit of Geometry. References [1] N.G. Van Kampen, Elimination of fast variables, Phys. Rep. 124 (1985) 69–160. [2] N.N. Bogolyubov, Dynamic Theory problems in Statistical Physics, Gostekhizdat, Moscow, Leningrad, 1946. [3] A.J. Roberts, Low-dimensional modelling of dynamical systems applied to some dissipative =uid mechanics, in: R. Ball, N. Akhmediev (Eds.), Nonlinear Dynamics from Lasers to Butter=ies, Lecture Notes in Complex Systems, Vol. 1, World Scienti8c, Singapore, 2003, pp. 257–313. [4] A.N. Gorban, I.V. Karlin, The constructing of invariant manifolds for the Boltzmann equation, Adv. Model. Anal. C 33 (3) (1992) 39–54. [5] A.N. Gorban, I.V. Karlin, Thermodynamic parameterization, Physica A 190 (1992) 393–404. [6] A.N. Gorban, I.V. Karlin, Method of invariant manifolds and regularization of acoustic spectra, Transp. Theory Stat. Phys. 23 (1994) 559–632. [7] I.V. Karlin, G. Dukek, T.F. Nonnenmacher, Invariance principle for extension of hydrodynamics: nonlinear viscosity, Phys. Rev. E 55 (2) (1997) 1573–1576. [8] V.B. Zmievskii, I.V. Karlin, M. Deville, The universal limit in dynamics of dilute polymeric solutions, Physica A 275 (1–2) (2000) 152–177. [9] I.V. Karlin, A.N. Gorban, G. Dukek, T.F. Nonnenmacher, Dynamic correction to moment approximations, Phys. Rev. E 57 (1998) 1668–1672. [10] A.N. Gorban, I.V. Karlin, Method of invariant manifold for chemical kinetics, Chem. Eng. Sci. 58 (21) (2003) 4751–4768, Preprint online: http://arxiv.org/abs/cond-mat/0207231. [11] A.N. Gorban, I.V. Karlin, V.B. Zmievskii, S.V. Dymova, Reduced description in reaction kinetics, Physica A 275 (3–4) (2000) 361–379. [12] I.V. Karlin, V.B. Zmievskii, Invariant closure for the Fokker–Planck equation, 1998. Preprint online: http://arxiv.org/abs/adap-org/9801004. [13] C. Foias, M.S. Jolly, I.G. Kevrekidis, G.R. Sell, E.S. Titi, On the computation of inertial manifolds, Phys. Lett. A 131 (7–8) (1988) 433–436. [14] A.N. Gorban, I.V. Karlin, V.B. Zmievskii, T.F. Nonnenmacher, Relaxational trajectories: global approximations, Physica A 231 (1996) 648–672. [15] A.N. Gorban, I.V. Karlin, V.B. Zmievskii, Two-step approximation of space-independent relaxation, Transp. Theory Stat. Phys. 28 (3) (1999) 271–296. D [16] A.N. Gorban, I.V. Karlin, P. Ilg, H.C. Ottinger, Corrections and enhancements of quasi-equilibrium states, J. Non-Newtonian Fluid Mech. 96 (2001) 203–219. D [17] A.N. Gorban, I.V. Karlin, H.C. Ottinger, L.L. Tatarinova, Ehrenfests argument extended to a formalism of nonequilibrium thermodynamics, Phys. Rev. E 63 (2001) 066124. [18] A.N. Gorban, I.V. Karlin, Reconstruction lemma and =uctuation-dissipation theorem, Rev. Mex. Fis. 48 (Suppl. 1) (2002) 238–242.
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CONTENTS VOLUME 396 G.E. Brown, M. Rho. Double decimation and sliding vacua in the nuclear many-body system O. Civitarese, M. Gadella. Physical and mathematical aspects of Gamow states
1 41
R.A. Duine, H.T.C. Stoof. Atom–molecule coherence in Bose gases
115
A.N. Gorban, I.V. Karlin, A.Yu. Zinovyev. Constructive methods of invariant manifolds for kinetic problems
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Contents of volume
405
doi:10.1016/S0370-1573(04)00209-1
