Renormalization Methods: Critical Phenomena, Chaos, Fractal Structures

,

— If p=p,-1, then (4_ 1 ) ' [2-,„.5(p,.,.._0] =1 at every point of the 2n-cycle. — If p = Ai n , then (f122 :)' [x„,i (p„)]= —1 at every point of the 2n-cycle. The evolution generated by fi2:. thus has a pitchfork bifurcation at p„; the fact that the 2'2 -cycle becomes unstable and is replaced by a 271 +-1 -cycle corresponds to a doubling of the period 2" asymptotically observed if p < pr,. Each branch s n ,i(p) of the 2n-cyc 1 e gives rise at p„ to two branches sn+1,i(p) and s n+1,j+2.(p) which are mapped to each other by

fr

— We conjecture that the values of the successive bifurcations form an infinite (increasing) sequence [pj]i> 0 which converges to a value p c , at which the attractor is no longer periodic (see figures 5A.1 and 5A.2). Its construction ( 5.2.2) shows that its structure is lacunary and self similar. The numerical computation of the Lyapunov exponent of the evolution shows that it is negative if p< pc and passes through 0 at p= pc , and that its envelope becomes positive for p> pc (figure 5.3). As p approaches Pc, the period of the stable cycle becomes longer, so that it becomes necessary to observe a trajectory over an increasingly long period of time to discern its periodicity; at p =p c , observation does not allow one to distinguish the asymptotic motion from a random motion. -

A similar type of behavior is observed for many one-parameter families [Fi,j p of .7": generically they display the same accumulation of period-doubling bifurcations. A quantitative study of the sequence of bifurcation values [pi 1 i>0 obtained on the numerically constructed bifurcation scheme leads to the following conjecture: the sequence (pi )i < 0 is infinite and

lirn 6 (p c — pi ) = A

0

The limit A exists and is non-zero, but it depends on the family [FI,]. The value pc on the family but even on its parametrization'. On the contrary,depnsotly the number 6 seems to be universal: 6 ,ez-i 4, 66920 In this situation, the aims of renormalization analysis are to prove this conjecture, to obtain the value of 6 analytically, and to describe the associated universality class.

0 is equivalent to limj —. (1,43 4 1 Al) = The statement lim3, S3( = A A(1 1/6). This second statement is more useful experimentally since it does not require knowing the limit value pc ; it can be tested by drawing the graph (j log(11 1 +1 the conjecture is true if the graph is a line, in which case its slope is log 6. 27

-

—

—

—

—

A comparative study of two typical examples

46

Relevance of the scenario This scenario was first described numerically on the family of logistic maps by Coullet and Tresser [1978a,b] and independently by Feigenbaum [1977]; its wide relevance was shown by experimental observations such as those made in normal liquid helium by Libchaber and Maurer [1980], and by Dubois and Berg4 [1981] and Dubois et al. [1981] in a Rayleigh—Benard experiment: a vertical gradient of temperature AT, applied from underneath to a bath of oil generates periodic convective structures within it, both spatially (cells) and temporally (waves). Measurement of the temporal period of the local state functions as a function of the control parameter AT reveals several period-doublings; the sequence of measured bifurcation values (ATi )i agrees with the conjecture ATi + 1 . These observations 28 , which were performed in systems whose evolution is not described by the logistic map, support the universal character of this "route towards chaos". The universality of this scenario is striking since it concerns not only the qualitative aspects (accumulation of doubling bifurcations), but also quantitative aspects such as the rate of convergence .5 which seems to be identical for all unimodal families behaving according to this scenario.

DETAILS AND COMPLEMENTS: THE SPECTRAL SIGNATURE OF THE SCENARIO

The presence of a period-doubling scenario in a family [FA ] , of evolution laws is revealed by striking spectral properties. Let 27r/wo denote the average duration of the temporal step. For Ln1 < p < ten , the spectrum of any temporal signal emitted along a trajectory contains all the harmonics cd,, 2 — nw 0 . At II = p n , a spectral line appears at c.,..),a2 c.,..4,44, revealing the period-doubling of asymptotic motion. We speak of a subharinonic cascade; its presence is the simplest experimental evidence of a period-doubling scenario. The spectra of FA4 n and Fp ,,+1 are similar, and can be deduced from each other by a scaling transformation which is independent of n. The spectrum observed at the threshold = p. c is chaotic ( 5.1.6).

2.2.4 Renorrnalization analysis

Outline The general principle of renormalization analysis of attractors of discrete dynamical systems is the transfer of the asymptotic limit n co to the iteration of an operator R. acting in the space of evolution laws (here .F). For this it suffices that Hf be conjugate to an iterate 14 , in such a way that a trajectory of kjn steps generated by f is related to a trajectory of n steps generated by RI f. The action of R on f thus reduces the apparent duration necessary to reach the permanent regime, and the study of the asymptotic regime n oo of the trajectories [f' (s0))>.0 is reduced to that of the asymptotic regime n oo for the sequence [Rnfin>0. By increasing the actual duration of a temporal step by a factor of k at each iteration, renormalization preserves only the long-term consequences of the dynamics while eliminating transient also the articles of Gollub and Swinney [1975], Linsay [1981 ] , Libchaber el al. [1982 ] , and the survey of Schuster [1984 ] . 28 See

2.2

The period-doubling scenario

47

effects. A comparison of the renormalized laws Rn f and Rng yields a comparison of the asymptotic properties associated to f and g: they are identical if the sequences [Rnf],> 0 and [fez g],, >0 have the same limit ya. The possible limits yo are the fixed points of R; these fixed points thus represent typical asymptotic behavior common to all the evolution laws in their basin of attraction and they are in this sense universal. The procedure sketched above enables us to globally compare the asymptotic properties of discrete dynamical systems, by analyzing their evolution laws rather than their trajectories; its main advantage is that it leads to explicit quantitative resuhs, which are deduced from the properties of the operator R and are thus universal.

Figure 2.4 - The critical map fp (s)= 1 — p cs2 and its first iterate The map fp (s)= 1 — px 2 and its iterate 42, are drawn for the critical value Pc = 1.4011550... at which the transition towards chaos by accumulation of period-doublings takes place. The intersection of the graph of f,,, c with the diagonal is the fixed point of the evolution; its coordinates (a, a) are given by a = [-1-F-V1 4/.1,]/(2p c) (a 0.5602..). The schematic representation evidences the self-similarity of the graphs: the restriction of the graph of f42, to the square [1 — p c , c 1] 2 is superimposable onto the graph of fpc after dilation by a factor of 1 — p, bringing the dotted square back to the initial —

one

48

A comparative study of two typical examples

Renorrnalization of the uniniodal maps The procedure is based on the self-similarity of the graph of the evolution laws at the onset of chaos (figure 2.4) and on the self-similarity of the bifurcation scheme shown in figure 2.3. The preliminary qualitative analysis of these similarities is crucial in order to construct an operator R adapted to the conjectured scaling laws and having a fixed point 9, for which the self-similarity is exact. Figure 2.4 shows that the graph of , restricted to the interval [1— pc , tic — 1] and dilated by a factor of 1— pc = foc (1) is almost superimposable onto the graph of ft„. This similarity naturally leads to the comparison of f with the map Rf(x) = (1/Af)f o f(x)f) where Af f(1). Here the factor of temporal decimation is k = 2, which is a natural choice in the study of period-doublings. Thus we define the renormalization operator by:

Rf(x) =

1 f o f(sAf) f

.A .t. = f[f(0)] = f(1)

with

Rf(0) =1

The relation of conjugacy linking Hf and f 2 is inherited by their iterates, in the form f, so the trajectories generated by Hf can be deduced from (Hf )r() = A I-i f 2n( z )) the trajectories generated by f via a temporal decimation (by a factor of 2) and a scale change (by a factor of Ai l with module I )tf 1 -1 > 1). This implies that their statistical properties, in particular the temporal averages, are analogous. The explicit relation

HThf(x) = A,T 1 f2R (A„s)

with A.

= f2 (0)

(1/1,2 1 < 1)

between the trajectories and the iterates of R shows how the iteration of f is transported onto the iteration of R. The bifurcation diagram given in figure 2.3 confirms the relevance of the choice of R, showing that the iterates lead to the same doubling scenario as fo. . They display cascades of bifurcations which are shifted with respect to each other: the j-th bifurcation of fr takes place at at the same time as the (j n)-th bifurcation of f,. since they both correspond to 2(11+j-1). R is thus well-adapted to the study of the selfa pitchfork bifurcation of 4, similarity properties of

fo.c ;

— the evolution at the onset of chaos, since by construction Rf o, —

the scenario towards chaos, since Rf Ai and

are comparable.

Analysis of the operator R shows that it has one (and only one") hyperbolic fixed point of R, which has stable directions and a unique unstable direction of eigenvalue > 1. It is explained below how analysis of the action of R on a one-parameter family [Fi ] i, of F proves that the eigenvalue 6 is the rate of convergence at which the bifurcation values (pi )j>0 accumulate (when they exist); thus it is identical for all the families presenting sTuch an accumu/ation, which is the case whenever they satisfy a transversality condition. This reasoning is schematized in figure 2.5. We compute = 4.69920... by numerically exploiting a truncated analytic expansion for

29

The trivial

fixed point 1/0) E 1 (With A

= 1) is stable since DR('çb)

O.

2,2

49

The period-doubling scenario

Figure 2.5 - Action of the renormalization operator R on a transverse family [Fi ] i, is a family of unimodal maps transverse to the stable manifold V3 of R at the fixed point 9. The manifolds [W] j >0 are constructed as the loci of the successive doubling bifurcations, so that F E 1411. They accumulate at V 8 , and the critical value pc =limi_,,,, pi of the parameter is obtained byby writing Ft4 e E V 3 (evolution law at the onset of chaos). The action of R is summarized in the properties limr,_,,, Rn Fi,c = cp and RF C The second expression gives the approximate relation PuRF„, where P' is the projection onto Vi', Linear analysis of this expression gives the scaling law p c — pi — 6' where 6 > 1 is the unique unstable eigenvalue of DR(cp).

DETAILS AND COMPLEMENTS:

THE ACTION OF THE RENORMALIZATION

OPERATOR

The domain of definition of the renormalization operator is determined by imposing the condition Hf E F for all f E VD, which gives

Do

ff E .T„ Af <0, f(A f) > 0, f2 (j) < —Ail c,7"

W0 of elements of F which undergo a pitchfork bifurcation define a manifold; thus, by construction of R, the locus of the successive period-doubling bifurcations is the family The set

[VVi]i>0 of manifolds given by Wi =

R - i [W0].

The fixed point 9 is obtained by plugging

its analytic expansion at x = 0 (with the constraint c)(0) = 1) and the associated expansion of A = io[9(0)] into the equation Rio = cp, and solving term by term. We can show that

60

A comparative study of two typical examples

the expansion obtained in this way converges normally, and conclude that a non-trivial fixed point so exists and is unique. The linearized operator at 90 is given by

[DR((p).gllx) =

[g(1)[x9o'

9o(x))+ g[cp(Ax)] + sof [9o(Ax)]g(Ax)]

It has a unique unstable eigenvector et, of eigenvalue 3° cS = 4.66920 .... The stable manifold of R can be characterized as the set of elements of 1. for which all the iterates of the renormalization are defined (§II.3) so that ,

V' = fl

0D

= 21% Di

since

Di

R 5 (V o ) C

We check that V =lim5..00 W5 which means that the manifolds [1'V]5 >0 accumulate at V. Let [1.11,], be a one-parameter family of elements of Y. The critical value p e of the parameter (i.e. the value at which chaotic characteristics appear in the evolution generated by Fo ) is obtained using the condition F4c C V' which ensures that F4c is related to the typical critical dynamical system (i.e. at the onset of chaos) whose evolution law is the fixed point so of R: lim R5 [F,1 ] i —>00

if and only if Foc E

By the very definition of W5, the sequence of bifurcation values [115 ] 0.o is obtained by By the very construction of R, we have R[Fj E W5_1. If I'm , is taking Foi C and if j is large enough, we can perform a linear analysis of the action of R on near In order to do so, we introduce the projection Pu onto et, parallel to the space E5 generated by the stable directions. It satisfies Pu o DR() Pu, and we deduce that — soj by neglecting the quadratic terms. Then we apply Pu to 6 PIRFoj — so] the relation R(Fp.i] gi_i; ignoring the curvature of W 1 -1, Pugi_i tends to 0 as j tends to infinity, which gives . Pu [Foi — so] •-•-.'s Pu[Foi _ i — (to] (for j large enough). Ignoring the non-linear terms arising from the curvature of V', we identify: ,

Pu [Fi,

Pu[F„ — Fp.c ] (11 — p c) Pt'

We deduce that p,j_i — p c is satisfied:

.5(pi —

(

E

L

c) 0(11— ) (Pc)

provided the following transversality condition

• Kallpiati)(Pc)] 0 0 To make this approximate reasoning completely rigorous, we would need to keep track of the quadratic terms coming from the non-linear action of R and from the curvature of the manifolds (W)jx) and Vs . This analysis is delicate since it requires the explicit equations of the manifolds; we refer the reader to the article by Vul and Khanin [ 1982) and to the monograph by Collet and Eckmann [1980].

30 The value of 15 is approximate since it cannot be computed without using computers and truncating the expansion, which limits the accuracy of the result.

A comparative table

2.3

2.3

51

A comparative table

The renormalization procedures used to analyze the critical behavior in the two examples detailed above are point-by-point analogous, as shown in the following comparative table.

1

One-dimensional Thing model

Period-doubling scenario

FORMALISM •

Extensive variable ("real space")

:

discrete variable

spatial: position zi = fa , j E Z microscopic length scale Az = a

N

sites,

ti = fa , j E Z microscopic time scale At = 7N steps, N >1 macroscopic time scale T = NT temporal: time

N >1

macroscopic length scale L = Na •

Translational invariance

j —* j +

1

(changing the origin of the indexation)

spatial (homogeneous system) •

Elementary (individual) state variable

local magnetization

spin si = s(z) E •

41 = {-1,+1}

Collective state

con fi guration

•

[s] = (si )1 <j
j E Z)

position at a given time si = z(ti) EX = [-1, +1]

trajectory (s.i)i
Direct coupling between nearest neighbors (j, j +1)

short range a

evolution map: xi + i = fo (si) short range r

Structure rule, determining the statistical distribution

reduced Hamiltonian N-1 , 7-1(N, K) = Ej= 1 si R. sj + 1 —

•

(labelled

(N constituents

pair Hamiltonian: hi(J)= — Jsisii-i •

temporal (autonomous system)

unimodal evolution law fi, (for example ft,(x) =1 — yx 2 )

Control parameter (organization increases with the parameter

)

K = JI(leB T)

A

relative importance of the direct

importance of the non-linear effects

coupling and the thermal fluctuations

(typically fi,(s) = I — tiz 2 )

•

Statistical distribution

thermal equilibrium

permanent regime (on the attractor)

Boltzmann-Gibbs distribution

invariant measure (see §II.4,

§5A.1)

52

A

comparative study of two typical examples

CRITICAL PHENOMENA The critical characteristics occur for

•

thermodynamic limit N oo (L = Na oo) and K If, = oo critical phase transition

the asymptotic evolution N (t = Nr co) and It p c

the

•

Divergence of the range

oc

onset of deterministic chaos

of the statistical correlations

correlation length (size of the connected domains of spins 8 = +1)

correlation time (period

of the stable cycle of 44 if

p < pc )

• Order parameter average magnetization by spin

envelope

of the Lyapnnov

exponent

RENORMALIZATION • Decimation N' =

N/2

(2j, 2j ± 1)

N/2 macrospins

N spins si =_ 82j

÷

N steps

N/2 macrosteps = sza.Xf where f = 1(1)

partial trace in the partition

function

—

iteration of

z(N,

f and scale

change z'

• Scale change preserving the minimum scale (resolution)

x/2—, A z ,

2;24+2-x3i 2

=a

t' = t/2 —* At'

134±2-t2i 2

7

• Physical invariants spin density 1/a

domain [-1, 1]

all spins of modulus 1

normalization f(s = 0) = 1

•

Conservation of the physical system via the

Z[2N,7-6 = C(7- ) N Z[N,7?-9-1

renormalization R

f 2N (s)

X(f)[ 7?-i]'1 ( 40t(f))

• Reduction of the apparent range of the correlations

[1ui ] =

(

7- l 2) (

• Renormalization r expressed via 1.[H4K)] E [r( )]

the control parameter R(fii)

fr(p)

• Critical point = unstable fixed point of r r(K,)= If, with IC, = 00 r(p) = pt, where pc depends on [fob,

2.3

A comparative table

53

• Reduced parameter 0, transformed by 1 during renormalization (6, = 0) The universal scaling laws are a consequence of the linearization F.(0) = 15 6 = '(9 = 0) > 1 is obtained from the linear analysis of i° at the fixed point 9

=e

F(e-K ) = •\//

•

=0

e —K gives 5 =

Pc r(P.i+1)' Pc Pi " 6 (PG where 8 = 4,66920 . . The scaling law satisfied by the range of the statistical correlations is

(6 ) — 0- ",

where

W(6)) = MO,

—

gives

6' = 2,

—

i.e.

y=

—

Pi+1)

(log 2)/(10g15)

REMARKS AND BIBLIOGRAPHICAL NOTES

The model of spins on lattices introduced by Ising [1925] was considered in the context of ferromagnetic transition by Dyson [1969]; for details on this model, its extensions and its study via renormalization, consult Mac Coy and Wu [1973], Baxter [1982] or the courses by Chandler [1987] or Goldenfeld [1992]; complete calculations for the two-dimensional model are presented in Plischke and Bergersen [1994]. For, the notion of deterministic chaos, we refer to §5.1 and to the introductory books by Devaney [1989] or by Baker and Gollub [1990]. The existence of chaotic behavior in the family of logistic maps was pointed out by May

[1976], [1991]. Conjectured by Feigenbaum following numerical observations of this family, the period-doubling scenario was described by renormalization independently by Coullet and Tresser [1978a,b], [1980] and by Feigenbaum [1977], [1978], [1980a]. This scenario will be considered in more detail in §5.2. Its complete analytic study is presented in Collet and Eckmann [1980]; a simpler exposition can be found in §5.2, in Schuster [1984] or in Peitgen et al. [1992]. One can also consult Coullet [1988] and, concerning the comparison proposed in §2.3, Bu and Mao [1982] or Lesne [1996].

3 Mathematical aspects In this chapter, we explain the application of renormaIization principles introduced in chapter 1; both spatial and temporal types of renormalization are considered, and we work in real and conjugate space. The essential step is to replace the study of a physical system S by the study of the action of a renormalization operator R on its structure rule 0(S). The analysis of the flow generated by R shows that the hyperbolic fixed points 0* of R corresponds to typical critical systems. Moreover it relates the critical exponents to the characteristics of R and of its linearized operator DR(e), thus proving their universality (§ 3.1). Following this, we give a probabilistic description of the critical phenomena: the "mean-field" approaches can be justified as laws of large numbers; the presence of giant fluctuations while the range of the couplings remains microscopic is equivalent to the critical divergence of the correlation length. The critical phenomena observed in systems of finite size are quite different from their ideal characterization given in the thermodynamic or asymptotic limit (§ 3.2). The difficulties encountered in simulating these systems and in numerically solving renormalization equations are partially overcome by a simultaneous use of the Monte Carlo method and of direct numerical renormalizations (§ 3.3). The use of a "renormalization group" is very effective if it is a Lie group, since in that case it suffices to study its infinitesimal generators; this group appears as the symmetry group associated to the invariance by

renormalization (§ 3.4).

3.1 Renormalization operators 3.1.1

Goals and typical approaches

Renormalization is introduced in the study of the large-scale (spatial or temporal) properties of a physical system S, in order to obtain a quantitative description of the singularities of its macroscopic quantities X(9) as a control parameter 6+ tends to a critical value 9,. We want to replace the leading-order results: lim X(61) = oo

0-4-8 c

or

lim X(0) = O

8-6.6 c

or

lim X ( 9) = X,

8—.6c

Mathematical aspects

56 by scaling laws: log X(0) Xe 19--8±o log I 6 — Oc

lim

—

= 7±

if X, E [0, oo[

If X,=00, then log I X(0) I replaces log X(0)— X, 1. The issue is thus to compute the critical exponents 7± and to describe their universality classes'', It is to be noted that the construction of the renormalization operator R should depend on a qualitative knowledge of the phenomenon, so that the fired points of R are the self-similar systems which exactly satisfy the scaling laws. For example, we shall study the conjecture [X(0) 0] by introducing a transformation R Rb ,K such that 1017 as 0 tends to is described by the function X(0), then RS is described by 0 if S X(b0). Showing that RS = S will ensure32 that the scaling law is satisfied in S and will give the value of the exponent 7= log K/ log b. Similarly, the fixed points of a renormalization R = Rb,b«,b ,y such that the function X(9, a) of S becomes —7 X(69, Pa) for RS are the systems such that X(0, a)-= ler x 1 9 1

Renormalization begins with a summation in blocks of n elementary constituents (usually called a coarse-graining in physics); it has the effect of a decimation, Le. it reduces the number N of elementary constituents (of scale a) by a factor of n and accordingly reveals the organization of these constituents at the scale an of the nblocks. It continues with a contraction of the length and time scales by a factor of k to preserve the apparent minimum scales. Such a renormalization, denoted Rk, weakens the apparent critical nature ofS by reducing the statistical correlation range by a factor of k in the renormalized system.

Renormalization methods vary according to the nature of the degrees of freedom of S, which can be

— spatial, as in statistical mechanics at thermal equilibrium (§2.1, §4.3). N is the number of elementary constituents whose size is fixed by the minimum scale a of the description; the (linear) extent of S is L= N i fda in dimension d. By assimilating a cell of volume (ka)' to an elementary subsystem of RkS , Rk reduces N by a factor of ka (hence n = kd here); a contraction of the lengths by a factor of k preserves the apparent minimum scale a. — temporal as in dynamical systems (§2.2, ch.5). Here N is the number of steps (of average length r) of the evolution and the extent of S is the duration T = Nr of the observation, which tends to infinity in order to reach the asymptotic regime; Rk reduces N and the time scale by the same factor k; 31 A diffeomorphism Y = F(X) corresponding to the transformation X(0) Y(0) of the observed state function preserves these exponents, which supports the conjectures of their universality. 3 2 The most general continuous solution is X(9) =f (log 8) where f is periodic of period log b, so bounded and infinitely covering its image, which justifi es writing X(9) 10i; this law is exact if Rt,1, b -rS b-yS = $ where log bi / log b2 is irrational since in that case f 2 must be a constant.

3.1 Renormalization operators

57

spatio temporal, as in critical dynamic phenomena (§4.4) or in diffusion problems (§6.2). Renormalization simultaneously reduces L and T according to L . The relevant value of a is also the value appearing in the scaling Llk and T law r(A) A relating the spatial scales to the corresponding characteristic times; this value can be known a priori or obtained by analyzing the iterated action of R. To conclude the renormalization, the decimation and the contraction of scales must be complemented by a transformation 01 1:?4,0 of the structure rule 0 of the system S, which compensates the loss of information on very small scales due to the coarsegraining and to the joint change of resolution a ka > a in the initial system. Rk4) depends on the short-range statistical correlations I < ka, internal to an elementary constituent of the renormalized system; the iteration of Rk eventually involves all the correlations. The final step is the determination of the action of Rk on the different state functions X of S. -

-

<> DETAILS AND COMPLEMENTS: "SECONDARY" TRANSFORMATIONS The action of the re-normalization operator Rk is frequently shifted to

— quantities deduced from the structure rule which express its consequences on the global behavior of S more directly. We mention for example the Boltzmann–Gibbs distribution associated to a Hamiltonian or the invariant measure associated to an evolution law. The advantage of transferring the renormalization to the statistical description of the system is that the probabilities are more easily related (via their moments) to the experimental or numerical results. — a family of coefficients [K] parametrizing the structure ride, which transforms the functional equations of renormalization into algebraic equations. However, even if the initial structure rule is parametrized by a single coefficient, the successive renorinalizations in general require an increasing (eventually infinite) number of parameters. In this case it is necessary to truncate in order to obtain a closed, solvable, finite-dimensional system of equations.

Once the scaling factor k is fixed, the values which must be given to the other (n, b, K .) in order to obtain non-trivial limits Rik quantitatively reveal the self-similarity of S by making the values of the associated exponents (log b k / log k .) explicit. The next step in our analysis is to study the flow generated by the iteration of Rk, in particular its fixed points. The macroscopic behavior of the indefinitely renormalized system will be non-trivial only if Rk is adapted to the organization on every scale of the statistical correlations between its constituents, and to its selfsimilarity. If this is the case, then Rk explains how cooperative structures appear in S, and the properties of Rk suffice to characterize the critical phenomena they induce. Renormalization gives an efficient and profitable global analysis but to the detriment of the microscopic description. At each iteration, information contained in the small-scale structures is lost because it is absorbed in the averages or in the local integrations used in the coarse-grainings. By construction, renormalization selects only those microscopic mechanisms which have repercussions on the higher scales; thus it reveals the universal macroscopic trends. Moreover, we could a priori eliminate all the details specific to the system with no effect on the global universal properties:

Mathematical aspects

58

— by taking projections to reduce the phase space (intermittency, KAM); — by realizing a spatial or temporal discretization (Feigenbaum); — by eliminating terms having only local consequences, or structurally unstable terms, keeping only the generic dominating terms and reducing them to typical forms (Moser). We note that renormalization preserves, and even accentuates the local nature of the couplings since only effective interactions between nearest neighbor macrocells33 are present in the sufficiently renormalized system. For this their real size, equal to lea after n renormalizations, must be greater than the real range A of the physical couplings. The local nature of the interactions is a necessary condition for the success of renormalization; it ensures that the successive coarse-grainings lead to a hierarchical description of the correlations as a function of their scale (which would be an incorrect picture in a system where physical couplings are present on every scale).

<> DETAILS AND COMPLEMENTS: REAL SPACE AND CONJUGATE SPACE Most of the time, renormalization is conceived in real space, where its physical and geometric meaning is more intuitive. The self-similarity of the fixed points of R can be expressed simply, even visually, and this guides the very construction of R. Another aspect of real space is that it admits a direct numerical approach in which renormalization is performed geometrically over a large number of independent random configurations of the system; the statistical analysis of the configurations renormalized j times makes it possible to give the action of Ri explicitly without any approximations other than those due to the statistical errors ( 3.3.4). In spite of these arguments, the analysis is frequently performed in conjugate space, for the following reasons (given in spatial terms): — The existence of a minimum scale Ax = a, discretizing any field s(i) describing the system in real space, does not destroy the continuity of the conjugate field but only bounds 27r/a}. its extent: g(0 is continuous in {q = lia 1 of Rk can thus vary continuously, making {flk , k> 1} into a Lie semi-group for which it suffices to analyze the infinitesimal generator (dRkicik)(k = 1) ( 3.4.1). —

— The structures or the mechanisms on scales smaller than the minimal scale a are easily identified since they are associated to the modes g > A; renormalization eliminates these modes from the explicit description while keeping track of their influence in effective parameters or in a stochastic term. — The separation of the spatial dependencies according to their scale, which is necessary to explicitly implement the coarse-graittings, is not well-defined in real space, whereas it is immediate in conjugate space. In conjugate space, decimation is replaced by a partition [g< Al k]U[g>11./ k] and the partial trace is replaced by the integration of the contributions

of the modes

J

(

g> Apo

into renormalized parameters.

— The use of conjugate space reduces the description of the system to that of modes ) with zero covariance whenever the microscopic statistics is homogeneous. In particular, 33

Or between consecutive macrosteps in a temporal system.

3.1 Renormalization operators

59

the modes g>Alk (written 6i) eliminated by the renormalization and the modes gi = < < >os >s o= << >so >6-s in terms of statistical averages). Tithe system is statistically isotropic, then g() depends only on g and the vanishing of the covariances (< g(Og(e)>= 0 if 4!) ensures that only integrals of the form f f(g)d appear; the dimension d of the space then becomes simply a. parameter since f f(g)d = f2d f(g)dg (where 1d = 2ird/ 2 [r(d/2)] -1 is the factor coining from the angular integration). In this situation it is possible to use perturbative methods with respect to c = d— X where is a particular dimension for which the solution is known.

Li

_Tom

cr

3.1.2

Transfer of limits

Renormalization does not change the nature of physical systems but only the perception we have of them. Thus, if S is a model with extent proportional to N, structure rule 0 and description by a global state function X(N, 0), then the renormalization given by

N NIk

— 1-1k0

X —) 1:1 k X

(Rk assumed invertible),

must satisfy X(N/k,R, k 0) Rk[X(N,0)], where X(N) = Tk [X(N1k)], and the transformation 71 is given by (TX)(., 0) -a- RiT i [X(.,140)]. The dependence of X on N follows by induction:

VNo, PEN,

X(kPAr o , 0) = R k- P [X(No, 740)1 = [TIXIINo,

The thermodynamic limit N —) oc is transferred to the iteration of 71:

lirn X(N, 0) = lirn R: P [X(No,1- P0)]= lira frf Xl(Noi

N—.00

15' —` 00

P-

1"50

If this limit exists, its value X00 (0). does not depend on No and is a fixed point of 71. The study of the asymptotic dependence on N of the state function X(N, 0) of a particular system S as N co is replaced by the study of the asymptotic flow generated by the transformation Tk in the function space {X(.)} - in other words, the study of the eggs is replaced by the study of the hen! One advantage of this point of view is that the universality of the results is automatically proved, since each result comes with the set {X(.)} of functions (and thus of systems) to which it applies. The physical interpretation of the renormalization associated to Rk and 74 ensures that the group-theoretic relations Rk i o Rk, = Rk i k, and Rk i o fl. = Rk i k, are satisfied whenever the renormalizations by the factors of k1 and k2 are well-defined (§3.4). If the dependence of Rk and Tlk with respect to k can be computed explicitly, we can use RPk = Rkp and R.: = Rkp to write:

lim X(N, 0) = lim RT,i1 [X(No, R-NO)] = 11m [TNX] (No, 0)

N, co

N—,co

60

Mathematical aspects

Fixed points and critical manifolds The preceding sections show that the renormalization analysis of a physical system S comes down to studying a functional operator R acting on a space containing the structure rule 0 of S and more specifically the asymptotic behavior (n . oc) of the trajectories [Rn0]>0 in The fixed points 0* E of R thus play a particularly important role since they act as attractors whenever they have stable directions. Indeed, by the very construction of R, each one corresponds to a system which is shows Rn(4)*) = exactly self-similar; in particular, the identity Ir(0*) = lir that the structure rule 0* gives access to its properties on every scale, including that of the thermodynamic limit (attained in asymptotic rules lin Rn(0), cf. 3.1.2). At each iteration, renormalization reduces the range e(S) of the statistical correlations in S by a factor of k. Moreover: — if OS)
(microscopic) order of the minimum scale a of the description; — if 4"(S)= oc, it will transform S (critical) into other, exactly critical models, from which the specific details having no macroscopic consequences are progressively eliminated. This reveals the self-similar structures of S and causes the structure rule oc, so that 0* is associated to a ir0 to approach a fixed point 0* of R; then typical critical system. The number N(S, a) of relevant degrees of freedom of a system S of dimension d observed with the minimum scale a is of the order of [e.(S)lal a : when (S) is macroscopic, N(S, a) is too large to consider analyzing the properties of S directly. One useful aspect of renormalization is that it reduces N(S, a) at each iteration by a factor of k a until the representative number of (effective) degrees of freedom becomes sufficiently small, when k -4 a. However, the remarkable strength of renormalization analysis appears only when the operator R has fixed points associated to critical systems. The preceding qualitative analysis of the action of R shows that such a fixed point 0' El) is characterized by the following criteria:

— It has stable directions whose associated stable manifold consists of all the systems which are exactly critical (among those described by some 0 G 4.) and whose successive renormalizations converge to 0*. — It has unstable directions along which the renormalization reduces the critical characteristics (in particular (0)) and increases the distance from 0 to 0*. The fixed-point relation Re = e implies that (0*)=(/--64,*)=4"(0*)/k. Thus the correlation length (0*) can take only the two extreme values (0,*) =oo for a critical fi xed point and 00*)=0 for a fixed point associated to a typical non-critical system. We will only consider the generic situation where R has at least one critical hyperbolzc fixed point 0* that is35 , a fixed point having no marginal directions, i.e. directions

See 11.3 for the definition and the construction of the stable manifold of a fixed point,. A stricter definition must be used in the infinite-dimensional situation, since then the spectrum of DR(0*) is not reduced to the eigenvalues; it always expresses the fact that a 34

35

sharp separation exists between the stable and the unstable spectral values but it also involves the norms of restrictions and of projections of D R(0* ) (Hirsch and Pugh [1970]).

3.1 Renormalization operators

61

whose eigenvalues have modulus 1. In this case we can perform a local analysis in the neighborhood of 0* of the flow generated by R (see figure 3.1).

Linear analysis of R in the neighborhood of a hyperbolic fixed point 4)* is a vector space, we decompose the structure rule E

If

o

o*

E

ci(0) ei E

DR(0*).e i = ei

where ej E (1)

The real number ci (0) is the component of 0 in the eigendirection ei of the linearized operator DH('). The stable components are reduced when we iterate the linearized renormalization and asymptotically vanish:

lim

n —Poo

DR(0T.[ci(0)ci]

=0

if 'Ail < 1

These stable components are said to be irrelevant or inessential since they describe characteristics of the system which have no observable consequences after a sufficient number of renormalizations, so a fortiori no macroscopic consequences. The unstable directions (l )il> 1) correspond to aspects of the phenomenon which are selected and amplified by the renormalization; they are directly related to the critical quantities, and thus are said to be relevant. One very important consequence of this is that the perturbations having only inessential components, identified by the fact that renormalization contracts them, do not affect the critical behavior and thus do not modify the universality class of the system37 . The significance of this linear analysis comes from a theorem on "straightening of the flow" in the neighborhood of the hyperbolic fixed point, which asserts the differentiable conjugacy of the flow generated by R and of the linearized flow in the neighborhood of 0*. This result can be made explicit via a non-linear analysis in the neighborhood of 0*, one of the steps is the construction of the stable manifold Vs of R in the neighborhood of the critical fixed point 0*:

-- { C V 3 E.

nErn Rçf =

= {(1) E 0,

V 71

Rn is defined}

VS is also called the critical manifold, since it contains the structure rules 0 of the critical systems. Let us add that the macroscopic behavior of a system of structure rule 0 C V 5, being described by = , is not only critical but also universal.

If 0 is a manifold of dimension d, we use a differentiable map to bring the neighborhood of (/). to Rd ; cf, Lang [1962] or Bishop and Crittenden [1964] for information on differentiable manifolds. 37 This argument will be used in § 5D.3 and § 6,2.5 in more concrete situations.

62

Mathematical aspects

cro E Eo

Figure 3.1 - Action of the renorrnalization operator R Let 4,0 be a subspace of structure rules where R has a unique hyperbohe fixed point qY " , i.e. such that the vector spaces Es and E." generated by the stable and unstable directions of DR(0") are supplementary and different from { 0 }. Then R leaves invariant two manifolds Vs and V" whose intersection is 0"; V' is called stable since it is tangent to Es at 0* and uniformly contracted by R; V' is called unstable since it is tangent to E" at cfr* and uniformly dilated by R. The critical behavior of a physical system controlled by the parameter p is of 00 describing it is related to the characteristics of R if the family [0 12] transverse to V 8 . Its unique intersection 04a. with V 3 , determining the critical value j.40,,„ is related to the typical critical system described by the universal function 0* via the formula lirrik,,„, Rk (0 0.) = 0'. The critical exponents are related to the eigenvalues of DR( 0*), which proves their universality in (Po. Typically, there exists a manifold E0 whose inverse images [En = Fe —T'Ecdn>1 accumulate at V". If DR(*) has a single unstable eigenvalue 8 > 1, the comparison of the family (0p,.,)n>0 defined by E En , with the family (crn)n>0 defined by on = En rl V" satisfying cr, = Ra1.,+ 1 enables us to show that the sequence of parameters (P)n>0 converges to poi, with

po, — pfi — (5 —n •

3.1 Renormalization operators

63

DETAILS AND COMPLEMENTS: FIXED POINTS AND EIGENVALIJES

Let (Rk)k E K be a family of renormalization operators parametrized by the scaling factor k G K C)0, oo[. The physical consistency of the action of these operators requires that they satisfy the group-theoretic relation Rk Rk, ;,= Rkk o for all 6, k E K. This implies that Rk and Rk o Commute, so for all ko,k E K, 0* E 4) and Rke are simultaneously fixed points of Rk o ; if one of the operators of the semi group has only one fixed point 0* in (I) , then this point is a fixed point of all the operators (Rk)kEK, If G = {nk , k > 0} is a continuous group (AC =]O, oop, then a fixed point of both Rk, and Rk, is a fixed point of Rkiikl for all integers n and p, so by continuity, for all Rk if log k1 / log k2 is irrational since then Z log kl f Z log k2 is dense in R = {log k k E K } . If 0* is a fixed point of all the operators (P? ‘ ...,k)k E K, the group law can be linearized to 0 4`; Lk, o 14 2 = Lk. 2 o Lk i = Lk= ik2 ) where Lk= DRk(e). Lk, and Lk, commute, so they have the same eigenvectors, which are thus independent of k E K. The eigenvalues Ai(k) satisfy Aj(k1))ti(k2) = A1(k1k2) for all kl, k2 E K. If K = {krd,n G N} or if K = [1, +c[,they are of the form .A5(k) = lei (where j is real or complex). If R admits a family (07 ) a of fixed points which depend differentiably on a real parameter a, then (ac ea )„=„0 is an eigenvector of DR(0: 0 ) with eigenvalue 1. In the case of a continuous semi group of renormalization operators (Rk)k>I, either the fixed point 0' of Rk o is a fixed point of all the (Rk)k>i, or (akRk)k,k o e is an eigenvector of DRk,(e) with eigenvalue 1; a hyperbolic fixed point, having no eigenvalue equal to I is thus necessarily a common fixed point of all the operators (R,k)k>1. -

-

-

-

3.1.4 Renormalization of a one-parameter family Let (Rk)k>i be a (semi)group of renormalization operators. We want to interpret the results of the analysis of the flow they generate in the phase space (I), in terms of scaling laws and critical exponents. With this aim, we consider a physical system S depending on parameters [A] = (Ai )i>o, which exhibits a critical behavior when a control parameter 9 crosses a value 0,; we choose Ao = 0 0,. The essential step is to move the action of Rk onto the parameters: we write rk([ill) for the parameters of RkS. FrOM now on we place ourselves in the parameter space. Let us determine the fixed points [A' ] = rk[A*]. We suppose there exists a hyperbolic fixed point [A], and perform the linear analysis of the flow generated by rk in A in the neighborhood of this point. Since the family (rk)k> i also satisfies the group law rk, o rk2 = rk , k2 , we see by the argument detailed above that the eigenvalues of Drk([A*]) are of the form (10))5>0. yo is real since we supposed that the critical phenomenon is controlled by a single real parameter A o , however, the values ('yi)i> i may be complex. Assuming without loss of generality that [A*] = 0 and choosing ;parametrization S [A] such that Dr k ([A*]) is diagonal, we obtain: —

Vk > 1,

[rkaApb =

Ai + 0(A 2 )

(

[A*] 0)

If Ao 0, then S is not critical: by construction, the renormalization increases the distance from S to the critical point, so that yo > O. Let us order the labels of the eigendirections in such a way that R'yj >0 if j<jo and Rj < 0 if j>jo > 0. The stable directions (of label j > jo) correspond to inessential parameters, and play no role in

Mathematical aspects

64

the critical behavior since the differences, from any system to another one, between the effective values of these parameters decrease as renormalization increases the real scale of the description. Conversely, the unstable directions (of label j< jo) are critical directions, as confirmed by the relation between the values (7i)i<j o and the critical exponents. If F is a state function which is regular at the critical point [Al = 0, we write it (approximately) in linear form to determine the behavior of its successive

renormalizations F([A1) = F*

EA, fi +0 (A 2) == F(r[A]) = F* + Ekn-hAi fi + 0 (A 2)

< oo and that the remaining terms are of the The regularity of F ensures that order of 0(A2 ). The renormalization increases the distance from F to its critical value oo (at least after enough iterations (n > no) so that the leading-order behavior is given by the contributions along the unstable directions j < jo ), except —

critical

if h = = fio , corresponding to a quantity F which is insensitive to the behavior (at [Al= 0) of the system S, or

— if A o = ...= Ai ° , which occurs when the initial system is already exactly critical, and is brought by renormalization to the fixed point [Al= O. In these two cases, F(rrki[A]) converges to F* as n tends to infinity. For quantities F which are sensitive to the critical behavior of the system S, we can typically show, using the physical interpretation of the renormalization (or in some cases the very construction of Rk), that for every real number k > 1, the action of Rk on S transforms F(S) into F(RkS) = k" F(8). If F diverges at the critical point, then cu. < 0, since Rk is designed to weaken the critical divergences. Whatever the sign of a F , the leading term in [A] of this relation is expressed in the form:

F(Ao, Ai,...) = k - "F F(le-Y 0 A 0 ,k 71 A1,...) This relation is valid for all k > 1 in the neighborhood of the critical point [Al = 0, and ensures that F satisfies the scaling law38:

F (A o , A 1 , . .)

t,b(A 1 A 0 7117° , .• • A • A 0-72

• • •)

[A] — O

where is an analytic function. Only the directions for which R-yi > 0 influence the /' tends to 0 with A o if R7j < O. critical behavior since the reduced argument AjA If for example 70 > 0 and 71 > 0 and R7i < 0 if j > 2, then F depends essentially only on Ao and A1, since its critical scaling laws are the ones obeyed by P(A0, F(A o , A 1 ,0,0 ...). They are expressed by P(A 0 , A i ) = A o"h° 7k(A / A0-71/= '°) where ik is an analytic function depending only on a single real argument, which must satisfy ik(0) =1 and 1,1)(z co) zc'P I 7 ] in order to obtain correct behavior in the cases (A 1 = "The results obtained here for a continuous (semi)group (Rk) k>i remain valid for a discrete (semi)group (RZOn>i, provided we allow an additional dependence of the reduced function V.7 with respect to y = log Ao, periodic of period -yo log k9.

Renorrnalization operators

Y.1

65

0, A0 0 0) and (ilo = 0, A 1 0). These two situations are described by the restricted Aal F 1-vi (A, scaling laws P(}10, Ai = CI) AoaPH° (Ao -÷ 0) and 1' (A o , 0). A typical example is that of the average magnetization M(71, H) of a magnetic medium S in equilibrium in an applied magnetic field H at a prescribed temperature T; the critical point is (7', - T = 0, H = 0) and we have Al (T, H = 0) (T - Tc r if T < A f(T c , H) I H1 10 or globally M(T, H) (T T c )) 11)[H (T - Tc )_ ° 6 ] with fl > 0 and §4.2.1). 8 >0 If we parametrize S by another family [s] = [s]([il]), the action of the renormalization involves an operator tk instead of rk, such that:

tk(IsiaAD)

[s](rk[AD

[e ] = [s]([ill) is a fixed point of tk and

Dtk ( [51).(9 [Wail; )1 (A ., =

(8[s]/8il5 )1 1, 1

The eigenvalues of Dtk([s1) are thus the same as those of Drk ([A1). This result is necessary since the eigenvalues determine the critical exponents, which are intrinsic characteristics of the critical phenomenon and must not depend on the formal framework (here the parametrization of S) used to study it.

DETAILS AND COMPLEMENTS: MULTICRITICALITY The theory of critical phenomena introduces the notion of multicritscality and its associated exponents, known as crossover exponents. Illustrated in figure 3,2, this notion appears if the renormalization operator R has two fixed points 01' and 4)'` of stable manifolds Vf. {0'} and V 1{0}, and if there exists a hypersurface E containing Ot and , meeting VI in the neighborhood of 0/ and contained in V in the neighborhood of 44 . If so, there exists (kc = (Pi E E or (pi eEnVf such that:

44

R'00 = 04' since 00 = cpi E VI, but if E f 0, lirrin , 03 RngS, = (Y; This is an example of multicritical behavior where the universality class of the system represented by O f is that of 07 or that of 44 according to the value of E The rate, in terms of iterations of R, at which the crossover between 07 and 0; actually appears depends on the parameter c. When this dependence on E takes the form of a scaling law, the associated if

c = 0,

exponent A is called the crossover exponent. Let us illustrate this point in the typical case where the structure rules 0 e 1. depend on two variables (or parameters) x and y and are renormalized by [Rk0](s, y) = Ic — a0(lcs, k cy). The fixed points common to all the operators (Rk)k>0 are the functions of the form 4'(x, y) = saik(yx,'), Given 1,bi and l,b2 associated in this way to two fixed points 01 and 0 4'2 , we can construct a regular function lif(z, t) such that kli(z, 0) = ki(z) and 'tV(z, oo) = 02(z). The family 0,(x,y) = za Ilf(ys - c , Ex -11 °) 77-. 07(x, y) and then presents multicritical behavior since

lim [Rk0,](x , y)

k

-co

lim xaxii(yx. - e , ck -11 ° x -1 ljA ) =- 44(, y) if 6 >

k

is obtained for different pairs (k, e), where the scaling factor k (measuring the "strength" of the renormalization realized by Rk or related to the number n of iterations of the operator Rk, if k = ktc,') varies according to k(c, co)

Rk0, =

66

Mathematical aspects

Multicriticality and associated "crossover" If the domain 4)0 contains a single fixed point 07 of R, which is moreover hyperbolic, then the universality class of a system S described in 4)0 by a one-parameter family (0 0 ) 1, transverse to Vf is the universality class of 01, Figure 3.2 -

and its critical behavior is related to the characteristics of DR(01) (figure 3.1). If 410 contains a second fixed point q of stable manifold V f {0;}, a transfer of influence (or in short a crossover) can occur between 01' and O *2. This is shown here on the trajectory starting at 0 a (where we have taken 0; E V to simplify the picture): to begin with, this trajectory feels the presence of 01, but it moves away from it along an unstable direction of 0"'i and reaches the zone of influence of (r2 . If the family (0 0,) i, passes through (I) a, then S belongs to the universality class of 04 and not to that of 44, as an insufficiently iterated renormalization would suggest: its critical behavior is described by the universal function 0'12' and the characteristics of MA0'0. On the other hand, the trajectory starting at Ob never feels the influence of g: even if S is described by a family (00 ) 0 passing through Ob, it still belongs to the universality class of 01'. In this situation we speak of mullicritacality, as the critical exponents differ according to the domain of 4)0 in which S is

described.

3.2 Mathematical study of critical systems 3.2 3.2.1

67

Mathematical study of critical systems

The probabilistic approach to a critical phenomenon

Physical couplings and statistical correlations The two examples in chapter 2 show how important it is to distinguish between:

the range A of the direct physical couplings between the elementary constituents. In the Ising model, A is the distance a separating the spins which are nearest neighbors —

on the lattice (i.e. the lattice parameter); in a discrete dynamical system, A is the time step At. It is a physical quantity, dependent on the microscopic (local or instantaneous) structure of the system. This range A does not vary much with the control parameter: thus it is not its value which determines the critical character of the system. We will always use a minimum scale a (defining the microscopic scale) greater than or equal to A to describe the system, so that only the nearest-neighbor elementary constituents (of size a) interact directly;

the range

of the statistical correlations. It diverges at T, = 0 in the Ising model and at the onset of chaos GI H) in the period-doubling scenario. Greater than A, 4" is a global statistical characteristic; a statistical organization of the constituents and of their statistical couplings39 is present on every scale from A to Two elementary subsystems separated by I ept,C, even if they have no direct physical interaction, will have correlated statistical distributions. Thus is the scale up through which —

the variations of the state of one of the elementary subsystems will propagate' from neighbor to neighbor through all the subsystems located at the intermediate distances, either via the short-range physical couplings, or simply because of the constraints imposed by the statistical equilibrium. In particular it is not possible to analyze the system using only a sample of size less than

Probabilistic description of the critical system A challenge in understanding critical behavior is the apparent contradiction between the microscopic scale of the physical couplings and the macroscopic scale of the phenomena they generate. The explanation of this paradox is based on the divergence of the range of statistical correlations observed in critical systems; it involves the cooperative behavior of the elementary constituents by which they are organized into a structure relating the microscopic and macroscopic scales. The details and the physical nature of the couplings are not in themselves important, as long as they induce this structure. Thus, the essential aspect of a critical phenomenon is not the physical couplings themselves, but the statistical coupling of these couplings, which explains why the phenomenon is controlled by the parameters on which the statistical distribution of the microscopic configurations depends, This assertion forms the basis of phenomenological scaling theories since it ensures that the only characteristic scale 39 The example of percolation (supplement 7A) shows that these statisticalcorrelations can be present even if there are no physical couplings between the constituents. This mechanism is often called the domino effect.

°

Mathematical aspects

68

of the global behavior is the correlation length the microscopic scale A plays no role, and the critical properties can be described via a single reduced variable rie (or tg in a temporal system) and macroscopic parameters. Thus we can study critical phenomena via an abstract formalism in the framework of probability theory, since as soon as a large number of degrees of freedom display cooperative behavior, it is the very existence and the statistical characteristics of this behavior which determine the macroscopic critical behavior, The existence of a common mathematical model reveals the universality of critical behavior, by displaying the formal analogy between critical manifestations observed in very different physical systems. In this model, the state of each of the N elementary constituents of the system S is considered as a random variable X„ (n = 1 . N); the essential ingredient is the description of the statistical correlations. The link with critical phenomena appears when we translate into physical terms the limit theorems giving results on the stochastic convergence of the sequences [X ] n>1 of random variables via the asymptotic behavior as N —fr•oo of the observable Xi

(or rather (1/N) E i=1 Xi). This probabilistic approach derives scaling properties as the universal consequence of collective behavior, and relates their exponents to the statistical properties of the configurations. It serves, moreover, as a guide to the analysis of critical properties via renormalization.

Critical phenomena and renorrnalization in probabilistic terms Let us consider a homogeneous system S; its boundaries play a negligible role in its thermodynamic properties, so we can assume that the random sequence [X ] = (Xi ) 1<< N describing4 I its N elementary constituents is statistically homogeneous. The variables (Xi )1 brings each Xi to a centered variable. The constituents of S present a self-similar collective behavior if there exists a scaling transformation relating the system described by n random variables to the one described by kn random variables grouped in n "k-blocks" of k variables. We show that this is the case if there exists a value )51* for which the random variable S,„([X],M) = i Xj converges as n tends to infinity to a random non-zero variable S([X]) with real finite values. DETAILS AND COMPLEMENTS: PROOFS

Set Sn [X] = B„E.7... 1 Xj: we must first determine for which real sequences

(Bn)n>1

the random variable Sk n [X] coincides for all the integers k and n with the result Sn [Sk [X]] of the coarse-graining into n "k-blocks", each described by a random variable identical to Sk[X], This can be written explicitly as

Skn = Bn

E

(4) 4

1
Bk,,

E 1<j
X

=E 1
Bk

E 1<j
'This probabilistic description can be either spatial or temporal. If the system is spatial, N labels a localized microscopic subsystem, and (1/N) 2_, ..7=1 X i is a spatial mean.

=

For a dynamical system, j = 1... N labels the successive temporal steps and (1/N) E7 1 Xj appears as a time average.

3.2

Mathematical study of critical systems

69

The consistency imposes the relation Bk.B, = Bk, (k, n G N i ), whose solutions are of the form 81.1 = n -0 (# an arbitrary real number). A k-block described by Sk([X], )6) =---

k -- flE21:=1 Xj can then be considered as a unique subsystem since the relation Sk,[X] = Sn [Sk] expresses the fact that the possible collective behaviors of the initial family [X] = (Xj)j>i and of a family [Sk] = (S) q > 1 of k-blocks are the same. This similarity corresponds to a scale-invariant physical phenomenon only if the asymptotic behavior of SN({X1, 0) as N oo is non-trivial, which restricts the set of possible sequences [X] and as a function of the statistical properties of [X]. fixes the value of the exponent

The value )3* = an, if it exists, is unique for the sequence [X] since S,([X],#)= n° *-0 S,([X], 13. ) converges to 0 if 13 > 0* (as n tends to infinity) and tends to +oo if < j3'. Thus it appears as a borderline value /3* = n - g3 E2n=o Xj = 01. This exponent )3* describes the most important aspect of the collective phenomenon since it gives the "effective amplitude" kk = A k . of a block of k constituents (elementary or not) of individual amplitude A. The transformation [X] Sk ([X], 13*) appears as a renormalization [Ski E Rg.,k([X]), implementing the method of "spin blocks" developed in statistical mechanics42 in a probabilistic context and emphasizing the self-similarity of the system. The form k -0 of the factor by which the amplitude of the k blocks is contracted follows from the group structure o Ro,k, = Rp,k i k 2 of the family [Ro k] k % N. of transformations. Suppose < X 2 > A 2 < oo. The real -

amplitude A'k E < (Ei=1 Xj ) 2 > 1 / 2 of a k-block is related not only to the amplitude A < XqXg >]1i1
010 2 = E ci (k

lil)

1c 2'5* A 2 ,

lCd <

A 2 SO (A

)2

< A 2k2

-k
This relation determines the value #* < 1 such that the apparent amplitude of a kIc - P • Xi +k( q _i)), converges to block, described after renormalization by A as k tends to infinity; #* is thus a macroscopic characteristic of the system since it depends on the global distribution of the sequence [X]. A sequence [X] which is the fixed point of an operator Ro01k0 describes a collective behavior which is exactly selfsimilar with )3*([X]) = /30 since in this case < Sk([X], 0) 2 >=< X 2 >= A for the finite value /co. The renorinahzation Roci k a thus integrates Me influence of the short-range correlations 0 < ko) directly into the definition of an apparent system, similar to Me initial system (< X 2 >—< S ([X] )30 2 > ) but whose real extent is ko times larger. One of its consequences is that the correlation range is reduced by a factor of k.

SI°

For statistically independent constituents, the central limit theorem gives the existence of a self-similar collective behavior of exponent 0* = 1/2. The random variable Sr,([X], 0* = 1/2) has amplitude A/Vii exactly equal to A <X 2 > 1 / 2 . Moreover, it converges in law to the variable 8 ([X)) of centered Gaussian law and variance < X 2 >. This asymptotic result remains valid if the constituents are correlated with

§2.1.2 and §4.3.l; in this analogy X, is the (algebraic) value of the spin located at the site labeled by j and Sk is the effective spin associated to a block of k "real" spins. 42 See

Mathematical aspects correlations C5 = < XiXi+i > decreasing fast enough (as a function of their mutual distance j) for the sum to converge: E± : Ci < oo. In this case, the limit in law e• Eio, 04 (in the absence of correlations, we find 800 ([X]) has variation K = K =< X 2 >), If, in the extreme case, strong correlations are present on every scale, being independent of the "distance" between the constituents (C5 E- A) , then A'n =nA and we must take /3* = 1. Another limiting case occurs when a sequence [X] presents maximal destructive correlations C = (-1)JA: then we have Al2.„ = 0 and A +1 = A, so that we must take fr= O.

<>

DETAILS AND COMPLEMENTS: GENERALIZED LIMIT THEOREMS

These theorems apply to statistically homogeneous sequences of random variables [Alai >1 which are weakly correlated in the sense that the correlations given by [Cn = < XjXj+ n >]ne z of the constituent j represented by Xj have convergent turns.

— Let us first study the generalization of the law of large numbers. Due to the weakening of the assumption on the statistical independence of the random values, we obtain a. weaker law, ensuring mean-square convergence rather than almost sure convergence. The variance of the centered random variable S N([X] or) Ar - "El
>=__

N-20

E

ck(N

- >

—N
summability 1(' 0 E+'>:c. IC < oo, we can write for arbitrarily small e: Ve > 0, 3N„ N -1 EIkHCkI< E N>lkl>N,

On the other hand, for

icki <

N > Nci = sup(N„ N,K0/), we get:

E iockl <

N,K o iN

<

so

N_ 1

Ik1ICk <2c

if

N>

Ikl
which proves the convergence result

lim N—•co

ck,(N

—N
-

= E Ck

K>0

< SN([X], cr) 2 >

—o0

Thus, SN([X], a) converges to 0 in quadratic mean if a >1/2. If the correlations are no longer summable but we still have E—N = 0 if 2a> 1 + 7,

— Under the same hypothesis of summ ability of the correlations, we can show that as N tends to infinity, SN([X], a 1/2) converges in law to the centered Gaussian law of variance This result generalizes the central lirrast theorem and expresses the fact that K=

cr,.

the statistical fluctuations of the density

N -1 EN . X. around its average Q are of the order 2=1

of (9(.IIC/N), as in the absence of correlations,

3.2 3.2.2

Mathematical study of critical systems

71

Fluctuations and correlations

Thermodynamic description and statistical fluctuations Let us consider a homogeneous sequence (Yi)j> 1 of random variables, a priori not centered (of average < Y > independent of j), each describing the value taken by a local quantity y in one of the elementary constituents of the system S (labeled by j > 1). The quantity kiki = is the macroscopic observable quantity representing the value averaged over the N constituents j = 1.. . N of the quantity Y. The basis of the thermodynamic description of S is the law of large numbers since this law implies the almost sure convergence of the random variable kN to its statistical average < Y In the thermodynamic limit (assuming that this law applies to the sequence (yi )j> i ), it thus seems legitimate to use only the deterministic value (called the "thermodynamic quantity") to describe S as it is perceived on the macroscopic scale, without describing its microscopic configuration precisely since it has no influence on the value of the observable quantity VN in the limit N co, However, to be reliable, this result must be completed by the description of the dependence on N of the fluctuations 837N = < (i'N — ) 2 > 1 / 2 since N always remains finite in the system S. Writing Xj = < Y >, it is easy to check that N— SN([X], )3= 1) behaves like Nraxi) -1 Sco ([X]) as N tends to infinity. The amplitude SYN of the fluctuations of the value kN observed around its average < Y > is thus of order < > 1 / 2 Nr([x]) -1 . _The The quantity y displays a critical character if its amplitude (the standard deviation of decreases to 0 (as N tends to infinity) more slowly than in the typical noncritical case where the N constituents are independent, i.e. more slowly than 1/VY. The collective behavior revealed by the convergence in law of S N [X]) to a nontrivial limit So. ([X]) thus generates a critical phenomenon if and only if 0*([X])> 1/2. The apparition of a critical phenomenon and its scaling properties are thus determined (like the value of 13*) by the nature of the internal statistical correlations, and it is not necessary to explicitly describe the physical mechanisms behind these correlations, nor the constituents between which they take place. This explains the universality of the critical behavior. 40, DETAILS AND COMPLEMENTS: CORRELATIONS AND CRITICAL FLUCTUATIONS

Let us detail the relation between correlations and fluctuations in the example of a spatial density m(f) assumed to be statistically homogeneous and isotropic. Empirical and dimensional arguments lead to the following expression for the associated correlation density:

C(f) <m(0)m(f) > < m > 2 = co 7.2—d— Y1 f f is an zqe

analytic function on R, normalized by 1(0) = 1, which behaves like f(z) at infinity, i.e. when z co. The characteristic length involved in this expression can be interpreted as the range of the statistical correlations, which is why it is called the (statistical) correlation length. For this density C(F) to be integrable in = 0, we must have n < 2. The asymptotic behavior of f ensures that f ic()idatf. < co, unless oc since then f (r I e) a 1: C(F) no longer decreases exponentially fast at infinity, but only

where

Mathematical aspects

72

geometrically (as / 2-61- g), which is too slow to ensure summability at infinity since q < 2. The non-summability of the correlations is thus a critical characteristic like the divergence = co (expressing the absence of characteristic length); this statement is confirmed by the fact that f C(f)d df is proportional to certain response coefficients of the system which diverge at the critical points. Note that f

C(f)d df

er(q = 0): the non-summability at

C(f) is equivalent to the divergence of its Fourier transform C(q) at the origin q = O. This singularity of a(q) appears only in a critical system, as shown by the scaling law, which is conjugate to the law satisfied by C: infinity of

Ô(q)

Qd

Co e - ')

(sin

yge

f(y)y i-,9 dy 4 .2-g7G(q) q n-2 g0,0

where the factor Od2r4 / 2 [1r(d/2)] -1 results from the angular integration. By the regularity / yg]f(y)y 1-1? (if < oo) and the fact that it is uniformly at q = 0 of the integrand [sin bounded by the absolutely integrable function f(y)y' - ' , we see that G(u) is regular on = oo. In R+, in particular at u = O. Consequently, is regular at q = 0 unless that case, i.e. at the critical point, we make the change of variables 6 = yqe in the above 1, gives 0(q) q71-2 g, where expression of C(1); the continuity of f at 0, with 1(0) g, = g(00) = fa (sin x/r)s 1- "ds is finite if 2 > r O. Away from the critical point, the

ye

am

>

rigorous definition of e in terms of

e = -[20(0)]thus

0(q)

is given by

(d2 , -,. ., /td.2,2) 2, = 0) = _& (0) /E,( 0 ) (

is the width of the peak of

where 6(q2 )

C(g);

0(q) at O. The advantage of this definition of e is that

it is based on the function 0(0 which is experimentally accessible since it is related to the scattering cross-section (of light or of neutrons) for a transfer qfinai - 4initiar q•

In the case of a system S which becomes critical when a parameter 0 passes through the value 9, = 0, the action of the renormalization operator Rk on the correlation function C(r,9) of a spatially homogeneous and isotropic density typically reads

[140(r, 0) icbC(kr, 1c -111'9)

where

(9)

ieri if 6

gc =

Indeed, renormalization contracts lengths, and in particular r and 4", by a factor of k; the scaling law 4-(9) 91 -2' then shows that the parameter Oik such that (01k ) = ke(0) is Eek = k-1111 0. An empirical scaling hypothesis on the asymptotic behavior of C(r,0) away from the critical point 0, 0 leads us to write C(r,O) , T2-d e(6) - " e -r g (9) as r tends to infinity; the value b q ± d - 2 is thus the only one which leads, for fixed r and 9 A 0, to non-trivial behavior lim k -[Rkg(r, 9). The study of the asymptotic behavior under the action of Rk thus gives the critical exponent q of the associated correlation functions,

3.2

Mathematical study of critical systems

73

Local fluctuatzons

Until now, we have described global statistical fluctuations, defined to be Ei. Y. describing the observable the difference between the variable 3-7N = (macroscopic) state of the system and its statistical average < Y >. We can also study the local fluctuations, i.e. the fluctuations [Xj = < Y >] 7 >i of the elementary state variables. These contain more information, since they give a complete (here spatial) microscopic description of the system by showing how the possible global fluctuation is spatially distributed. They enable us to describe the spatial structure of the inhomogeneities on every scale; on the scale nA they are described by the family [Z," -= n E7=1 Yo+ih>0 . The essential result (proven below) is that the typical spatial extent of the local fluctuations is equal to the statistical correlation length C. We already know that the criticality (at 9 = 0,) of a system is equivalent to the non-summability of the correlations between one constituent of S and all the others; it is reflected in the divergence of the correlation length E(9) at the value 0,. The observable consequence of this divergence is the increase of the typical spatial extent of the local statistical fluctuations: they become "critical", involving larger and larger domains of S, until they become directly perceptible on the macroscopic scale at 0,, A typical physical example is the critical opalescence observed in the critical liquid-gas transition ( 1.2.1). DETAILS AND COMPLEMENTS; ESTIMATION OF THE LOCAL FLUCTUATIONS

Let A be the microscopic scale of a spatially extended system

S (in Rd ) which is

statistically homogeneous. We subdivide S into elementary constituents of volume Ad, labeled by Zd and described by identical random variables [Yff ] le z d . The state of the subsystem occupying the ball B(i=ATt o , a) is described by:

2(a) =

E

37,1

(spatial extent a)

< Z(a)>= Af(a)

An E B( 2 1a)

where the sum is over the Ar(a) (2aP)d constituents contained in 13(,a). By statistical Afto. Let denote the homogeneity, the distribution of 2(a) does not depend on characteristic length of the correlations C = - 2 ; these can be written C.-0AM where the auxiliary function C- (l), dependent on a dimensionless variable C(ft) ft, has characteristic scale IlAftli = 1. If these correlations are summable, we estimate their sum. if

K

by:

R;(")lddi < °°

K

E cf,

vAid'atot

The generalized central limit theorem applies to

where

<

E'(tt)ddft

SZ(a) = .Z(a)- <2(a)>

dProbPr(a)]-1 /2 8Z(a) = z) 7-- (27rK)-1/2 e-22/2xdz K

Ct ot

if a

Prob (SZ(a)l > zn/Ai(a)) <e/2K

The random variable 6Z(a) describes the inhomogeneities perceived on the scale cs: the inequality IbZ(o)I > 201-(a) means that the realization Z(a)/Ai(a) of the spatial average

74

Mathematical aspects

in the domain of extent a differs by at least 2( from its statistical average < Y >. In this situation we speak of a "fluctuation of spatial extent a and of amplitude E" since, neglecting the possibility of individual differences 4 3 < Y > I greater than kt, at least *A1.(a)1(k 1) elementary variables must be all greater than E or all less than < Y > — E. This situation occurs with a probability greater than Oexp[-160.Af(a)1 K] exp[- 4 2 ad /Ct ot 4r] and smaller than expf-2e2adlôt0td.l. The spatial (linear) extent characteristic of the instantaneous fluctuations of amplitude is given by the length i(e, e) Ôt10/. tdc -2/ d, proportional to the correlation length e, where ri depends on the probability threshold e -161 above which an event is considered to be observable, —

3.2.3

Mean-field theories and the law of large numbers

The term "mean -field theory" originally referred to an approximation introduced in the analytic study of ferromagnetism to compute the statistical quantities describing the observable macroscopic properties of a system of N spins, in thermal equilibrium at a temperature T and embedded in an exterior magnetic field Ho ( 4.1.3). Its principle is to replace the explicit interaction hi between the spin j and the N — 1 others by a spatially homogeneous term h(< M>) depending only on the statistical average < M> of the total magnetization which we want to compute. Each spin thus feels an effective field of intensity II0 h(< M >), which formally plays the role of a uniform constant exterior field. This procedure extends to any family of N interacting subsystems described on the microscopic scale by a family (Yi) i <j )[ Yi ] formally analogous to the term describing the influence on the subsystem j of a given deterministic homogeneous exterior field (independent of j). This field is said to be mean because it does not depend on the configuration of the N subsystems but only on the global statistical average < M >, coinciding with a thermodynamic state function. This approach has two major advantages: .

— it reduces the analysis to that of N independent subsystems; — it allows an empirical approach since < MN > is observable on the macroscopic scale: we can introduce an adequate coupling term K(< MN >)[.] without knowing the interactions between the subsystems. Here the terminology "effective field" seems more appropriate than "mean field". A first condition for the validity of this approach is that the N subsystems must have an analogous environment, so that the associated N "real" coupling terms have realizations which are not too spread out around one average value with which we 43 1f such an individual difference is observed, we begin by describing the fluctuations of amplitude kf, before setting them aside to analyze the fluctuations of smaller amplitude; thus we give a hierarchical description with decreasing amplitude and increasing spatial scale.

3.2

Mathematical study of critical systems

75

identify K(< MN >)[.]. In particular it is essential that the system be (almost) homogeneous on the microscopic scale. Another condition is that each subsystem must interact with many others, so as to effectively feel an average influence, depending only on statistical quantities according to the strong law of large numbers. For instance, for a lattice model, the mean-field approach is valid only if all the sites are geometrically equivalent; the greater the number m a of nearest neighbors of a site, the better this type of approach works. As ma grows with the dimension d of the lattice (of given geometry), there exists for each model parameter d a critical dimension d, above which the mean-field approach gives exact results. From this qualitative discussion, it appears that mean-field theories give poor or even wrong results in the neighborhood of critical points, because of the inhomogeneities on every scale of the critical configurations, and also because of the importance of the cooperative behavior of the constituents. This is confirmed by the mathematical interpretation of the mean-field theories, which shows that they depend on statistical laws whose failure characterizes critical phenomena (see below). The inadequacies of mean-field theory in the study of critical systems sheds doubt on the validity of all the classical methods of statistical mechanics, and shows the necessity of taking statistical fluctuations into account and approaching the description of their organization in a way which is simultaneously global and multi-scale. This is exactly what renormalization methods provide. DETAILS AND COMPLEMENTS: PROBABILISTIC JUSTIFICATION OF THE MEAN FIELD

Translating the mean-field approach into probabilistic terms and examining in this context the conditions for its validity reveals the reasons for its failure to correctly describe critical phenomena. Let us present a typical problem. In a system SN of N identical particles, the state of a particle i is described by its fixed position ft: and one or more additive variables si; [s] = (si)i

where the statistical average < > is taken over the distribution of the configurations [s]. Under the hypothesis that it is spatially homogeneous, BN no longer depends on i. One

Mathematical aspects

76

of the first weaknesses of this approach appears here: it smooths out all the microscopic inhomogeneities since each particle feels exactly the same field BN. In order to apply it, the system of particles should a priori be very homogeneous, even on the microscopic scale, so that the homogeneity of the effective field fiN does not seriously skew the situation. The collective behavior resulting from the organization of inhomogeneities cannot be described via meanfield theory; its failure is flagrant in the case of critical phenomena, where inhomogeneities exist on every scale. A second approximation uses the macroscopic size of the system to —

identify BN and its limit:

i. e.

lim B —N

Bc

Bi(N,[s])

?p m< . Bi(N,H) >

N-1.00

In this form, the mean-field approximation is the formulation in physical terms of the law of large numbers; it makes it possible to replace the random variable Bi(N, [.1) almost surely — by the statistical quantity Bco . This method will then be valid under the same conditions, i.e. a constraint on N (namely N 00) and, assuming that the function J(.) is bounded by Jmax < co, a hypothesis of summability of the statistical correlations between one particle and all the others:

E

Vk E Z,

1<si+ksk>—<s>217.

EICi

H1<0 < o

j-=—oo

j = - 00

If this is not the case, the system is critical and the strong correlations induce a structure on every scale, which thus cannot be neglected. The (generalized) central limit theorem improves the approximation by giving the order of magnitude of the error obtained by replacing Bi(N, [s]) by the "mean field" Itiw; under the same hypothesis of summability (ko < 00), it states that -IrVtBi(N, [s]) — B oo ] converges in law to a centered Gaussian law of variation here bounded by 44 2 az.Ko , so

SBN

<

[hoc —

The statistical differences between

2 ( Nj.))1 > 1/2 < 0(Jrnax \Mo/N) Bi (N,[s]) and boa are estimated by:

Prob[[sb i-Bi(N,[8] ) bool >

1513k

Ts,T

K0,17.2),„n, NA 2

An alternative, more empirical approach starts from a phenomenological expression The same sequence of statistical arguments can then be applied, under the same hypothesis of summability of the correlations Ko < co, so that we can assume that h does not depend on the particle i nor on its position f, and replace miy([si) by its thermodynamic limit ik. The "mea,n field" is then h(6). The fluctuations (MN =< ih(mN ([ s]) — > 1 / 2 are bounded by (sup liel)(5mN = C9(1/A/7 ). The probability of observing statistical differences greater than A between h(m N ([s]) and h(Fneo ) is bounded above by Ko sup Ih'VNA2.

hi(mNasn replacing Bi(N, [s]).

Mathematical study of critical systems

3.2 3.2.4

77

Critical phenomena and finite scaling laws

Strictly speaking, a critical phenomenon is a notion which is well-defined only in the thermodynamic limit (if they are spatial) or in the asymptotic regime (if they are temporal). Indeed, the divergence of a correlation length or time can be observed only in a system whose extent L or T is infinite and the singularities characterizing the critical point 0 = 0, are those of thermodynamic or asymptotic quantities:

X(0) = urn X(L, 0)

= lim X(T,0),

or

T-a.

since the quantities X(L, 9) or X(7 1, 0), typically defined as integrals of bounded functions over a finite domain, are regular with respect to O. Two EXAMPLES Spatial example: the partition function of a system of N spins c = +1, of Hamiltonian H, in thermal equilibrium on a one-dimensional lattice of parameter a is given by DETAILS AND COMPLEMENTS:

E

Z(N, )3)

e -Orl(N.E6l)

(0 reciprocal temperature)

It is a finite sum of 2 N positive and bounded terms, so the free energy per spin F(N, /3) = — (1/N/3) log Z(N, )3) is regular with respect to /3, even at = fle . The singularity appears only in the thermodynamic limit F„0 (/3 ) (N/3)' log Z(N, —

Temporal example: Let [f8 ] 6, be a family of transformations of a set X, which are analytic and whose derivatives are also analytic with respect to a parameter O. The quantity

7N(0,xo).

E logimfzcsom 0<j
ou

so E

uo
is a finite sum of N finite terms so it is also analytic at O. However, the Lyapunov exponent oo is absolutely not regular with respect to (0) obtained in the asymptotic limit N

(see figure 5.3).

In an experiment or a simulation, L is necessarily finite. The statistical correlations far from the critical points have a short range 4" < L: in this situation no finite-size effects can be perceived, since the system SL is simply the juxtaposition of subsystems of size which are statistically independent and whose individual states directly reflect the macroscopic state of S. The densities and the intensive quantities observed in any sample of size greater than coincide with those observed in the thermodynamic limit. As a parameter 0 approaches its critical value 0,, the range 4-(0) of the correlations increases until it is of the order of L: the statistical fluctuations become crucial so that the random nature of the observed quantities XL can no longer be neglected. Only their realizations can be perceived experimentally or numerically; their distribution depends on L and they cannot be identified with the (deterministic) quantity X,,, predicted for a system of infinite extent. It is thus of much importance to describe

78

Mathematical aspects

analytically the statistical averages gr, and in particular the dependence on L of their differences with the limit X. 0 DETAILS

AND COMPLEMENTS: ESTIMATION OF THE STATISTICAL FLUCTUATIONS

If LE1c4- , the system of size L (in dimension d) appears as a checkerboard consisting of kd statistically independent cells of size described by a family (xj) 1<j. Using the statistical independence of the variables (xj ) i .=< (XL— < XL >)2 >, we give the following upper bound for the random difference 1X00 2 c.)d

Prob [Woo — XL1> El<

T

1.x.0 —X,, < 0 [ 71 -1/2 (e/L) d,2

where 7 is the threshold of probability below which the event is considered as not observed. The finite size effects, increasing with 4.1 L, become more marked as we approach the critical point. -

Finite-size sealing The description of the critical behavior as 0 0, of a global quantity A(L,O) in a system of size L is based on the following arguments. (i) In the thermodynamic limit L

Acc (64)

(0

Do, we observe the scaling law (7 > 0 )

—

(ii) For L < cc, A(L, 0) is a regular function of L and 6; (iii) At the critical point 0 = 0,, we suppose 44 that A satisfies a scaling law

A(L, 0,) , La (whose exponent

a is not yet determined).

(iv) The critical behavior has a unique characteristic scale, equal to the range 19 — 9,1' of the statistical correlations. These hypotheses indicate a dependence for A of the form 45

060

A(L,0)

—

L' F[Li lv(0

—

0,)]

where F(z) has characteristic scale Az = 1. Then (ii) ensures that F(z) is analytic." The comparison of limi„„o AL (0) with A 00 (0) shows that F(z) must behave like lzr, 10 9CI this scaling law in at infinity and that Œ = -y/v. Written for L = (0) —

—

"This phenomenological hypothesis is known as the finite size scaling hypothesis, 45 The general form would be A(L,8) ,--, La" F[L'I'(8 B e )] which must be reduced to eliminate the second characteristic scale 1(0 , 10 (according to iv). 46 If X is a random variable, F is a random function each of whose realizations is analytic; the sequence of arguments remains valid since AL converges almost surely towards A. -

—

—

3.3

79

Numerical methods

finite size coincides with the thermodynamic scaling law, which shows that the latter is valid if the size L is greater than the range 00) of the correlations. When L <(0), the fact that F is analytic at z 0 implies that:

A(L,0)

Div F(0)+10 — 0,IL (1 +7) Ir F' (0) + • • • si L 1111 r0

—

0,1< 1

The critical exponents we want to determine occur in the expression of the dependence of the global quantities on the size L. Thus they can be determined experimentally or numerically by studying (L, 0) A(L , 0). However, it is necessary to know 0, beforehand in order to exploit the finite-size scaling laws, which limits the precision of the results.

3.3 3.3.1

Numerical methods Simulation of critical phenomena

The principle of numerical simulation is to reproduce as faithfully as possible the microscopic, even molecular structure of the physical system S, and then the interactions and elementary mechanisms which determine its state of equilibrium or its global evolution. The statistical treatment of a series of observations made on this reconstruction identical to the system then gives the desired macroscopic properties. One of the main difficulties is the distortion caused by the (unavoidable) use of a discrete, finite model to reproduce a real system S which is continuous and very large, even infinite if the thermodynamic limit is considered. Since we have a lattice with at most N d sites, we must make a tiling of S with Nd cells. Fixing the size A (in real size) of a cell limits the dimension of the simulated system to L = NA. We take A large enough such that the system is statistically homogeneous on the scale A. However, A must remain small enough for a direct simulation of the cell to be possible, which gives the restriction L < L max.. The range 4" of the correlations gives the scale, starting from which the statistical fluctuations become negligible; all the realizations of S are then identical and homogeneous observed on the scale C. If L the simulated sample faithfully represents every system of greater size L, and the intensive quantities XL (which are a priori random) describing its state can be identified with the thermodynamic (deterministic) quantities X. Thus 1,„„, is the condition for a simulation of the thermodynamic behavior of S. Without this condition, the statistical fluctuations cannot be neglected. They are present on every scale and induce critical characteristics; the simulation must take them into account, and this is impossible without truncation to a lattice of finite size. The results of the simulation remain random in this case; by realizing a great many independent simulations one can obtain statistical averages, but the information contained in them cannot be decoded except by a theoretical analysis of the finite-size effects and usually by assuming that the position of the critical point is known. Thus, simulations give results which are difficult to exploit in the neighborhood of critical points: it is impossible to study critical phenomena on a simulation. In particular, it is impossible to deduce the position of the critical point or the value of the critical exponents from a simulation without using a finite-size scaling theory (§3.2.4).

80 3.3.2

Mathematical aspects Periodic boundary conditions and renormalization

To avoid the numerical limitation on the size L of the lattice, the simplest solution is to impose periodic boundary conditions: we "glue" the opposite edges of the lattice, so that we can place all the points of the space on it, just as we "roll" up the real line around the unit circle; this surjective correspondence is of course not injective since the points = s2, sd) and i+kL=(si l-k i L,s2+k2L,„,xd-1-kdL) coincide for every d-uple k of integers. This procedure is equivalent to performing a simulation on an infinite checkerboard each of whose squares is identical to the lattice observable on the screen. Another advantage is that it suppresses boundaries hence boundary effects; all the points have now the same geometric and statistical environment, which is a necessary condition for the validity of the notion of thermodynamic limit (1.3.3). The image of the checkerboard, which we already saw in §1.2.3 and 9.2.4, emphasizes a hypothesis whose validity is implicit: the correlation length must be much smaller than L, and the correlation time r must be much smaller than the average time taken to cross the lattice, so that the identification of the different points (i knk e zd does not lead to any ambiguity. If this is the case we can really consider the cells as statistically independent, identical superimposable domains, and describing one of them describes them all, so that the macroscopic system can actually be reconstructed by juxtaposing them. Renormalization is a natural approach in this context; it reduces the range of the statistical correlations by a factor of k, but preserves the step A and thus also the maximal size L ma, of the simulated lattice, and in n iterations, it brings the initial system to a system of apparent correlation length k'e which is small enough compared to L„ az. for it to be legitimate to use periodic boundary conditions. Thus, what can be effectively simulated is the sufficiently renormalized system, whose quantities are related in a known way, via the renormalization equations, to those of the original system.

3.3.3

Numerical resolution of the renormalization equations

To perform a numerical renormalization analysis, it is first necessary to parametrize the structure rule by a countable family [K] of real numbers, in such a way that the functional equations used to determine the fixed point (R0* = e) and the eigenvalues (DR(0'),e= A e) can be expressed as countable systems of algebraic equations. This simplification is unfortunately insufficient for a numerical resolution, because the parameter space is still (except for certain exceptions such as the Ising model) infinitedimensional: even if can be parametrized by a finite number of real numbers, its successive renormalizations will depend on an increasing number of parameters, which will become infinite at the limit, i.e. for the fixed point 0*. Thus it is necessary to replace the exact but infinite system of renormalization equations for [K] by an approximate system of finite dimension obtained from the family [K] by truncation and closure of the initial system. The underlying hypothesis is that it is possible to order the parameters [1( ] = (Ki)j>0 in such a way that the influence of the terms in which they appear decreases rapidly with the label j; the truncation is then justified in the framework of a perturbative computation. The more parameters we actually keep,

3.3

Numerical methods

81

the more effective coupling terms remain, increasing the faithfulness of the description of the correlations in the initial system. We can test the validity of a truncation [Kj = (Ki) 0 .( 1 <j0 by carrying the resolution up to the order jo + 1 to check that the results are not noticeably modified. However, the rapidly increasing complexity of the computations is prohibitive in implementing numerically these improvements and justifications. In the following paragraph, we show how this difficulty can be avoided,

3.3.4 Renormalization and the Monte Carlo method

Objectives and principles of the Monte Carlo method The study of many physical phenomena requires the computation of integrals of the form A = fx f(i)p(i)drn(i) where the phase space X is contained in RN; rn is the Lebesgue mesure of RN , pM a probability density and f(i) a local state function. A can be interpreted as the statistical average <1(X) > of a random variable X with values in RN and of distribution p. For example, A can be a global state function of a system S in thermal equilibrium at a temperature of T 11k B)3; the point G X represents the microscopic state of S, and its statistical weight is given by the Boltzmann-Gibbs distribution pW 46) -1 e'3) . If rn(X) < oo, the natural approach to the numerical computation of A consists in discretizing X into q identical cells of volume 7) in RN, centered at points (4,i ) 1<j< ir, and approximating A by the finite sum:

A, = v

E

f(i,j)

p(,,j)

where

qv = m,(X)

1<j <9 Then one must show that A, converges to A as y tends to 0, and estimate the error terms IA —AI. If ni(X) = co, X must be approximated by an increasing sequence (Xv) v of sets Xv C X of finite volume V, which can be subdivided into a finite number Viv of cells of fixed volume y. Then A(X) is approximated by A,(Xv), and the smaller we take y and the larger we take V, the better the approximation turns out. Since the number of cells must be quite large to obtain a reasonable precision applying this procedure numerically tends to be long and costly, increasingly IA — so as the dimension N of X is large. Moreover, a large number of points of X have small weight p(i) and thus contribute to A in a negligible way, and this fact is not exploited in the above computation . In this context, we introduce the Monte Carlo method, another technique for numerically computing the integral A, using the arithmetic mean a n f(xi) of a random sequence [f(ii )]i< j< „ where each point j is selected in X according to the distribution p. This method is based on the strong law of large numbers applied to a sequence (fC5 )j>1 of random variables with values in RN , independent and obeying the same law p; it asserts that the random variable A, =n -1 E)n., 1 f(Xi ) converges almost surely to <1(fc)> = A as n tends to infinity. Thus we can approximate A by a typical realization a„ of A n , with a precision which increases with n. Statistical relations allow to estimate the difference between a, and A (71 < 1 being fixed):

Prob [IA— Ani >

<

Mathematical aspects

82

where c(n, n, [i;ii i <j‹,) is computable from the sequence [j]1<j<, of observations and from statistical tables; its value increases as n decreases, so that it is necessary to compromise between the precision c and the reliability n. The problem is now to make the sampling according to a given distribution p47 . By extension, the actual numerical algorithm presented below is also called the "Monte Carlo method". DETAILS AND COMPLEMENTS: SAMPLING OF A DISTRIBUTION p

The density of probability p(i) can be written in the form of a Boltzmann—Gibbs distribution Cexp[ EM where can be interpreted as the configuration of No = N/k particles in the space Rk , actually on a finite lattice isomorphic to a subset of 7., k since any numerical computation requires bounding and discretizing the set X of values taken by 1; E(Z) is then the effective energy of this configuration. We define an elementary move to be a random isotropic move of a particle randomly selected on the lattice, of length chosen equiprobably between two bounds and A2 > 1. In order for the choice of the elementary moves not to induce any statistical bias, we impose periodic conditions for the moves reaching the boundaries of the lattice; in this case it is possible to show that the "free random walk of the particle (made up of these elementary moves without energetic constraints) is ergodic, so it visits all the sites of the lattice with the same frequency. Starting from an arbitrary configuration i (0 ), the method consists in selecting an elementary move, then in testing it according to a energetic criterion of admissibility. Namely, the motion of a particle j which leads from the configuration i(n ) to the configuration f/ (with X n) = yi if i A j) is: E )) < 0, then i' (4.1 ) = — accepted with a probability of 1 if AE(n) = — accepted with a probability of exp[ AE(n)] if AE (n) > 0 (then )= otherwise the move is rejected and the configuration at step n +1 will remain i.:(" 1 ) = —

(

—

n

. ( " 4-1

It is then possible to show that the distribution of the random configuration converges to the desired distribution ° p() = Cexp[ E()] as the number Ar of moves considered (equal to the number of steps of the simulation) tends to infinity. Simple and inexpensive, this "Monte Carlo method" has the additional virtue of inducing only statistical errors due to the finite number Ar of steps of the simulation: the distribution pexp of the simulated configuration is identified with p only in the limit Ar oo. For finite At. , the ergodic theorem and the strong law of large numbers give only approximative results and we can estimate the difference between p and pezp directly from the results of the simulation, using large deviations results° . Thus there exist internal tests of the quality of the sampling. —

equiprobable configurations f in X and weighting them by p(t) would give poor results, since the zones of X contributing significantly to A are not uniformly distributed: the arithmetic mean of this uniform sampling of the integrand converges very slowly to the statistical average A. 48 Because of the necessary discretization of X, we actually obtain a distribution p, defined on a lattice (v 1 IN Z) N and constant on each of its cells (of volume y); but here, the choice of y does not affect the number of steps of the simulation: to start with we fix it small enough for lp — pv 1 to be smaller than the desired precision. 49 See Ellis [1985] and Sokal [1989 ]. 47 Selecting

3.3 Nurnerice methods

83

Used jointly with the Monte Carlo method, renormalization principles help to avoid the difficulties arising from the finite size of the lattice while preserving the statistical qualities of the simulation 5° .

The large-cell renormalization method 51 We first explain the principle of large-cell renormalization in the case of the Ising model on a square lattice (i.e. indexed by Z2 ), depending only on the dimensionless coupling constant K; the reduced Hamiltonian of a configuration [(7] of N x M spins is given by: NM 7-t(N , M,K,[a])= K EEmi,j(cri+i,j cri,j+1)

= +1)

i=1 5,1 The idea is to replace the analytic renormalization (extending in two dimensions the one-dimensional scheme exposed in §2.1), which is difficult to exploit because it makes new coupling terms and thus new coefficients appear at each iteration, by an effective renorinalization based on simulations. Using the Monte Carlo method on an L x 2L lattice, we successively place m configurations of 2N2 spins on it, according to the Boltzmann-Gibbs distribution of probability associated to 7-1(2N, N, K). Separating this lattice into two L x L cells, we assimilate each of them to a unique macrospin of of value *1, determined for each of the rn configurations by the majority rule: cr' = +1 if the total magnetization of the cell is positive in the configuration we are considering. The advantage of this extreme reduction of the system to a single pair of -1Ccrli a has the same macrospins (cri, c'2 ) is that its effective Hamiltonian H' form as the initial Hamiltonian; the effective coupling constant K' satisfies tanh K` = < cr'i cr>, where the average is over the Boltzmann-Gibbs distribution associated to H'. We compute < cr'i cr; > as the arithmetic mean of the m values oli o; deduced from the m simulations where m is taken large enough for the law of large numbers to be applicable. Statistical relations using only the results of the simulations and the number m relate the confidence interval to the precision of this estimate, which is thus perfectly controlled by internal means. The dependence K i (K) can be made explicit by applying this procedure for various values of K; these can be limited to values near K. The step whose justification is the most delicate is showing that the relation aKtiaK Lilv (near the critical value K,), which is valid for analytic renorma,lization 52 of scaling factor L, remains valid here and makes it possible to numerically compute the critical exponent u of the statistical correlation length 4".

50 See for example Swendsen [1982], [1984] or Binder and Heermann [1988]; a collection of detailed examples can be found in Binder [1979], [1987]. 5 ' See Friedman and Felsteiner [1977] for some of the original work on this subject. 52 Using the relation K = J1kB 71 where J is constant, we can write T - Tc.rd IK -KJ'. The action of the renorrnalization e(K1 = 4v(1‘)//, gives K`-.K . ...., (K-K)1, 11 ' (equality of leading order terms); the relation PI-, valid in the neighborhood of Kc, follows from it.

alcvax

84

Mathematical aspects

Direct numerical ("small-cell") renorrnalization Let us give a sketch of this second procedure53 , remaining within the restricted framework of a system of No spins of reduced Hamiltonian 7-(0)(No, [ai m ); let 4,113,y) be the associated Boltzmann—Gibbs distribution. Whereas the analytic renormalization operator RI, acts on the structure rule H ( °) or on a family [KM] of real numbers parametrizing it, here we apply renormalization directly to the configurations. The Monte Carlo method gives m configurations [cr(°) ] sTo distributed according to the probability law 40) . On each of them, we perform the geometric renormalization, illustrated in figure 1.6, and represented by Rk. Each configuration [(YOU obtained after j iterations contains Ni =k —i dNo spins and by construction obeys the law p(i) associated to the Hamiltonian renormalized j times ?ILO= Rik. H (0 ). The advantage of this procedure is that we obtain m configurations whose distribution is controlled by the renormalized Hamiltonian n(r) without ever needing to explicitly work with the successive renormalizations of the Hamiltonian 7-(°), and thus without needing to approximate them; we implement here an exact renormalization. The only limitation on r comes from the decimation: we need N, = No k — rd > 2. The only error at this stage is the statistical error (which can be estimated) due to the finite number of steps of the Monte Carlo method used to make the selection of the initial configurations [cr(°)]sTo . The procedure continues by choosing an expression with J real parameters [K ] E IC) approximating 7-f ( r) , then determining the values [K ( r ) ] E ICJ leading to an approximate distribution P(') compatible with the observable characteristics of the distribution of the m configurations renormalized r times. The approximation linked to the projection onto a. subspace ICJ of finite dimension J of the parameter space IC,„, thus intervenes here only once, in the well-known statistical context of the estimation of the parameters of a law of a given form; in this case it is known how to estimate, or at least to bound the associated errors. If the action of the renormalization does lead to convergence to a fixed point [1.C*] in IC O3 , then the leading term of the matrix M (r ) of the partial derivatives NI (r) = a KS' ) 180) is constant in the neighborhood of (K1 and can be identified with the restriction to KJ of the r-th power of the renormalization operator linearized around the fixed point [Kw]. Its eigenvalues are of the form er7 , where the values taken by 7 are related to the critical exponents in a simple way. Thus we have developed a numerical method54 to compute an approximate expression for these exponents, in which the errors can be estimated internally, from the results of the simulations,

"See for example Swendsen [1982], [1984].

"The analytic foundation given by the study of the flow generated by the renormalization operator Rk in the space of structure rules remains essential since it ensures the existence of a fixed point, justifies the expression of the critical exponents and proves their universal character,

3• 4 Renormaiization groups

85

3.4 Renormalization groups To every renormalization operator TZ acting in a space 0 of structure rules, we can associate a semigroup g+ = n > 0} with composition law 0, called "the" renormalization semigroup55 . It can be extended to a group g {Rn.,n E Z} if and only if R. is invertible on (I) or on a domain V C 1. In itself, this group does not contain any more information on the renormalization than what is given by the operator 7Z. However, the study of its structure enables us to use notions from group theory and the analytic tools connected to them to describe the action of renormalization. In particular, via an isomorphism (typically a parametrization of the renormalization operators by one or more scalars), we will attempt to relate g to a group whose structure is known and whose properties thus also apply to Ç.

3.4.1

Group law, Lie groups and infinitesimal generators

One parameter groups (or semigroups) -

In the simplest case, the factor k > 1 of a scale change" occurring in a renormalization suffices to parametrize it. In general, this factor can be chosen arbitrarily in a set IC C J1, -Foof, where no value is particularly privileged for intrinsic physical reasons. Internal consistency requires that all the renormalization operators associated to the scaling factor k coincide with R,k , which leads to the following grouptheoretic structure: if k1 G IC and k2 G IC, then k1k2 E IC and nk, 0124,=14 2 0 '741 =14 1k2 It ensures that the choice of a particular value of k does not cause any bias. In particular, R I is the identity operator. (k, x) must thus be a commutatave semigroup57 , isomorphic to the semigroup (Ç +, 0) of renormalization operators. Having specified the isomorphism ki R,k parametrizing g+, we can replace the study of the asymptotic behavior n –+oo of the sequence [7V: 0 ]>1 of iterates of a given operator Rk o by that of the behavior k (in AC C R) of Rk . Since the semigroup IC 0 {1} is a subgroup of ([1, x), it can –

— be equal to ({kR, n — or be a dense subgroup in ([1, oo[, x); — or be the trivial subgroup ([1, cc[, x), in which case we study the regularity of the dependence of 74 on k. 'We often speak of "the" renormalization group; in fact there are as many groups as there are different renormalization operators, i.e. as there are domains of application of renormalization. These groups naturally have resemblances and they are studied in a similar way; still this does not really justify the use of the singular! 56 The nature of such a change of scale is not important: it suffices that there exists a physical quantity X(0) depending on the structure rule ¢ such that X(/24¢) = kX(0) and all the other scaling factors are of the form k*, where a single choice of the exponents a leads to a non-trivial asymptotic renormalization (k ■• 00). 57(g+, 0) is also isomorphic to („7 , -1-) where J = (log k, k E iC} (setting R., = The only possibilities are J = {O} , J = R+, J = ) 0 Z or J dense in R. —

86

Mathematical aspects

Lie groups Describing a renormalization method via the semigroup structure of g+ does not turn out to be particularly fruitful unless (g+, 0) is a Lie (semi)group, i.e. can be equipped with a differentiable structure compatible with the group law. Supposing from now on that g+ can be extended to a group g, we define its dimension d to be the (topological) dimension of its differentiable structure. The study of the renormalization is then reduced to the study of the d linear operators spanning the tangent space to g at Idt , called the Lie algebra A of g and having the same dimension d. Acting on the same space 4) as the elements of g, they are called the anfinatesama/ generators of the group because the group can be reconstructed via these generators, and they reveal all of its remarkable properties. To begin with, let us give a detailed description of a toy model'. Let a renormalization operator be defined on a function space (I) of analytic structure rules F(x, y) by:

(Ric F)(x , y)

E

G = {Ric, K E RI

eKb F (e -K x , e - y)

We endow 4) with the topology of simple convergence, and the vector space (containing G) of operators acting in 4) with the topology of weak convergence; then we check that K RK is an analytic isomorphism from (R, ±) to (Ç, o) which transfers the commutative, dimension one Lie group structure of (R, -I-) onto (Ç, 0). The Lie algebra A of g, also of dimension one, is written A RA where A is the infinitesimal generator of the group G: A

(d7ZA I dK)(K = 0)

VF E

acting on (I) according to the formula

(AF)(x,y) = bF(x,y) - aj (x , y) - ay a„ F (x , y)

The knowledge of A is sufficient to reconstruct the operators (RK)K>o:

VK E R,

ItK

= eK A

where

eK A

= d4. E Kn An ra>1

n!

More generally, every renormalization group G (RF)KER equipped with a differentiable structure such that the parametrization K RK is a differentiable isomorphism of (R, +) in (Ç, o) is a one-dimensional Lie group with infinitesimal generator A = (dRiddif)(If = 0). The relation RK = exp(KA) shows that A and Ric are simultaneously linear or non-linear.

Infinitesimal generators and hnear analysis We must make a careful distinction between an infinitesimal generator and a linearized operator. 58

A concrete example is given in §6.2, in the framework of stochastic diffusion,

3.4 Renorrnalization groups

87

— The linearization Di() at a fixed point 0" of 7Z is defined even if R is an isolated operator. It is linear and acts in the space T.4), tangent to the space 4) at 0*; it coincides with R. if R. is already a linear operator. — An infinitesimal generator can be defined only if g is a Lie group (so a fortsori a continuous group). It acts in the same space 4? as the elements of g and is linear only if the elements of g are. Let us continue the study of the Lie group G = {RI( , K G R} of Lie algebra A= AR. The relation Ric =exp(KA) shows that 0* E 40 is a fixed point of all the elements of G if and only if A(0*)= O. Differentiating the group law RK, 0 ER-2 = Ric 1 +.7,c 2 in 0* shows that the set of linearized operators .Co.. = {LK E DRK(0*), K E R} is a Lie group isomorphic to G. Differentiation with respect to K (in K = 0) and differentiation in 0* commute, so that the Lie algebra of 4. is A lp = Thus we have' LK = exp[K DA(0*)j. The element e,,f E To.l. is an eigenvector of DA(*) of eigenvalue -y if and only if it is an eigenvector of each operator LK with eigenvalue e7-K. The stable directions of LK do not depend on K > 0 and they coincide with the unstable directions of all the operators (LK)Ic<0 and with the directions having strictly negative eigenvalues (-y 1: determining the fixed points common to all the operators R E g means finding the elements 0* E whose images under each of the d operators (A 2 )i=1„.d spanning the Lie algebra A of Ç are simultaneously zero. The linear analysis of the elements of g near one of these fixed points 0* can be deduced from the spectral analysis of the linearized operator (Diti(0`)) 3= 1 .d. Thus, in order to perform a linear analysis of the action of the renormalization operator in 4), it suffices to know the Lie algebra A, i.e. a basis of A; it is not necessary to study the elements of G explicitly. It is only for the non-linear analysis and the explicit construction of the stable and unstable manifolds that one must look at the operators R E Ç .

0

DETAILS AND COMPLEMENTS: GROUPS

G

OF DIMENSION

d> 1

A renormalization depending on d independent scale changes is parametrized by the d associated scaling factors k = (k1,. • . , kd) E ICd; the set G+ = (Ri)j,- Ek, (equipped with the law o) is a semigroup isomorphic to (7Cd, xd) where k x d =.(k1 k1, . . . , k ded ). Every operator Rk can be expressed as a commutative product Ri=74i.„1 R-1,k2 ,1...1 O ... 0 of elementary operators, which corresponds to decoupling the operators associated to each of the d scale changes and making their independence explicit. If K = a0, +00[) d , we want to know if (g, 0) is a Lie group, in which case its Lie algebra is generated by the d infinitesimal generators )1 < j
=

= 1)

= expRlog

only makes sense to write this if the fibered structure of the action of DA(e) over A is made explicit. Since A(0') = 0, the iteration formula occurring for example in the exponential is DA(IP)I DA' 1 (0) o DA(). We cite Lang [1962 ] for more details on this notion of the exponential and on the eigenvalues of such an operator DA((r). 59 It

Mathematical aspects

88

(0. ) = for j = 1... d, 4)* is a fixed point of all the operators of Ç. The for j = 1. .d. If ooD d } is a Lie group of dimension d and the associated set Ere = DRE (* ), k E Lie algebra is spanned by the family [DA5(4)*)11<1
3.4.2

ri

• ej i<,
Transformation groups and associated symmetries

Group theory appears naturally in physical problems via transformation groups. Each element g of a group g of transformations is defined by the concrete modifications it induces on a physical system S. At the same time, it transforms each state function F(S) of S according to the rule TF,g [F(S)] = F(gS). The group law carries over as well: TF,g , oTF,g = Trigl ,g ,. If S is left invariant by a transformation g g then all the state functions F(S) are invariant under the corresponding transformation TF,g . The invariance of S under the action of all the elements of G is an analytic reformulation of the existence of symmetries in S. The invariance of S under a subgroup of g reflects the existence of more restricted symmetries. Note that if the symmetry of S is only statistical, it is not revealed by invariance of the state functions, but in the invariance of their probability distributions.

c

0

DETAILS AND COMPLEMENTS: THE MOST FREQUENT EXAMPLES

G=

— The group of spatial translations lot° E G), where (G, +) is a subgroup of (R d , +), acts 011 the spatial functions f () by [TY0f](x) = f[01-.0(x)] = i5 0) since the point ± of the translated system 9 0 S coincides with the point ± io of the initial system S. Invariance under the action of G reffects the homogeneity of S if G = R', in which case the local state of S is identical at every point, or just its spatial periodicity if G = dZ. — The group of temporal translations h o t t + to, t o E K1, where (K,+) is a subgroup of (R,+), acts on the temporal functions 9(0 by [Tto gi(t) = g(r_t o t) = g(t —t o ). Invariance under this group implies the stationarity of the state of S if K = R, in which case the instantaneous state of S is independent of time, or only its temporal periodicity (of period to) if K to Z. — The group 0(d) of rotations of Rd is associated to the isotropy of a system S inscribed in the space R d . It acts on the scalar fields by [T,-.A.]("±) = 14(r .1 [14) and on the vector fields by [Tr r[17(r -1 [±1)1 since a point of the transformed system rS cornes from the point r -1 1 of the initial system S. — The permutation groups are associated to properties of symmetry under permutations of the constituents of S.

homotheties are associated to the scale invariance of S: a scaling law X(k1.4) = k a X(1.4) will be revealed by an invariance Tk,k.X = X where [711c,k4X1(p) —

Groups of

3.4 Renormalization groups

89

An explicit resolution of the equilibrium or of the evolution of S is at best approximate and numerical; it depends in an essential way on the specific details of the model. On the contrary, the symmetry properties gS = S and Tg F = F involve only the way in which the quantities F describing S depend on its parameters, which relates their study to the description of the universal properties of S. A renormalization group can be compared to other symmetry groups encountered in physics: it is a group of transformations acting in a space 0 of structure rules; the invariance of E 0 under the action of its elements reflects the self-similarity properties of the system described by 0 and quantitatively reveals the associated scaling laws. The renormalization operators are constructed according to the symmetry properties they are supposed to reveal. This point of view unifies the presentation of the action of renormalization on the one hand, and the consequences of the symmetries of the system on the other. DETAILS AND COMPLEMENTS: INVARIANCE UNDER RENORMALIZATION

— The abstract renormalization (74 F)(s , y) m F (x k', yk — a) describes the scale y —4"y, F )1c 6 F. Invariance under renormalization expresses the transformation er scale invariance F(ks, ka y)= k b F (s, y). -

—

— In the one-dimensional Ising model ( 2.1), invariance under renormalization Rfl of the reduced Hamiltonian is equivalent to statistical invariance of the spin system with respect to the transformation K RCK tanh -1 [(tanhK) 21) of the reduced coupling constant completed by a contraction a a/2 of the lengths. The invariance is statistical since it does not involve the realizations of the configuration [8] of the spins, but rather its probability law P(71, [s]) = e —.H([i]) /Z(7-1). The renormalized system has apparent correlation length [R(K)] = e(K)I2; the invariance under renormalization is reflected on this observable statistical quantity by the equation for the fixed point = parallel to R(K*) = K* The solutions are the trivial fixed point (K = 0, = 0) and the critical point (K = 00, = 00).

the period-doubling scenario (§2.2) on the phase space [-1, 3 ], the invariance under renormalization Rf* = f* of an evolution law f* is equivalent to: —

In

for all s E [ LI] —

Rf*(s):€

f* 0 f*(.As)

where A = f (1)]

(s)

reflecting the similarity of the graph y = f*(x) of f to the graph of f* 0 f*, — A stationary Mark.ov diffusion in Rd is described by a process (X t ) f >0 specified by the probability of transition p(i , t) ( 6.1.3); we observe its mean-square displacement 0 2p(x, y,t)d dg, The renormalization operator Rk ,K transforms p and TNP,t)= fRdils consequently D according to 06.2.0 —

t)

k ap(kt, k9, Kt) =

j, t)

D(Rk,KPi t) = k -2 D(p, Kt)

The invariance under renormalization Rk,Kp = p, expressing the statistical invariance of t—;.tIK implies the scaling law the trajectories under the scale transformation D(p,t) = D(log t) t 2"" where 7 = log If/ log k and DO is log K periodic, -

Mathematical aspects

90 Marginal directions of renormalization and symmetries

Let us show that if the physical system S to be renormalized presents a continuous symmetry, reflected in the invariance of its structure rule under the action of a semigroup of transformations (L)„,>i, then a fixed point 0* of the operator of renormalization R may possess an eigenvalue of modulus 1. Renormalization affects only the perception we have of S, not its intrinsic properties; in particular, it should preserve its symmetries, so R must commute with all the operators (T4.> 1 so that they simultaneously preserve O and R. For all a > 1, Ta 0* is thus a fixe,] point of R since RT,4* = Ta RO* = Ta *, Differentiating this relation with respect to a gives = (0.11).=1¢)* : either (3ci n a -,..10 * is zero if O* is a fixed point of all the (7:0,2 >i (in which case the system described by Os presents the same symmetries as S, or it is an eigenvector 6° of DR(Os ) with eigenvalue 1, said to be marginal. A marginal eigenvector reflects the trivial action of R along the curve of fixed points (LO*).> 1 . This result can be generalized to a symmetry group having a Lie group structure: if .B belongs to its Lie algebra, then either BO* = 0 or it is a marginal eigenvector (of eigenvalue 1) of DR(O*).

3.4.3

Representations and symmetry groups

We can study the action and the properties of renormalization via those of the representations of the renormalization group G, in particular when considering a physical system possessing symmetry properties. A representation of a group (G, x) on a vector space E is a map [g E G Mg E GL(E)], with values in the group GL(E) of automorphisms of E into itself, which preserves the group law M91 oM9,= M91x 92 ; its image M(G) is thus a subgroup of GL(E). The dimension of the representation is defined to be the dimension of the vector space E. Two groups (G,xG) and (H, x fr) related by an isomorphism have the same representations: if [g F + Mg ] is a representation of G on E, then (0 is a representation of H on the same vector space E, having the [h Nh same image N(H)=M(G), and injective if and only if M is injective. From a physical point of view, the representations of a group can be interpreted as the observable manifestations of its action. A good example is the transformation TF, g [F(S)] of the (observable) state function F induced by the action of a group G on the systems S. The operators (71F,9 ) 9E G are linear and satisfy the group law TF, 91 oTF, 92 TF,g is a linear representation of (G, x) on the vector space E in which F takes so g its values. -

DETAILS AND COMPLEMENTS: REPRESENTATIONS OF A LIE GROUP

Let g = {g o , 9 E R} be a one-parameter group isomorphic to (R, ±). A correspondence g 0-4. Mg of g in GL(E) is a linear representation of this group on a vector space E of 1v19 M(g 9 )] is a linear representation of (R, +). If (g, 0) dimension d if and only if [0 is a Lie group and if M is a diffeomorphism, then M(G) = {Me, 9 E RI is a Lie subgroup 60 Similarly,

(0,T),=. 0 0* is an eigenvector of DR(T„,,e) of eigenvalue 1.

3.4 Renorrnalization groups

91

of dimension I of the Lie group GL(E) (of dimension d2 ). Its Lie algebra is RL where

L =

dMg dg

TIT (0)=

dM

(go).--(0) g (go )4 dg dg (I)) = —

go = e" and

where A = [dg6ld9j(0) is the infinitesimal generator of G. L is not necessarily invertible 61 . In a more general case, the image M(g) of a Lie group g, of Lie algebra .4 under a differentiable and injective representation M is a Lie group of the same dimension n, which forms a subgroup of the Lie group GL(E), so that we must have dimE > 071. If (A 1 , , A n ) spans the Lie algebra A of g, we construct Li = (dMg idg)(go).Ai: the family (L1,..., L n ) 1 .. rd defines spans the Lie algebra L C EndE of M. The correspondence [Ai —> Li, j an isomorphism between the algebras A and L, extending the notion of a representation of a Lie group G to its algebra A, so that the representations of g can be studied via those of A. For example, a subvector space F of E is invariant under each endomorphism (M9 )9€ g of an injective and differentiable representation M of g (isomorphic to R") on E if and only it is invariant under the finite number n of any set of operators spanning L.

Representations of renormalization groups Let G be a renormalization group acting on a set 0 of structure rules. The most TR which describes obvious example of a representation of g is the correspondence R how the renormalization transforms a state function f, i.e. TR[f(0)] = f(R0); here we take E to be any vector space containing the values {f(0), e 0} of f. Moreover, most often it is the action of TR and not that of R which is actually observable, since the structure rule 0 is not in general perceived otherwise than via the state functions, whose value AO) it determines. The representations of g have remarkable properties if there exists a subset O sym C I) of structure rules invariant under all the operators of a symmetry group T. The elements R of G, restricted to 0,y ,,, must commute with each element T E T since renormalization preserves symmetries. Let s be a representation of a group S of transformations containing T Ug on a finite-dimensional vector space E. Let t denote the restriction of s to T and r the restriction of s to Ç. Because of the group law ssi o = sso s, satisfied by the definition of s, the commutativity of every pair (11, R) implies that of the associated linear operators tT and rR: thus t and r are representations of 7- and g on the same vector space E, having the saine invariant sets and being diagonalizable in the same basis. Another consequence of the commutativity Rn 0* where of the elements of G with those of T is that if (iS 0, y, and if the fixed point 0* belongs to cil sym , the structure rules [T0]Te7 all belong to the same universality class (that of e) since they have the same asymptotic behavior under the action of renormalization. Conversely, systems with different symmetries must belong to different universality classes since the symmetries of their limits under the iterated action of renormalization are different, so the limits themselves cannot coincide. In this the Lie algebra of GL(E) is all of End(E) since if B is any endornorphism of then e B is invertible with inverse e B. sl Indeed,

—

E,

Mathematical aspects

92

way, the universality classes can actually be labeled by their symmetry group. This labeling is frequently used in field theory and in those problems of statistical mechanics which can be formulated in the framework of field theory; thus, Z2 = —1, +11 designates the class of the ising model. The orthogonal groups 0(n) and the quotient groups 0(n)/0(k) also occur; they are related to properties of symmetry of critical systems under rotation. DETAILS AND COMPLEMENTS: "LANDAU" THEORIES Let T be the group of transformations associated to the symmetries of a system S and acting on its parameter of order in. The group T has a representation O on the vector space of functions f of the variable m, defined by a relation VT E T [071](m) = LT[f( 71-1 rn)]

LT is a linear operator, such that the symmetry of f can be expressed simply by f(Tm) LT[f(rn)], i.e. OT f = f for every element T E T. To be a state function of the system, a function f of the order parameter m must be symmetric, i.e. belong to the intersection of the eigenspaces of eigenvalue 1 associated to the elements 01, of the representation as T runs through T. This constraint plays a crucial role in the construction of a phenomenological free energy 1(m) depending on the order parameter rra and with empirically adjustable coefficients. This construction is the preliminary step of the theories where

of phase transitions known as "Landau theories", with reference to the phenomenological analysis of ferromagnetism proposed by Landau 62 ( 4.1.3, Landau [1950]).

REMARKS AND BIBLIOGRAPHICAL NOTES ■•■■■••IPORMIPIR,

The advantage of a renormalization operation to study scale properties and obtain universal results is emphasized in Kad.anoff [1976 ]; we also mention the discussion of the mathematical foundation of renormalization methods by Gawedzki [1986]. The linear analysis of a renormalization operator, revealing the essential directions and the inessential parameters, is presented in Fisher [1974] and in Ma [1976]; the notions of "crossover" and its associated exponent in the case of renormalization used in statistical mechanics can also be found there. For a complete stiidy of linear operators and their properties, we refer to the books by Dunford and Schwartz [1958) and Naimark [1967]. The use of renormalization in spatial or temporal real space is commented in Hu. [1982], while Burkhaardt and Va,n Leeuwen [1982 ] survey the various applications in their introduction to a collection of articles on this theme. The probabilistic approach to critical phenomena has been considered essentially only in Jona-La.sinio [1975]. On the other hand, the role of the collective behavior manifested by elementary systems in the apparition of critical properties is frequently discussed; see for instance the collections of articles edited by Haken [1974], by Stanley [1975] and, more 62 A

complete presentation of this topic can be found in Goldenfeld [1992].

3.4 Renormalization groups

93

recently, by De Masi and Presutti [1989). Concerning finite-size scaling, one ca,n consult the collective books edited by Cardy [1988]; the articles collected in Privman [1990] deal more specifically with the numerical difficulties caused by finite-size effects and the use of finitesize scaling for exploiting simulations of critical phenomena. The advantage of exploiting the ideas of renormalization in these simulations is described in Ma [1976]. The advantages of a direct numerical realization of a renormalization operation appears in the work of Reynolds et a/. [1980 ] , using a "large-ce11" technique in a percolation problem. Concerning the Monte Carlo method, the "historical" article is that of Metropolis et al. [1953]; apart from a very accessible exposition of the basics of the method, it gives a rigorous presentation of its statistical foundations. A complete exposition of various algorithms and of the statistical analysis of the samples they provide is given in the course by Sakai [1989 ] . Applications to various problems of statistical mechanics can be found in the collections of articles edited by Binder [1979], [1987) and in the book by Binder and Heermann [1988 ] . Finally, let us cite the two articles by Swendsen [1983], [1984 ] , which recapitulate the relations between renormalization methods and simulation methods, in particular the Monte Carlo method. For details on the basics of group theory used in the context of renormalization, the reader can consult the reference book of Hamermesh [1962 ] or that of Tinkham [1964 ] for the general notions, Sattinger and Weaver [1986] or Georgi [1982 ] for Lie groups and Lie algebras, and Miller [1972 ] of Armstrong [1988 ] concerning symmetry groups. More marginally, let us cite Bredon [1972 ] on transformation groups, Helgason [1978] studying Lie groups in the context of differential geometry, and Racah [1961] for the role played by symmetry groups in physics.

4 Statistical mechanics Renormalization can be applied to the critical phenomena studied in statistical physics. The example of ferromagnetism (§ 4.1) illustrates the characteristics common to all critical transitions; these are the characteristics which cause the failure of mean-field theories and direct perturbative studies, making the tools of renormalization essential (§ 4.2). The basic principle, introduced in §2.1 for the Ising model, is formulated in real space (method of "spin blocks"). The computations are actually made explicitly in conjugate space, in a perturbative form ( 4.3). These tools also apply to the dynamic aspects of critical phenomena ( 4.4). Diagrammatic techniques are detailed in supplement 4A. In supplement 4B we introduce the magnetic critical transition associated to spin glasses. The necessary basics of statistical physics are recalled in appendix III.

The

4.1 4.1.1

example of ferromagnetism

Magnetic media and ferromagnetic transition

A substance is said to be magnetic if certain of its atoms 63 have an intrinsic magnetic moment called spin. To gain some intuitive understanding of the behavior of spins, we can identify them with small magnetic sticks with fixed positions, sensitive like compasses to their mutual influences and to the external magnetic fields. Magnetic substances are studied a great deal, for various reasons (Mattis [1988]): — their properties are observable: it is easy to vary the control parameters, namely the temperature T and the magnetic field H, and to measure their magnetization and specific heat; — they exhibit critical transitions; — in studying them, we encounter various types of elementary interactions, differing by their sign, their range Or their degree; — they motivate the theoretical study of spin systems since it suffices to describe the statistical distribution of the configurations of their spins to be able to describe their macroscopic magnetic properties. 63 These are atoms of metals such as Fe, Co, Ni; the spins are those of the electrons of an incomplete shell (f or d) of these atoms.

96

Statistical mechanics

Paramagnetic and ferromagnetic phases Following the example of the critical liquid—gas transition explained in chapter 1, we will present the critical transition encountered in ferromagnetic substances: it corresponds to the passage from the ferromagnetic state, characterized by a spontaneous non-zero magRetization Mo , to the paramagnetic state, where magnetization appears only in the presence of an external magnetic field H O. The control parameter is the temperature T; the transition temperature 7', is called the Curie pond". This transition is easily observed: in the absence of a magnetic field, an iron rod is attracted by a magnet at a low temperature; it is no longer attracted when it is heated until red-hot (T> Tc ) and abruptly becomes attracted again while cooling.

<> DETAILS AND COMPLEMENTS: A QUALITATIVE EXPLANATION OF THE TRANSITION Ferromagnetic substances are characterized by interactions tending to align the spins. We adopt a semi-classical description", limited to binary couplings between the spins. The Hamiltonian describes the joint effects of the (classical) Coulomb repulsive interaction V between the spins gi and gj and of the (quantum) Pauli exclusion principle, which does not allow the two spins to be in the same state (i.e. to have the same position and orientation). This principle forces the spins to be farther from each other when their orientations are parallel (11) than when they are antiparallel (n). Consequently, V(11) is smaller than 17(11) since V decreases as the distance between the spins increases, and this is reproduced by the term In the absence of an external fi eld and if T > T, the ferromagnetic interactions between the spins are not intense enough to dominate the thermal motion and align the spins: their orientations are spatially disordered and fluctuating; the total magnetization is thus zero and the system is in a paramagnetic phase. However, the system displays domains, called Weiss domains, in. which the spins are all aligned so that their spontaneous magnetization is non-zero: their size is limited to about No (T) spins where No (T) increases as T decreases to T. When T decreases towards Te , the correlations become strong enough for the different Weiss domains to join up, and finally they coalesce to form a single domain at the value T T c : the system becomes ferromagnetic. Below Tc , the physical couplings between the spins counterbalance the thermal motion sufficiently Co align the spins and the spontaneous magnetization tends to the non-zero value Mo(T), characteristic of the ferromagnetic phase as the number of spins tends to infinity (figure 4.1).

Critical aspects of the ferromagnetic transition The spontaneous magnetization Mo(T) shown in figure 4.1 is called the order parameter since it displays the transition and also the apparition of a long - range order, given here by the alignment of the spins in a macroscopic domain. The remarkable

"Named after Pierre Curie, the first person to discover it. in the quantum relations, this description is valid "Obtained by taking the limit h when its scale is much greater than quantum scales. It gives formally classical relations but takes into account, via correction terms, certain specifically quantum effects such as the Pauli exclusion principle.

4.1

The example of ferromagneti8271

97

point is that this transition is abrupt, whereas its intuitive explanation would suggest a gradual modification. It presents all the features of a critical transition: — the magnetization m per spin at vanishing (magnetic) field passes continuously from zero to non-zero values as T decreases and passes through the value Tc; — the susceptibility when the field vanishes, given by diverges at Tc ;

xo = (aMlaH)(T,H= 0),

— the specific heat Co (per spin) at vanishing (magnetic) field is discontinuous at T =Te and its derivative with respect to T diverges at T = Tc ; — the intensive quantities m, xo and Co satisfy scaling laws with respect to IT—T, at H = 0 and with respect to H at T =T,, the exponents measured experimentally appear to be universal, depending only on the dimension d of the substance and on the number n of effective components of the spins; — the observation of neutron scattering in the substance gives access to the spin-spin correlation functions since the scattering cross-section is proportional to their Fourier transform; it shows the divergence of their characteristic length (or their characteristic time if we start from a situation out of equilibrium); — the transition is associated to a spontaneous symmetry breaking: the substance is invariant under inversion of all the spins if T >71, (symmetry Mo —M0 ) since Mo = 0, but it is no longer invariant if T
Figure 4.1 - Spontaneous magnetization

Mo(T)

transition at 71, between the paramagnetic phase (P) and the ferromagnetic phase (F) is of second order because the order parameter Mn(T) varies continuously while crossing the critical point; the vertical tangent at T, shows on the other hand that the derivative (aM0/o9T) diverges at T. This behavior is quantitatively described by the scaling law: The

(71, — Tr (for T < Tc ) Mn(T) with 0 < /3 < 1 M(7 is the maximal value (or saturation value) of the magnetization, reached at T = 0 when all the spins are fixed in a perfect ferromagnetic order.

98 0

Statistical mechanics DETAILS AND COMPLEMENTS: CRITICAL EXPONENTS

In an infinite medium, the scaling laws of the transition are given by (we set

0=

(T —

e(9) e•J 4-0101—fr which

expresses the divergence of the range e of statistical correlations between the spins. 6 is the interatomic distance, and away from the critical 4-0 , which explains the appearance of Weiss point, we have e eo. If 0 < 1, then e' domains above the critical point (i.e. for 0 > 0), which increase in size as 0 approaches 0+; — the law

7 obeyed by the susceptibility x in the limit H 0; — the law x(B) — the law C CoI9 obeyed by the specific heat per spin. C(T). The exponents are identical for T < T, and T > 71, but the prefactors are not. — Below 71, we also have Mo(T) A191° where A is a constant. Dimensional arguments and the scale-invariance hypothesis lead to the relations a = 2 — id and oil- 2,3 = 2 (called "hyperscaling" relations). If N is bounded, the global quantities eD (T), Co (T), m(H,T) remain regular at T; we speak of "finite-size effects" which mask and weaken the criticality ( 3.2.4).

4.1.2 Langevin theory

and Curie's law

A paramagnetic substance acquires a total magnetization fi=xfi (statistical average) in the presence of an external magnetic field H, such that M .if > 0 (as is indicated by the prefix "para"). The susceptibihty x, which is a priori a tensor, reduces to

a positive scalar if the substance is isotropic. The simplest model for expressing x as a function of the characteristics of the substance is the Langevin theory, which assimilates the substance to a system of N spins (carried by the magnetic atoms) without interactions. It leads to Curie's law =C /T (where C is a constant depending only on the nature of the spins) which predicts a spontaneous magnetization only at T=O. The schematic explanation of this law is that the spins in thermal motion become fixed as T approaches 0, and aligned because of the presence of an external magnetic field. The magnetization takes its saturation value M*(1/)= N pH at T = 0, when all the spins are parallel; for T> 0, the thermal motion disorganizes the alignment and leads to a magnetization value smaller than ile(1/). Without applying a magnetic field, we obtain M = 0 for every value T> 0 in this theory where the spins do not interact with each other.

DETAILS AND COMPLEMENTS: CURIE'S LAW

Each of the N spins of the substance, of modulus S and magnetic moment 66 tiS, feels the influence of the local uniform field67 B = p o (ff + A71) where A71 = x ri. The partition

For electronic spins, =DAB where the Bohr rnagneton of the spins taB is related to the e < 0 of the electron and to its mass m by pB = eh/2m; the abnormai gyrornagnetic factor or Lande factor g is a quantum corrective term whose value is g = 2 for an electron, while the classical description gives g =1. 67 In the international system of units, po = 4 10-7 SI. 66

charge

4.1

99

The example of ferromagnetism

Z(N,T,171)= E[s] exp(—/..t T3 .3i /ki n at the temperature T, where the sum is over the configurations [s] = (j )i<j< jv of the N spins. The thermodynamic relation M = k E,T(8ZI8B) gives Brillouin's formula: function is

M(N ,

if) = ( N

28

[(28 + 1) coth(z(2S + 1)) — coth

where z

=

1.18 2k B T

For a weak field, we obtain Curie's law giving the susceptibility X per spin:

X =

po S(S + 1)p2 = C 7-, 3k R T

This law is satisfied if the concentration of magnetic atoms is weak enough so that the interactions between the spins can be ignored.

4.1.3

Mean - field theory and the Curie—Weiss law

Various theoretical approaches have been proposed to explain the phenomenon of ferromagnetism and to reproduce the transition from the paramagnetic phase. The approach known as "mean-field theory" is a phenomenological theory replacing the real interactions between the spins by the influence of a "mean" field Hint (M) depending only on the total mean magnetization M. In the presence of an applied field If ,z,t , the effective field felt by each of the spins is thus identified with licif =Ifert Ifint(M)•

0

DETAILS AND COMPLEMENTS: THE ADVANTAGE OF MEAN-FIELD THEORIES

It is difficult to describe the state [s]= (sj)j=i,..., N of the N spins

precisely because the problem is self-consistent: indeed, the state of a spin si depends on the states of all the others and influences them in its turn. The approximation obtained by neglecting the influence of the spin si on the N — 1 others is too crude: by repercussion, it is equivalent to neglecting all the interactions between spins and therefore reduces the situation to the Langevin model, i.e. precisely the one whose excessive simplicity we are trying to improve upon. The advantage of a mean-field approach is that it replaces the local magnetic field hi describing the real couplings felt by the spin si and depending explicitly on j and on the con fi guration [s], by a mean field Hint(M) depending only on the statistical average M. Each spin thus feels the same effective field Heif = ext+ Hin t(M); the statistical state of the population of spins is then computed as if Heff were an external fixed field.

Scaling law for magnetic susceptibility The simplest model takes 1-l(M)=aM where a > 0; plugging I I eff into Curie's law, we obtain the Curie-Weiss iaw68 :

A 1 (11„t,T) =

T) where

71, =

aC >0

"In 1907, Pierre Weiss proposed this phenomenological description extending improving the previous results of Pierre Curie.

and

100

Statistical mechanics

The susceptibility when the field vanishes xx.0(T) = [aM/01/,,t1(Hext only diverges at T = T, but more precisely obeys the scaling law:

f

= 0,T) not

XH.o(T) (T, - 7 1) -7 where

1.0(T) (T T c ri i

7 = 7' = 1

if

Scaling laws for magnetization and specific heat Taking Hir,t (V)= aM is valid only if M is weak and in fact insufficient at He = 0 and at T0 . To reproduce the saturation of the local field coming from the short range of the spin/spin interactions, we adopt the following model, justified by symmetry arguments (we write H H„ t ): Hff(H,

M) = H ± Hint (M) = H M(a - bM 2 )

where b> 0

Again, we plug Heft into Curie's law: giving M M(H , T) M (1 71/T +bCM 2 /T) CHIT We find M(0, T0 ) = 0 and the value of the susceptibility when the field vanishes -

given by the Curie-Weiss law, but we can describe the behavior of the magnetization at T = T, or at H = 0 in a better manner: the explicit resolution of the equation M= M(H , T) shows that M satisfies the scaling laws: if T

M(H =0,T) (T, -

< Tc

where 3 = 1/2

6=3 The part of the specific heat which is of non-magnetic origin is regular at T = T, so we M(H,T=Tc ) ,,, 11/1 1 / 8

where

can restrict ourselves to studying the contribution C, due to the magnetic component Um = I eff (M). M of the internal energy: Ç Um (I/ = 0, T > T,) = 0

U,(H = 0,T < T,) = aM(H = 0,T)2 = - a(T, - T)/bC SO Cm (1/ = 0,T) [8umi8T] ff ,0 is discontinuous at T = T, since C,,(0, Tc+) = 0 and Cm (0, T;) = a/bC. -

0

DETAILS AND COMPLEMENTS: EXPRESSION OF THE FREE ENERGY

We can also exploit the mean-field approach in the framework of classical thermodynamics, by adding the contribution —Hint(M).M to the free thermodynamic energy F11 = 0 in the absence of an external field. At a fixed temperature T < 710 , the curve M Fri, o (M,T) has two symmetric minima, which correspond to the stable macroscopic state of the system; the associated non-zero values ±1V10(T) give the spontaneous magnetization at the temperature T; the phase is thus ferromagnetic if T < T„. On the other hand, the only stable state above T, is the paramagnetic state in which the spontaneous magnetization is zero.

4.1

101

The example of ferromagnetism

Figure 4.2 - Free energy FH=D(M, T) as a function of the magnetization M for different temperatures

Advantages and limits of mean-field theories A positive aspect of this theory is that it effectively predicts a ferromagnetic 0, with experimentally observed properties: transition at T, — the spontaneous magnetization MHO(T) is non-zero below

Tc ;

— the susceptibility at vanishing (magnetic) fi eld xH =0(T) diverges at

71;

— the specific heat CH-0 (T) at vanishing (magnetic) field is discontinuous at

-

M(H,T =

Tc ;

has non-linear behavior with respect to H;

— the scaling laws and the critical exponents are universal, and they are independent of the substance whenever this substance presents a ferromagnetic phase. The negative aspect is that the values 7=71 =1, V1/2 and 6=3 of the critical exponents given by the theory, noticeably disagree with experimental results: thus the theory is qualitatively valid but quantitatively incorrect. Its major deficiency is that it does not take into account the inhomogeneities of the local field induced by those of the configuration of the spins. Thus, it ignores the consequences of the spatial statistical fluctuations of the microscopic state of the magnetic substance and the possible multiscale organization of these fluctuations. The presence of "giant" fluctuations (of macroscopic size), characteristic of critical phenomena, is reflected in the case of a ferromagnetic transition by the existence of Weiss domains (of macroscopic size) in which the spins are aligned. These domains play a major role in the properties of the transition: for example, the low probability that the thermal fluctuations induce a simultaneous flip of all the spins of a domain s = +1 (a necessary step for the coalescence of the domain with a neighboring domain s = —1) explains

102

Statistical mechanics

the divergence of the relaxation time towards statistical equilibrium when T is near T. This divergence, common to all critical phenomena, is known as critical slowing down' . Moreover, since it ignores the individual spin/spin couplings, mean-field theory can predict nothing concerning the correlation functions of the spins and their possible cooperative behavior. It is in the hope of improving these inadequacies that research has become oriented towards the use of renormalization.

4.2 4.2.1

Critical points Goals of a theory of critical phenomena

Strictly speaking, critical phenomena are observed on the macroscopic scale in extended systems of volume L 4 in R.4 , consisting of a large number N of elementary constituents (or "particles") of microscopic size (much smaller than L). Their global description is thus possible only in the framework of statistical mechanics. We will consider here only systems in thermal equilibrium at a fixed temperature T. The adequate formalism is thus that of the canonical ensemble (iIII.2). The role of the temperature T in critical behavior is intuitive since T controls the thermal motion of the particles, so also the spontaneous fluctuations of their configurations; it is quantitatively reflected in the dependence on T of the statistical distribution of the microscopic configurations (Boltzmann-Gibbs distribution). i,From now on, we will use the reduced (dimensionless) temperature 0 = (T — T,)IT, where 7', is the "critical temperature" at which the critical transition is observed in the thermodynamic limit [Li—, co , N L -4 = const]. The system can also be influenced by uniform constant applied fields, which we will call H, modifying their definition if necessary for the critical value to be H = O. To describe critical properties it is necessary, first and foremost, to determine the scaling laws satisfied by the various thermodynamic quantities X(0, H) of the system which are identified (in the thermodynamic limit) with the statistical quantities calculated in the canonical formalism. These scaling laws quantitatively express the singularity of the quantities X at the critical point (0 = 0, H = 0); away from this point the functions are, on the contrary, regular. Among these observable quantities, two are particularly significant: the order parameter M with n components, which displays the transition, and the correlation length 4", whose divergence indicates its critical character,

<>

DETAILS AND COMPLEMENTS: THE PHENOMENOLOGICAL APPROACH

This approach is based on a scaling assumption supported by experimental observations and which asserts that the transition is described in the thermodynamic limit by the following scaling law:

Li ng (T, H)

Fsi,,,g (T, H) = let 2—a Y (HO -61 )

where

0 = (T

—

Te )IT,

"See §4.4. It totally differs from the observed behavior when T is very far from if T < Tc , the order (the alignment of the spins) is quickly restored after a local and Tc., then on the contrary it is the disorder which is instantaneous perturbation. If T quickly restored after the momentary application of a field (inducing an alignment of the spins).

4.2

103

Critical points

fsing (T,H) is the singular part at (T = T H = 0) of the reduced free energy per particle. The essential point of this hypothesis is that the scale function Y depends only on a single variable z = HO—° , which is equivalent to its invariance under the scale transformation [H k H , 0 > k0]. We assume the function Y to be continuous and even differentiable at x = 0, so that the scaling law satisfied by Li ng at vanishing field is fsing (T, H 0) Y0 0 2—a . We observe that the function fs ing (T, H) is regular if 0 tends to O for fixed H 0, which means that: —

Y(s)

1.0 2— ' )/ °

(x

±oo)

so that

fs i„g (Tc , H)

1111(2 a)/A —

Starting from the classical thermodynamic relations:

S = — (49F/OT) H=, te and

C=T

we obtain the singular part of the specific heat of a field, whose derivative diverges at a

> —1:

whicmplesta

Co(T) E Csing(H = 0,T)

(a

> —1)

We similarly observe scaling laws for the order parameter at vanishing field: Mo(T)

M (° H = I3 )

AM P

(0 < < 1)

and for the linear response coefficient (or susceptibility) at vanishing field:

Xo(T)

(" MH)Te(H = O, T) Bier/

The quantities f (or F), Mo , Co, and xo are global quantities involving the structure of the system as a whole; the preceding relations define the critical exponents a, /3, 7 and A of the transition. In a more general case, these exponents can be different on the right and on the left of 0, zr.. 0; by symmetry, they are always identical on both sides of H = O. A usual order parameter is the thermodynamic quantity M:

M = — kaT Of sing' (910T,cte (9, H) This relation introduces conditions of consistency between the exponents: = 2— a—A

and

7

+ 2A — 2 = A —

These two relations express "hyperscaling" and their experimental verification is a first test of the scaling theory. Explicitly:

M(9, H) — 03 11 (110)

M(0, H) MolHi lf

with 6 =

13

can be experimentally verified that the exponents a, #, 7, 6 and A depend on the dimension d of the space, on the number n of components of the order parameter M and on the way in which the elementary constituents are organized, but not on the details of their couplings (of microscopic range) nor on the local structure of the substance. In this sense, the scaling laws above are universal: it suffices to know certain generic data (values of d and of n, nature of the cooperative effects) to obtain the quantitative information given by the values of the critical exponents. It

Statistical mechanics

104

Relevant description of the critical properties The presence of a critical phenomenon is revealed by particular properties of the macroscopic quantities, and is thus observed on the corresponding- scale. The problem is to understand how short range physical couplings can have consequences which are perceptible on much higher scales, and to determine the minimal conditions on the microscopic structure of the system for it to be critical. Thus, a theory of critical phenomena must fulfil the following objectives. — It must prove the validity of the thermodynamic scaling laws starting from the statistical description of the particles. Thus it will fill in a deficiency of scaling theories which postulate these laws; such theories are macroscopic and purely descriptive whereas a convincing theory must relate the microscopic model to the macroscopic observations. — It must express the critical exponents and the universal functions appearing in these scaling laws as functions of the parameters of the microscopic model and explicitly give the associated universality classes. — It must estimate the correcting terms to be introduced when thermodynamic limit is not reached ("finite-size effects" §3.2.4).

the

We present below a comparison of the ferromagnetic (§4.1) and liquid—gas (§1.2.1) transitions, which clarifies the similarity between their formalisms and their properties, critical or not according to whether the "external field" (or whatever quantity plays an analogous role) is or is not zero, and which shows the relevance of a unified presentation of phase transitions.

4.2,2

Ferromagnetic and liquid—gas transitions

Physical frame

Ferromagnetism

Liquid-gas transition

Control parameter

temperature T

temperature T

_ Exterior field

applied magnetic field H

external pressure P

Critical point

H = 0, T = T,

P = P,, T = T,,

Order parameter

total magnetization (statistical mean) M(T, H)

differences in massic volume Iv y — v L 1(T, P)

4-2

Critical points

First order transition -

105

at T = T*(11) if H 0 0 discontinuity of M(77, H)

Second order transition

at 7' = T, if I/ = M(T, H = 0) continuous at T,

Susceptibility

Xo(Tc)= — 1 7, (// I

-

Specific heat

am

=

0)

=

+cx)

C(T, H)=(aulaT)(T, H)

ac

(Tc±o)=±00 H =O =0

Free energy

dF = —SdT — H dM

Critical fluctuations

fluctuations of magnetization

Correlations

spin-spin correlation function

4.2.3

at T = T"' (P) if P P, discontinuity of v(T, P)

at T = T, if P = P, [vv— viST, PO cont. at T, , 1 av xcgc) , - - aPITc(Pc )= +oe

Cv(P,T):÷:(auiaT)(P,T)

ac,

(1,±0)=±00

aT I P=P.

dF = —SdT — PdV fluctuations of density

two-point correlation function

Limits of the methods preceding renormalization

Any study of critical phenomena has to tackle the following intrinsic difficulties.

Experimentally: The divergence of certain state functions makes observation difficult. Due to fluctuations of large amplitude and large spatial extent in the microscopic configurations of the system, measurement is not reliable for accessing to the value of statistical means (which coincide with the desired thermodynamic quantities). Also, the phenomenon, and thus its measurement are very sensitive to noise since the longrange statistical correlations cause the perturbations to have repercussions on every scale.

Numerically: The description of the critical properties is plagued by finite-size effects related to the impossibility of simulating thermodynamic behavior on a lattice of finite size, and of reproducing the divergence of the range of the statistical correlations and of certain state functions at the critical point (§3.2.4).

106

Statistical mechanics Analytically:

Critical phenomena clearly mark the limits of "classical" methods. — Critical properties follow from the organization of the elementary constituents, which can be revealed only by an overview of the system, in the whole space and on every scale: it is impossible to limit critical systems to a sample (either in real or in conjugate space).

— The divergence of the statistical correlations makes it impossible to reduce the system to disjoint, statistically independent constituents. Moving the study from real space-time {(27, t)} to conjugate space {(q, ci.))} reduces the problem to zero-covariance modes when the statistics are homogeneous and stationary, but shifts the difficulty to the presence of a singularity at the origin (q ---› 0, c,.) --). 0). — The organization of the microscopic phenomena makes them perceptible on every scale, so that it is impossible to take them into account via some effective uniform deterministic parameters. All methods based on separation of scales fail. — A striking failure is that of the mean-field theories, presented in §4.1.3 on the example of ferromagnetism and in §3.2.3 in a general probabilistic framework. Indeed, an effective field which is uniform and depends only on statistical quantities cannot reproduce the consequences of the organization of the elementary constituents or those of the inhomogeneities of their configurations. — The fact that the thermodynamic limits at the critical points are singular makes it impossible to exchange the analysis of a system of finite size L with taking the limit L --. oc. — The presence of "giant" fluctuations and of important correlations means that cumulative expansions are not very useful (§I.2): the irreducible moments measuring the departure of the distribution of the statistical fluctuations from Gaussian statistics cannot be treated as correcting terms. — The critical divergences invalidate the direct perturbative calculations of the partition function: typically, the expansion which is valid far from 7', converges more and more slowly as T approaches the critical point T=71,. Its terms, taken individually, diverge at T=T,; there are balances leading to a finite sum, but more and more terms are needed as T approaches Tc , so that an actual analytical computation is impossible.

0, DETAILS AND COMPLEMENTS: SCALING THEORIES The inadequacy of the first analytical approaches led to the development of phenomenological descriptions called scaling theories, in which certain scaling laws are postulated, based on experimental observations and dimensional arguments. From this it is possible to deduce new scaling laws and to make the universal relations (hyperscaling) between the observed exponents explicit ( 4.2.1). The main weakness of these theories is that they require external ingredients to choose the adequate scaling hypothesis, to justify it a.nd to determine the value of the exponents appearing in it. Another deficiency is that it gives no access to the universality classes; the universality of the results remains conjectural or at -lization analysis, performed in the space best only experimentally observed. Only a renormof the structure rules, makes it possible to construct the scaling laws, to prove their validity and to determine their universality class, i.e. the set of perturbations of the initial system

4.3

Renorrnalization techniques

107

which do not affect its critical behavior. This last result justifies a posteriori the choice of models which could appear excessively simple, even unrealistic. 0

4.3 Renormalization techniques 4.3.1

Methods in real space

A sketch: the method of spin blocks Before the development of renormalization methods, the idea of assimilating a block of spins to a unique effective spin was used to construct relevant phenomenological models. The essential step is the replacement of certain microscopic details without long-range consequences by their effective influence on higher scales. Indeed, to reproduce the macroscopic and universal properties of the system, in particular the possible critical properties, it suffices to specify the characteristics controlling the cooperative behavior of the spins. In the case of a microscopic system consisting of spins placed on a lattice of parameter a, we construct a more efficient model by increasing the minimal scale a —+ ka of the description. The elementary subsystems, now at a distance of ka from each other, are not isolated spins but blocks of led spins assimilated to a single "macrospin" . As long as ka is greater than the range of the real couplings between the spins, the adjoining blocks will interact, which justifies the hypothesis that couplings are reduced to nearest neighbors, which is often adopted in the study of spin systems on lattices.

The contribution of renorrnalization Sketched in figure 4.3, the method of spin blocks was extended by Kadanoff, who by iterating it was able to construct a first scaling theory of ferromagnetism (Kadanoff [1966], Gawedzki [1985]). It contains one of the essential points of renormalization analysis: the taking into account of the microscopic couplings via effective terms on a higher scale. The specificity of renormalization compared to these early approaches is that it completes this operation with a scale change which brings the system to a new system which is similar to the initial one and in particular has the same minimal scale; thus the operation can be iterated and the study can be shi ft ed from the transformed system to the manner in which it is transformed under renormalization.

0

DETAILS AND COMPLEMENTS: EXPLICIT COMPUTATIONS IN RENoRMALIZATION

Once the spins are grouped into blocks of kd spins, each block being assimilated to an "effective spin" which can take only two opposing values, the two steps in the renormalization procedure are as follows. A transformation K fk(K) of the (dimensionless) coupling constant describing the average interaction between the blocks. If the initial spin-spin couplings have range qa, they will have an apparent range sup(a, qk — noa) after no renormalizations and only neighboring blocks will interact; this result supports a model involving a single coupling constant K describing the interactions between nearest-neighbor spins, after performing no renormalizations if necessary. —

Statistical mechanics

108

k) and of the amplitude of the spins to conserve densities. Their invariance ensures that the renormalization modifies only the model used

— Scale changes

of the lengths (by a factor of

the for describing a fixed physical system. It is a mere extension of the procedure used in the study of the one-dimensional Ising model; the "blocks" are segments, each containing a pair of spins (d = 1, k = 2). The subsequent analysis is similar. The correlation length satisfies e(fk(K)) = so that the fixed points of fk are either trivial ((K)) = 0) or critical ((./C c ) = oo). A fixed point K, corresponds to a typical critical system, invariant under renormalization and thus self-similar, consequently satisfying scaling laws, for example:

4.(K) — 1K— '
where u = log k log ifl(K,)1

Renormalization weakens

the apparent critical nature of the system by decreasing the correlation length by a factor of k. Consequently, (K) > 1 and thus K, is an unstable fixed point70 of fk and Jr > O. The reduced coupling constant K is typically of the form J1k8 T, in which case the scaling laws in K are transformed into scaling laws with respect

to

71, via K — K, (T, — lk

Construction of renorrnalization in real space The method of spin blocks can be adapted to other systems in thermal equilibrium encountered in statistical mechanics. It gives the basic idea underlying the renormalization techniques used in (d-dimensional) real space to describe the thermodynamic properties of the systems. A Hamiltonian HN is defined on the space of configurations of the N-particle system; these configurations are denoted by [iiN. An operator RK acting on the reduced Hamiltonian 7-1 = Hlk i3 T is constructed according to the following principles.

The action of RK reflects a "summation by packets" of Kd particles; (RO-t)(N, .) is thus a functional of 7-1(KdN,.). —

— The n components of the elementary quantity § (relative to only one particle) are modified by factors of b1,...,bn whose choice is imposed by the symmetries of the problem.

— Ric preserves the partition function Z(N, RO ) = Z(KaN, 7-i) so that 91 and

RK91 describe the same system but perceived at different scales. — Each Kd-block is labeled as a single macro-particle, which is equivalent to performing a decimation preserving only N out of every K d N particles; it is then described by $(g=4) /bi where j= 1 . . n is the index of the component and 6, E Zd the label of the site on the lattice. A partial trace of the partition function over the configurations of the KdN N particles eliminated by the decimation yields an implicit expression for RKH: —

exp [(RK91)(N,[g]N)j=

E

exp (7-i(K d N, [11N]

[8 11,:ci N—N

one-dimensional Ising model is a limiting case: K,= +co is marginally stable since ifi:( 1(c)1 = 1; consequently I/ = co and (R") diverges faster than a power law $(K) e2K. 70 The

109

4.3 Renorrnalization techniques

(a)

• ——0 Ills — — *

•- — — 0 I • — — 0 • — — 0 • ——* • ——* • ——* M +

--

M.1

_1 r

, ,--ie ,__, ...

•F

+

•t-,___,

II

II

...., L —,

_

- I I

1--"--1. I .H. .1 -‘ •i--'•'. Iii •t--i•

I

ilk.__•+lk_ — • I

(b)

I

II

a

I

I

III. — — •

I

lk — — AO

(c)

lu

II

a

Figure 4.3 - Spin blocks for a square lattice (Kadanoff's method) The points represent spins located on the sites of a square lattice whose parameter a is the minimal scale of the problem. Each of these spins has an energy, depending on the external magnetic field and described by a "free" Hamiltonian hp ; it also has an interaction energy, related to the couplings existing between nearest-neighbor spins and depending on a coupling constant Jo. The principle lies in a summation by packets of k x k spins shown as square cells drawn in boldface in the diagram (here k = 2); the horizontal and vertical line segments represent the interactions between the different spin blocks, while the dashed segments represent the interactions internal to (a)

each block. (b) Assimilating each k X k block to a unique spin, we contract the lengths by the factor k to preserve the number density of sites; the apparent minimal scale thus remains equal to a (which corresponds to a real scale of ka). The interactions internal to a block are taken into account in the "free" Hamiltonian of the block which becomes h l ; similarly, the elementary interactions between spins belonging to different blocks are grouped into interactions between blocks, with a new coupling constant J1-

(c) The iteration of the procedure leads to "macrospins" of k 2 X k 2 real spins, of "free" Hamiltonian h2 and coupled by binary nearest-neighbor interactions, with coupling constant J.. Only the phenomena perceptible on the real scale k 2 a are still explicitly described; the phenomena on smaller scales are taken into account in the effective parameters J. and h2.

Statistical mechanics

110

— The Hamiltonian FIK9i is made as similar as possible to 7-1 by adequately choosing the factors b1 , — The lengths L are contracted by a factor of k> 1 to preserve the densities. If n = 1 or if b1 bn =b, the relation Ns`k d IL d = IONsILd gives the value k=Kbi ld. The operation can be summarized as follows.

s(i)

s(i)/bj

j = 1, ..., n

L —) k

N K —d N • --+ RKH

Once RK is constructed, the analysis is performed according to the standard scheme presented in (§3.1.3). It begins by seeking a hyperbolic fixed point 7-t* describing a typical critical system. Then the linear analysis of RK around 91 gives the stable and unstable eigenvectors, having the physical meaning of respectively inessential and critical dependencies; the associated eigenvalues are related to the critical exponents. It concludes with a non-linear analysis to construct the critical manifold of 71 4` and to detect possible multicritical behavior, which achieves the determination of the universality class of the associated critical phenomenon (53.1.4). A renormalization procedure is always conceivable. However, its applications are actually limited by insurmountable technical difficulties 71 For this reason, methods developed in conjugate space and lending themselves to perturbative methods are often preferable (54.3.3).

4.3.2

Fields and path integrals

The renormalization procedure sketched in the previous paragraph is more fruitfully implemented in the framework of field theory, the original domain of renormalization, in which many well-understood techniques are available. In the case of spins g with n components and norm equal to 1 placed on a hypercubic lattice of dimension d and parameter a, we first replace the discrete configurations [gajac z 4 of the lattice (with normalized elements g,l ) by a continuous field §(i) of variable x E Rd and with The summations ad E_ Ez, over the sites are replaced by space integrals values in fR, ciai; the Hamiltonian becomes a functional of the field g(i.):

a / = N(, T) = l where the integra,nd A(, ±) depends a priori not only on gM but also on the others values taken by the field g, most often via the derivatives of as the interactions between the spins are usually supposed to be "local", i.e. very short-range. This integrand A(g, ±) includes a (local) additive term w (. (X)) corresponding to a weight n Only the one-dimensional lsing model detailed in § 2.1 leads to a an exact, solvable algebraic equation for a unique coupling constant K as the renormalization flow stays in the associated one-parameter family of Hamiltonians,

4.3 Renorrnalization techniques

111

factor in the Boltzmann—Gibbs distribution 72 designed to be very small when the norm of the field g(l.) varies in the domain of values very different from 1 which are artificially introduced by the continuous limit. The summation over the different configurations [ga ] Gez, , involved for example in the computation of the partition function Z(11), is replaced by a path integral whose "variable" is the field §: ,

Z(7)

jexp[ 71(§,T)] -

instead of

Z(7 0 -

E

expf_H([ii, 11)]

[Yelet z, The expression chosen for the Hamiltonian is empirical in the sense that only a comparison of the theoretical predictions with the experimental observations can discriminate between the different models. We prefer an effective expression with adjustable parameters, whose form is simple enough to perform analytical computations, to an exact but very complicated expression. To avoid the delicate justifications of the discrete-to-continuous passage in real space, we can construct the model in conjugate space, specifying the reduced Hamiltonian 71 as a function of the conjugate field g&), of variable E Rd , which is continuous even if it is obtained by Fourier transform of a discrete configuration (§III.1). The existence of a minimal scale a is reflected in the support of the field î which is then reduced to q
71) =Rilo(T) q 2)IrsUll2 + Ai(TWOH 41c1d4 q
is a frequently used model; the conjugate field î (with n components) is normalized in such a way that the coefficient of 41211:0)H2 is equal to 1.

((> DETAILS AND COMPLEMENTS: CORRELATION FUNCTIONS Among the indications of the presence of critical behavior, one of the most reliable and the most easily observable is the divergence of the range of the statistical correlations, defined qualitativel y as the characteristic length of the correlation function r. For a homogeneous system of reduced Hasniltonian 7-i, we define:

r(71,) =

J

i(0)§(t)P(7i, .3')

where

P(7-1,$) , exp[—N(s)]/Z(n)

is then related to the asymptotic behavior r i) e--1Ix11/F H) (as tends to infinity). One of the reasons for working in conjugate space is that (?-t) is better defined there; under (

To preserve the average <s > = 0 and the variation < ligii 2 > = 1 of the real spin, w depends only on the norm s=ligii and satisfies fa„ 32 eXP[—W(3)](1 713= 1. Typical choices are w(s) = $ 2 /2 (Gaussian model) and w(s) = ai s2 a2 3 4 (o4 model). 72

112

Statistical mechanics

the hypotheses of isotropy and of homogeneity of the system, the Fourier transform of can be written f( q2), from which we define e by:

e = [f(q 2

F(1)

)]_ 1 [d1l(q2 )/ d(q2 )1(q =

4.3.3 Renormalization in conjugate space Let us suppose from now on that we have constructed a reduced H.amiltonian 7-1, which adequately reflects the essential properties of the interactions described with spatial resolution a. In conjugate space, 7-1 is a functional of a field OW, with values in R.' The modes q> A , 27r/a, excluded from the model since their wavelengths 27riq are smaller than the minimal scale a, are taken into account in the parameters of 11 or in a stochastic perturbation called "noise", which by construction only has components q 1, they give the modes of the renormalized system. The modes excluded by the truncation are taken into account in a modification of the parameters or of the noise, Renormalization then involves scale changes by factors chosen to ensure that the observable physical invariants are conserved and that the renormalized Hamiltonian is as similar as possible to the initial Hamiltonian. The first condition means that renormalization does not modify the physical system but only its description, which varies with the spatial resolution. The second is necessary for the Hamiltonian to converge to a fixed point 7i4 under the action of renormalization. It is exactly these constraints, imposed on the scaling factors by the convergence condition, which finally reveal the scaling laws and the critical exponents of the system, .

(>

DETAILS AND COMPLEMENTS: EXPLICIT COMPUTATIONS OF RENORMALIZATION

The integration over the large wave vectors will be done via, a partial trace, by

decomposing:

0(0 = 00() + 0(4) The functional integration on

since

q<

with

f ocP(4)

0 is done by exploiting

=

o

if if

q > Al k q< k

the fact that:

00

and so have disjoint supports. Since the support of p does not meet the domain Alk, we can exchange the functional integration f and the integration fq
transformation

0 01 of the field is given

by:

6(q) = f [00(0)]

q
4.3 Renorrnalization techniques f

113

is chosen so that the renormalized. Hamiltonian ni(01) has a form as similar as possible

to the initial Hamiltonian No(0), in the hope of revealing self-similarity properties. The renormalized Hamiltonian is implicitly defined by:

exp [hi (C )] =

exp [7-1 0 (0 0 (p)1 (09

This operation can (and must) be iterated. After n renormalizations, the wave vectors occurring explicitly correspond in the initial system to vectors q < Ak - n: renormalization 0 ( fi gure 7). The apparent acts like a magnifying glass placed over the singularity q

renormalization reduces the critical character. Technically, correlation length is 4"„ the transformation No NI explicit; it needs the use of the most delicate step is making diagrarnmatical analysis (also known as graphical analysis) detailed in supplement 4A, which shows how to obtain the analytic expression of the renormalized Hamiltonian Ni in the form of a perturbative expansion. Once the renormali2ation equations are explicitly known, the study ends with the typical procedure presented in § 3.1.

0

It is a fruitful exercise to explore the analogies obtained by reformulating, in the context of field theory, the determination of the statistical equilibrium state of an extended system and the adapted spatial renormalization. Every problem requiring the computation of a path integral 73 of the form:

43) = exp[—H(i3 , (b)] with 70, =

(i5 , i)dd

where the integrand A(0 , i) has a singularity at 8 = )3c and i = 0 can be analyzed by a renormalization method analogous to the one applied here in the framework of statistical mechanics, where

4.3.4

Z

is a partition function.

The main universality classes

The study of systems of spins clearly reveals the role played by the two dimensions parametrizing the models used: the dimension d of the underlying real space and the number n of components of the order parameter. Thus, both experimental observations and analytical arguments of renormalization show that the universality classes of critical phenomena at thermal equilibrium are specified by (d, n) and by more qualitative constraints defining sets of physical systems of the same nature (sign, form and range of the physical couplings, geometry and symmetries of the system, integrability of the local quantities, and so on.) The physically interesting situations and those for which we have exact results are displayed in figure 4.4.

'For example, an average with respect to a functional density mechanics (with respect to the trajectories 0).

4, or an action in Lagrangian

Statistical mechanics

114

<>

DETAILS AND COMPLEMENTS: THE ROLE OF THE PARAMETERS

d

AND

n

n can be interpreted as an index of symmetry of the order parameter; for spins, n is the number of components (Stanley [1968]). — For n=1, we have Ising spins of fixed direction (so determined by their orientation ±1). — For n =- 2, we speak of the XY model or XY spins. — For n = 3, we speak of Heisenberg spins. These cases n = 1,2,3 are encountered in magnetic substances. The case n = 1 also corresponds to the superfluid transition of helium, to the order/disorder transition in alloys and to the critical transition of simple liquids. — The case n = ao is remarkable since it corresponds to an exactly solvable model (for every dimension d) called the spherical model. — The statistical description of polymer conformations, where the polymer chain is considered as self-avoiding random walks, corresponds to the case n = 0 (§6A.4). — An exactly solvable formal case is the Gaussian model n = —2 (§4A.1). If d is the dimension of the real space, the eases d = 1,2,3 correspond to observable situations. However, integrals of the form f f(i)dq and f g(q)d' appear in the computations, where the isotropy of the system implies that the integrands f and g depend only on the moduli x and q: we replace (i =

or 0

dd i

by Sid

L

z d—i dz where 1.2d

= 2(7r) d12 [r(d12)] -1

r

The Euler function can be continued 74 to R in such a way that the computations can be formally continued to non-integral values of d. For d = 1, exact solutions are possible. An exact but partial solution exists for (d = 2, n = 2) (Onsager [1944]). We intuitively understand that d plays a. role: — in "steric" arguments, by fixing the number rnd of nearest neighbors of a site. The larger is md, the more valid is the procedure of replacing the real influence of these neighbors by that of an effective homogeneous field (3.2.3); the inadequacy of a mean-field theory for the Ising model, ruling it out completely in dimension d = 1, decreases as d increases, and its results are exact if d > d = 4; — in the scaling laws, since dilation by a factor of k >1. of the lengths is reflected in the diminution by a factor of k —d < 1 of the densities; — in the integrability of the spatial functions: if d decreases, the integrability at x = 0 increases and the integrability at s > oo decrea.ses75 . If OW depends only on the modulus x of t, we have L,(dx). E L(d) < > IOI P E Li(d d i) 10129sd-1 —

e

Perturbative expansions and renormalization In the computation of the partition function in conjugate space, n and d appear simply as numerical coefficients, which can take non-integral and even negative values. The continuous dependence on d and n obtained this way is usually differentiable or even analytic, which means it is possible to use perturbative approaches starting from values (d*, n*) for which the solution is known. The most frequently used expansions F(n 1)=n! if n is integral; if z >0, r(z)-a-f:3 t 2-1 e —f dt and f(z-1)=F(z)/(z — 1). 75 :C a is integrable at 0 if and only if a > —d, at infinity if and only if a < —d.

4.3 Renorrnalization techniques

115

are in powers of c = 4 — d (for fixed n) or in powers of 1/n (for fixed d). However, the direct perturbative calculations lead to expansions whose terms diverge at the critical point, because of the singularity q = 0 of the spectral components. To be conclusive, they absolutely must be used as complements to a renormalization method: it is not the partition function (specific to a Hamiltonian) but the renormalization equations (expressing the transformation of the Hamiltonian or more simply that of its parameters) which must be expanded around (de, 71, * ), to an order which is limited only by the complexity of the computation. The result is a truncated expansion, thus an approximation, but it explicitly gives the critical exponents of the phenomenon under consideration.

c=0 Spherical model

c= 1

c= 0

n = oo 1/n

=3_

Heisenberg spins XY spins Ising spins Polymers

n=1

• n = 0 Or .sager

n = —2

Gaussian model

d=1

d=2

d >4

d=3

Figure 4.4 - Summary of the various situations

(d, n)

The main models are represented as a. function of the dimension d of the underlying real space and of the number n of components of the control parameter; they are interesting either for their physical relevance or for their calculabi li ty. The bold lines underscore the cases for which an exact solution exists. The parameters with respect to which a perturbative expansion is conceivable are f = 4 d and 1/n. —

Perturbative expansions and renortnalization In the computation of the partition function in conjugate space, n and d appear simply as numerical coefficients, which can take non-integral and even negative values. The continuous dependence on d and n obtained this way is usually differentiable or

116

Statistical mechanics

even analytic, which means it is possible to use perturbative approaches starting from values (d*, re") for which the solution is known. The most frequently used expansions are in powers of c = 4 — d (for fixed n) or in powers of 1/n (for fixed d). However, the direct perturbative calculations lead to expansions whose terms diverge at the critical point, because of the singularity q = 0 of the spectral components. To be conclusive, they absolutely must be used as complements to a renormalization method: it is not the partition function (specific to a Hamiltonian) but the renormalization equations (expressing the transformation of the Hamiltonian or more simply that of its parameters) which must be expanded around (d', ri*), to an order which is limited only by the complexity of the computation. The result is a truncated expansion, thus an approximation, but it explicitly gives the critical exponents of the phenomenon under consideration.

4.4

Dynamic critical phenomena

In this paragraph we introduce the study of the dynamic properties of a (spatially extended) system in the neighborhood of a critical point, introduced previously as a state of stable thermodynamic equilibrium presenting a set of remarkable static properties. We will concentrate here on the non-equilibrium properties. Our goal is to prove the necessity of renormalization methods in the analysis of such a situation. The techniques are designed to analyze spatial and temporal dependencies jointly, and thus to obtain a global spatio—temporal description of the critical phenomenon. Achieving such a program is of extreme importance, because it gives an idea of how to approach the vast and difficult domain of statistical physics out of equilibrium.

4.4.1

Qualitative aspects

Spatio—temporal phenomena Spatio—temporal phenomena are naturally divided into four classes. The first two concern the behavior of the system, developing spontaneously and irreversibly from a situation of non-equilibrium. We may observe: — the relaxation towards thermal equilibrium at a temperature T (near the critical point TO imposed by a thermal reservoir; it is described by the relaxation tames of the order parameter and other macroscopic quantities; — the evolution of the fluctuations of the order parameter, described by the spatio—temporal correlation functions or by their Fourier transforms, called the dynamic structure factors; the accent is laid on their dependence with respect to the temperature T of thermal equilibrium. The two following classes cover all phenomena taking place in the system, initially in thermodynamic equilibrium and submitted to an external influence. Relevant studies focus on:

4.4

Dynamic critical phenomena

117

— the response of the system to a time-dependent external excitation, for example an oscillating electric or magnetic field; the analysis is performed in conjugate space via response functions describing the modification of each spectral component of the density of the order parameter as a function of those of the excitation. — the transport phenomena occurring in the system; its state will for example influence the dynamics of a test particle, or the conduction of heat and electricity. The relevant quantities are the transport coefficients occurring in the expression of the flow of the transported quantities; typical examples are the electric and thermal conductivities, viscosity, and the diffusion coefficients. We could also consider the propagation of waves in the system; to analyze them it is necessary to determine and the their dispersion relation, describing the relation between the wave vector propagating in the system. frequency 44) of a mode exp(—iwt —

Critical slowing down The proximity of a critical transition T = 2', induces anomalies in each of these four classes of observable properties: singularities appear in the temporal correlation functions, the transport coefficients and the response functions. The most typical is the divergence of the relaxation time 7-, towards equilibrium. It is explained by the fact that the statistical correlations between the elementary constituents have macroscopic range and induce cooperative behavior on every scale if the temperature of equilibrium is near to its critical value T. Thus a whole organized structure, present up through macroscopic scales, must be restored after perturbation; this requires very long times, since the range of the physical couplings, which is not very sensitive to temperature, remains microscopic at T =71 T. Because of the divergence of r relaxation is no longer described by an exponential decrease e - tfrc but by a power law t'. Experimentally well-observed, this dynamic property is characteristic of critical phenomena; it is called critical slowing down. Another "signature" of a critical dynamic situation is the divergence of the temporal range of statistical correlations, supplementing the divergence of their spatial range already observed in the static situation. In conjugate 0, w —> 0) of the dynamic space, this corresponds to the singularity at the origin (q structure factors. ,,

0

DETAILS AND COMPLEMENTS: QUALITATIVE EXPLANATIONS

The term "domino effect' is used to indicate the coherent organization of the microscopic couplings which explains the static and dynamic critical properties, as it gives a much greater range to the statistical correlations than the range of the direct interactions between the constituents. — The perturbations are amplified and reflected on every scale, just as we observe on a row of dominoes. This is reflected in particular large-scale properties of the spatio-ternporal correlation functions. The typical example is that of a one-dimensional lattice of spins s = ±1: turning over one spin forces the neighboring one to flip, and so on all down the line. — Because of its organized structure, the system presents greater "inertia", behaving like a "gelatinous mass" in its relaxation towards equilibrium. Even the slightest local evolution has repercussions throughout the system, just as it is impossible to shift only a single domino

118

Statistical mechanics

in a row. The stabilization after even a local perturbation or after a modification of the parameters is global and does not occur by means of independent local relaxations; this is reflected in the divergence of the relaxation time. The typical example is that of a spin system, which for T > Tre near Te presents vast zones (the Weiss domains) where the spirts are lined up; the spontaneous rearrangement of this very inhomogeneous structure, caused uniquely by the thermal agitations of the spins, towards a situation of thermal equilibrium at another temperature T' > T, near 71 takes much longer than in a non-critical system where the orientations of the spins are disordered and not highly correlated. — The slowness of the relaxation means that the system remains fixed with respect to external influences which vary too rapidly for the whole of the collective structure to follow. This is reflected in the anomalies of the response functions. An example is a spin system which is insensitive to the variations of an oscillating magnetic field at a temperature lower than its "freezing' temperature ( 413.3).

— The existence of a static organized structure modifies the propagation phenomena in the system, This is reflected in the transport coefficients and the diffusion laws. Typical examples are the diffusion of a test particle and the electrical or thermal conduction in a fractal structure (§ 7A.4). — As in the static case, dynamic critical phenomena are associated to a spontaneous namely that of temporal reversibility (t t); we typically observe phenomena of freezing or hysteresis. breaking of symmetry,

—

— As in the static case, the critical dynamic properties are not consequences of the physical nature of the individual couplings but of the statistical characteristics of their organization and its large-scale consequences; thus we expect these critical properties to be universal. In other words, what is relevant is the existence of a domino effect and not the form or the color of the dominoes.

Description of dynamic critical phenomena Concerning the thermodynamic and asymptotic behavior of extended systems, a theory of dynamic critical phenomena must: — describe the different dynamic universality classes; — determine the associated dynamic critical exponents, — establish their links with the static critical exponents, — make the universal functions occurring in the scaling laws explicit. The study concludes by determining the corrections needed to take into account the finite size of the system and the finite duration of the period of observation.

4.4.2 Renormalization methods

Spatio-ternporal formalism Dynamic critical phenomena are first studied analytically on a mesoscopic scale A; the dynamic criticality is perceived in the statistical properties of the density s( ,t) of the order parameter M. On the scale A, the microscopic mechanisms of the evolution

4.4

Dynamic critical phenomena

119

are perceived as random, and integrated into the "noise" terms, which by construction have only spectral components q < A 271-/A. Consequently, s(, t) is a stochastic process, and the statistical averages (denoted < >) are taken with respect to the global (spatio-temporal) probability distribution. — In order to visualize the organization of the different scales and their relative importance in the macroscopic dynamics, the first step is to decompose s(±, t) into modes 4(0, of spatial scale 27r/q, via Fourier transforms of the spatial variable E at the fixed time t On a large temporal scale and in the thermodynamic limit, we can neglect boundary effects and suppose that the statistics are homogeneous, isotropic and often even stationary 76 .

— Homogeneity implies the vanishing of the covariances of the modes S(t):

< Sq(t)Sr(e) > = (27r) d 6d (g. g') Ef(g, t t')

Rd

with

C(0,t)ddi and C(i.,t) = < s(0,t0)8(0 +

+ to) >

The statistical decorrelation expressed by this property favors the study of the evolution of the modes S(t) rather than the study of the density s ( , t) in real space. The correlation function C(i, t) (an n x n matrix if M and thus s has n components) does not depend on o by homogeneity of the statistics, nor on to if the statistics are stationary. — To describe the instantaneous properties of the system, it suffices to study the

spectral density E(g) of the field s (where Tr is the trace of the matrix Ô):

2E(q) =

(27r) -d

f Tr [a(qt7i,t = 0)] dd-1 5 11011=1

(q >0)

The isotropy of the statistics ensures that Ô(g, t = 0) depends only on the modulus q of g; in this case we can take the angular integral f dd- ifi leading to a factor of 2 (70 (1/ 2 [ [" (d/2)] -1 . — To describe the transport phenomena, we introduce the current density fl, t) of the order parameter M, whose flow across a surface E represents the quantity AM having crossed E per unit of time. The evolution of the modes ST (t) is described by mesoscopic phenomenological equations, which involve only the modes q < A 27r/A. They must preserve the symmetries of the system and are chosen as simple as possible, eliminating inessential terms; this reduction is moreover supported by the fact that we are seeking only universal properties. Since the critical phenomenon is not a consequence of the microscopic couplings but of their cooperative behavior, we expect that the non-linear couplings between modes on different scales, generating a structure relating the scales, should play an important role.

7 ' Except,

naturally, in the study of the properties of relaxation of the statistical distribution towards the distribution describing the thermal equilibrium of the system.

Statistical mechanics

120

Analytic formulation In this context, the principal features to be analyzed in order to perceive the existence of (and to describe) possible dynamic critical aspects are the following.

—) oc, t —) oc) behavior of the correlations — The large-scale t), or equivalently, the behavior (as t —>oc) of the temporal correlation C(4, t) of the 0) of the structure factors C(4, u.)) mode S(t), or again the behavior (as q 0, c<.) and the behavior as q --+ 0 of the spectral density E(q). — The typical relaxation time towards thermal equilibrium at a temperature T near Te , which coincides with the relaxation time r(T) of the slowest mode S(t) (and for this reason is called the critical mode), towards its value S,jc (eq) at equilibrium:

S(t)

Sfc(eq)

Ae-ti-rc(T)

— the response functions r(4, w); if we limit ourselves to the linear response, they determine the spectral components -.9-14.,c.s.)) of the system to which we apply a weak excitation f starting from its state of equilibrium:

rii(4,e,w)fice,coddit ± 0(111112)

0).

i=1 ft — The transport coefficients, relating the spectral components of the current density j to those describing the external influences or to those of the gradients imposed by the initial and boundary conditions.

DETAILS AND COMPLEMENTS: THE DIFFERENT APPROACHES

Four approaches, of increasing complexity but also increasing efficiency, were successively proposed to perform this analysis. • The conventional theory explaining critical slowing down is Van Hove's theory. It assumes that the different coefficients controlling the equations of evolution (for example the transport coefficients) remain finite when T = Tc ; the argument is based on the fact that they reflect mechanisms with very short range (smaller than A), and concludes by analogy with the temporal correlation functions C(4, t) on small spatial scales (q>.A. ,-, 270) which remain finite even at T = T. This approach ignores the "snowball" effect induced by the non-linear couplings between the modes, whose influences are reverberated from scale to scale, up through the scale of the statistical correlation length. This theory, in which the critical divergences cannot, by hypothesis, come from contributions related to the dynamics of the system, describes only the repercussions of the static critical properties on the dynamic ones. For example, the relaxation time Tc (T), estimated here as the ratio of the static susceptibility x(T) with a kinetic coefficient, diverges like x(T) at the critical point Tc ; the divergence of Tc (Tc ) is reproduced, but the exponent is inexact more often than not. • A more correct theory is the theory of mode coupling; it describes the non-linear interactions between the modes, which lie at the origin of the critical phenomenon. The resolution is based on separation of the components ST (t) into "slow" modes, which are explicitly described, a,nd "fast" modes (with microscopic characteristic time), which are taken -

4.4

Dynamic critical phenomena

121

into account, after a statistical average, in an "effective" noise term on a mesoscopic scale, or in a modification of the parameters of the problem. This reduction has the additional justification that critical properties are manifest only on large temporal scales. • The preceding analysis is supplemented by a phenomenological scaling theory, in which scaling laws are postulated from the experimental data. The exponents are computed using other relations satisfied by the quantities occurring in the scaling laws. • The most recent theories use renormalization methods. They unify the two preceding approaches, prove the hypothesis of scale invariance and describe the universality classes. Apart from their constructive and demonstrative power, they have the advantage of providing a global vision, revealing the physical reasons behind the the apparition of dynamic critical phenomena.

Outline of the main idea The basic idea, which actually preceded renormalization, is to eliminate the modes Mt) whose relaxation times are short (i.e. the "fast modes") from the description: indeed, since the fast modes describe temporal behavior which is imperceptible on the macroscopic scale, they play no direct role in the critical dynamics. However, the statistical correlations between the different scales in the neighborhood of the critical point make it impossible to neglect them completely. They must be taken into account in stochastic terms (or "noise") added to the equations of evolution of the other modes. This noise cannot be assumed to be decorrelated in time unless the relaxation times are short enough, which will not be the case as the critical point is approached. The new aspect of renormalization is that it includes this step in an iterated process which transfers the study to the transformation itself, enabling us to compare the respective behaviors of different systems and to obtain universal results. The procedure begins with a partial resolution of the equations of evolution of the different modes, expressing the "fast" modes (A/k < q< A) as functions of the "slow" modes (q < A/k) associated to the critical behavior and to the noise terms. This expression is plugged into the relaxation equations of the slow modes; the procedure continues with a statistical average over the components q > Aik of the noise; its contributions are integrated into a modification of the parameters of these equations. A scale change by a factor of k >1 to re-establish the extent q < A in conjugate space concludes this typical renormalization procedure 77 .

<> DETAILS AND COMPLEMENTS: RELAXATION TIME Confirming observations, renormalization analysis shows that the dependence on temperature of the relaxation time towards thermodynamic equilibrium r(T) at the temperature T of a system initially out of equilibrium can be expressed in the form of the (T — Te ) where scaling law re (T) > O. The divergence at Te of re (equal to the relaxation time of the slowest mode) which it describes quantitatively reflects the critical slowing down. The scaling properties observed then lead to an expression for the relaxation time rf of any mode 4- in the form:

T) = e(TY e[qe(T)]

with

M1)

IT—TI

"A more detailed description of this procedure in the case of fully developed hydrodynamic turbulence is given in supplement 5D (§5D.3).

122

Statistical mechanics

Ç(T) is the static correlation length. 0 is an analytic function which is strictly positive on R and independent of T. Taking q = 0 gives the condition of consistency re ,.. xi/. At T = Tc , we obtain the reduced scaling law To-,T,), q -z . The real number z > 0 is called the dynamise exponent Experimental results show that the dynamic universality classes are more numerous than the static classes. We obtain z = 2 — n in the case of the Gaussian model (§4A.1). For the other models, direct computation is impossible, and it is better to use renormalization methods to compute z. Moreover, these methods make it possible to justify the conjectured scaling law and to determine its universality classes.

o

REMARKS AND BIBLIOGRAPHICAL NOTES

It is impossible to give a complete survey of all the articles on critical phenomena and renormalization methods in statistical mechanics. We limit ourselves to some basic references, and refer the reader to the bibliographies in those references for more specific orientations. First of all, we cite the books by Itzykson and Zuber [1985], Zinn-Justin [1990] and Rivasseau [1991] on quantum fi eld theory; the methods of quantum field theory under li e the application of renormalization to statistical physics. Phase transitions and critical phenomena are studied in the reference work of Stanley [1971] and more recently in Aizenman [1986]; let us also mention the conference proceedings edited by Domb, Green and Lebowitz [1972-88] and by Busiello et al. [1987], as well as the collective book edited by Stanley [1975]. The statistical tools used in field theory are explained in Itzykson and Drouffe [1989]; on the subject of path integration, it is useful to consult Schulman [1981] or Kleinert [1990]. Specifically concerning renormalization methods, the articles of Gell-Mann and Low [1954], Bogoliubov and Shirkov [1959], Kadanoff [1966], and Wilson [1971a,b], [1975a,b] are of historical importance; other examples of problems solved by renormalization can be found in Brezin et al. [1974], 11976], Wallace and Zia [1974], Kadanoff [1975] or Le Guillou and Zinn-Justin [1977]. Let us also cite the conference proceedings edited by Gunton and Green [1973], the survey by Amit [1978] and the study by Collet and Eckmann [1978] on hierarchical models. More pedagogical books are the introduction by Ma [1973], the survey articles of Fisher [1974] or of Wilson and Kogut [1974] and more recently, the introduction by Hu [1982] and the courses by Goldenfeld [1992] or Plischke and Bergersen [1994]. The principles used in the application of renormalization are discussed in in Gawedzki [1985], [1986] and in Kadanoff [1976], [1990 ]. Ma [19761 and Halperin [1976] are introductions to dynamic critical phenomena, containing a detailed presentation of the various possible approaches, from the Van Hove theory to renormalization methods. An historical markland is the collection of papers edited by Budnick and Kawatra [1972 ]. A typical example is recounted very accessibly in Ma and Mazenko [1974], [1975]. The fundamental reference article is still the survey by Hohenberg and Halperin [1977], which presents a complete (and therefore rather technical!) study of the different dynamic universality classes via renorrnalization. A very different approach, in real space, is proposed by Mazenko and Valls [1982]. A more recent review about the dynamic renormalization group can be found in Barabasi and Stanley [1995].

Supplement 4A Graph analysis Graph (or "diagrammatic") analysis is used in the perturbative computations of statistical mechanics and field theory. It completes the renormalization explained in § 4.3.3, by making possible the explicit computation of the path integrals realizing the partial trace in conjugate space. The role of the graphs is only figurative, since they correspond to analytic expressions in a one-toone way. However, using them makes it easier to enumerate the terms, to estimate their order and to compute the logarithm of their sum, which gives a perturbative expansion of the renormalized Hamiltonian.

4A. 1 The Gaussian model and perturbative expansions The diagrammatic analysis presented in this supplement is the one used in conjugate space78 to compute an explicit expression for the renormalization 7-6 —> Ni of a Hamiltonian = 0 ph o , obtained by adding a perturbation of amplitude

no

proportional to p to a quadratic Hamiltonian

=

(a 1 (.)

6, 1(00(0(X - 0 (270d

given)

Every Hamiltonian obtained as a spatial integral of local and quadratic terms with respect to the real field 0(4 such as OW' or 11N(I:)112 can be expressed in the form 4 9 °) . The order 0 is thus the Gaussian model where the wave vectors, bounded above to reproduce the existence of a minimal spatial scale, are rationalized so as to vary in fq < r}. The term pho, with respect to which we take the perturbative expansion, contains only even terms in 0, by symmetry'' . Thus it can be expanded in powers of (1) in the form

E f ...1

1 00

1/ 0(0) = (—

2j

r aj (ii ) 42 • • . fi,j)

-1r

2i dcl oil (Ho(q,)) [11_ (27r)d

If 0 has n components, then ai is a totally symmetric tensor of type (n2i, I). The term of order 25 corresponds to a local term in real space only if ai 42 . • • 42.0=

oh, -

( 2 7r)do(4-1+ 4-2 + • • ' + q21) ai Oil i 42 . — Çr2j —1). Although we are dealing here with analysis in conjugate space, we omit the symbol for simplicity. The subscripts 0 and 1 refer respectively to the initial system and to the renormalized system; the superscript (0) denotes the 0-th order of the perturbative procedure. The Harailtonians 7i are dimensionless quantities. 78

—

In a system of spins, this symmetry corresponds to the invariance of the Hamiltonian under simultaneous inversion of the orientations of all the spins.

Graph analysis

124

Renormalization equation for the Hamiltonian As detailed in § 4.3.3, the renormalized Hamiltonian is implicitly determined by taking a partial trace of exp[--No(0)] over the components v, associated to the large wave vectors q>7rIk:

00( ) +90(0

with

0 { (15°C4) 40(0 = 0

if if

7r/k < q < 7r q < w/k

The scaling factor k >1 is chosen. The vanishing of the product so(60 and the quadratic (o form of 7-1,(3°) imply that 7/0(0) = 17(0)() + nÛ)(0). The perturbative procedure starts by rewriting the renormalization equation exp[-9-1 1 (0 1 )] = fcp exp[-9 1 0(0)] in the form: e -Ni(01) = e - 7-0(00) e -Nnco) e -mh,(4,0+(o) co where the renormalized field 0 1 , independent of the inessential components is related to O D by a scaling transformation: q < 7r

01(0 = f [00 WO]

(k > 1 fixed)

f is chosen to preserve the physical invariants and the normalizations. After using the integral expression of ho to take the series expansion of exp(-ph o ), the integrations which is thus performed over the wave vectors are exchanged with the path integral ço ' first. Let j(r) denote an r-tuple (j1, j2 • jr ) of integers greater than or equal to 2. We can make the contributions of the different terms (-pho)"/(r!) explicit as follows: e -Ni(01)

=

(0) (150)

00

EE

A r [00) j(r)]

5(r)

:72 - .ir)

r=0 {Ar )}

where the expression for the different terms (A,.), >0 is given by:

Ao

(0o, i(r)) =

e-71(0°)( w ) =

exp

1 2 fq>c,

(27r)d

f 7n=i

r 2j,

x J0 e-'(°)(P)

r1[00 (r) +40(r)] m=1

[ r

II m=1 1=1

d d-rn qi

d (27r) =

Diagrammatic representation In this complicated formula, it is still necessary to expand the products j(00 (4/ ) 90(0)), then take the path integral j. In order to enumerate and estimate the different terms, we need to introduce a symbolic representation for them. We represent each term by a "graph" (or a "diagram"), also called a graph, containing

4A.2

Analytic expression of the graphs

125

in its graphic characteristics the information necessary to reconstruct its analytical expression, contributing M times to exp [-9-6(0 1 )] where in is the multiplicity of the graph. The graph is said to be disconnected if it consists of several disjoint graphs, in which case it represents the product of the terms associated to each one.

4A.2

Analytic expression of the graphs

As shown in figure 4A.1, the graphs consist of vertices (points where a certain number (here necessarily even) of branches meet) of links joining these vertices and of branches having one free end. — The order 0 graph, i.e. without any vertices, has value 1. Each free branch arriving at some vertex carries an entering wave vector [0 1 (4)]. (q <11c) and is associated to the component 00( ) = —

, meet, is — A vertex of type j, where 2j branches carrying the vectors associated to the factor p(27r) d{5 a (qi + + - • - 42i)ai(41, 42 - • .42j-i) proportional to the amplitude p of the perturbation; a graph with r vertices thus has order r in the perturbative expansion. — A link joining two vertices80 , carrying the (entering) vectors j and 2, is associated to the propagator <1 42 L (270d6cl (1 fa1(41)r i • — The integration over a wave vector carried by a link joining two vertices has when the integration is domain air < q < ii, but the domain is restricted to q < over a wave vector carried by a free branch.

— exp [-7-i 1 (0 1 )] is the sum of all the graphs, connected or not, counted with their multiplicities (computed without orienting the links and considering the vertices as though they are indiscernible), then multiplied by the factor A o exp [-743) (00)1.

Expression of the renormalized Hamiltonian A major advantage of this diagrammatic approach is the very simple relation between the perturbative expansion of exp [-Hi (01 )1 and that of 7-11(01). A direct combinatorial study shows that - n1(01) is equal to the sum of all the connected graphs, counted with their multiplicities, added to log A0-7-4(°) (00). The renormalized Hamiltonian N1(01) is thus obtained with no computations other than a simple selection among all the terms contributing to exp [-91 i (01)1. The graphs with no free branches and the term log A0 give constant contributions to the renormalized Hamiltonian 711(0 1 ), identical for all the fields 01, i.e. for all the configurations of the renormalized system: these terms do not appear in the normalized functional 0 1 ). Thus they do not have any observable physical probability density' consequence and can be ignored. 'Possibly identical, as in the disconnected graphs in figure 4A.1 81 A simple functional extension of the Boltzmann-Gibbs distribution.

126

Graph analysis

Disconnected graphs:

Connected graphs:

x

7 (1) m1 =8 >

( 3 ) rn3 =

(5)

36

ms = 72

(7) m7 = _.k

7i< m2 = 48 (2)

9>IC.< *

Xm7k

m4 = 72

(4)

48

(6)

IZ:

36 rZ,

=

•

free branch

m

vertex

link

s = 12 (8)

=

propagator

Figure 4A.1 - Diagrammatic analysis Shown here are the order 2 graphs, Le. those with 2 vertices, of the c4 model obtained by quartic perturbation of the quadratic Hamiltonian of the Gaussian model: the vertices have four branches. The number mi indicates the multiplicity of the connected graph (j). The expressions of the graphs (5) and (6) are given below as an example.

DETAILS AND COMPLEMENTS: ANALYTIC EXPRESSION OF THE GRAPHS IN FIGURE

4A.1 The graphs of type 5 give a, total contribution of:

72p2

(2703d

de% fqi>TrIk

f

fq 2>rfk,

3 d 3 0 ( 3 ) 0 (—)

0
X

— 2)a 2 ( 3 ,., —0) a2(i1, am1)zi1(42)a1(-1-12)

while the graphs of type 6 give:

48p 2 fqi >roe

(270 3d

Jq>w fk

dd421<13
ddis 00(43)00(-43)

q2, li3)(12&1,

(Mai (4-2)di

2,

4i + q2 + 42 + 43)

411.3

Approximations and results

127

4A..3 Approximations and results This perturbative method gives the expression of the renormalized Hamiltonian h i up to an order 0(tir) which, although finite, can be as high as the complexity of the computations permits. 'H i has the same form as 9-t o since the sum of the graphs with 2j free branches gives a renormalized coeffi cient ajl) ; this coefficient is a function appearing in 9-(0 . Thus the of 2j — 1 wave vectors and replaces the coefficient zgi renormalization can be expressed as a functional operator:

[ a? ) ]

> o = 1?"

: (Rd)2I - 1

>o

C)

The analysis of the action of R. runs into a major difficulty: except in exceptional cases appear at every order j, even if only certain of the aj(,°) are ( 2.1), coefficients [an

2 >o non-zero. Because of this, renormalization only works if the contribution of a vertex with 4 branches is bounded by a small parameter' c and if the contribution of the other types of vertex decreases as the number of their branches grows. The order of the coe ffi cient kii(1) ] then increases with j, so that we can truncate the expansion of the j>0

Hamiltonian. Various approximations lead to closed algebraic equations for the leading order term, making it possible to compute the fixed points of R. Linear analysis around the critical fixed points then gives a perturbative expansion of the critical exponent (g 3.1).

DETAILS AND COMPLEMENTS: SOME POSSIBLE APPROXIMATIONS

0) are relevant in As only the spectral components of small wave vector (q in the appearing integral expression of critical behavior, we can replace certains terms the Hamiltonian by approximations in the neighborhood of = O. The validity of this approximation is supported by the fact that after n renormalizations, only the renormalized spectral components (for k >1 fixed) appear explicitly:

0.(4) = r [00 (o - 11

with

q<

7r

associated to initial components restricted to { q < kir}. To replace the integral functional equations appearing in the expression of the renormalized Hamiltonian by algebraicequations, one possibility, known as the approximate recursion formula, is to ignore the dependence on the moments of the functions [a.(i nj>2, which are assimilated to constants [a.r ) (q =0)1i > 2; thus we find ourselves in the situation where a 0) (q 2 ) = q 2 c/ o . The function f occurring in the renormalization of the field is constructed to give the normalization aa (1 1) /a(q2 ) = 1.

e = d 4, where d is the dimension of the dimension above which results given by mean-field underlying space and d, = 4 is the critical calculations are valid. 82 A case which occurs frequently is when

—

128

Graph analysis Another approximation of the analytical interpretation of the graphs is obtained by: — setting all the wave vectors carried by free branches to zero; — replacing the internal wave vectors by an average value go ; — replacing the integral f ...fq> ,(27r) —d ddii by a constant value C; — keeping only the graphs whose number of internal links at each vertex is even.

The positive aspect of these approximations is that they allow to compute (approximative) contributions from graphs of arbitrarily large order.

This diagrammatic technique is encountered in many other domains, for example in the study of turbulence when a stochastic stirring force is added to the NavierStokes equations (§ 5D.4). It is only really useful as a complement to a perturbative approach: the order of the terms is then determined graphically, and the perturbative character makes it possible to keep only a finite number of graphs. This gives interesting analogies between problems of very different physical origins: whenever the diagrammatic expansions are identical, the results can be transposed from one situation to another. The value and the interpretation of the coefficients carried by the vertex and of the propagators vary but the hierarchical structures of the expansions and the way in which the different contributions cooperate or compensate each other are identical and thus lead to the same exponents. An example is the analogy between the Landau—Ginzburg model for n=0 and polymers considered as self-avoiding random walks (§ 6A.4).

REMARKS AND BIBLIOGRAPHICAL NOTES Complete computations in the framework of diagrammatic analysis, as well as an exhaustive study of all the models and their physical interpretation, are presented for example in the article by Wilson and Kogut [1974]. Consult also the general books by Toulouse and Pfenty [1974], Ma [1976], Collins [1984] or Brown [1993]. On the subject of path integrals, some useful references in the context of diagrammatic analysis are Feynman and Hibbs [1965], Schulman [1981 ] , Kleinert [1990] or the recent monography by Veltman [1994 ] ; see also Itzykson and Zuber [1985] or Ryder [1985], which is devoted to the statistical tools used in field theory.

Supplement 4B Spin glasses "Spin glasses" are magnetic substances presenting a phase in which the orientations of the spins are (nearly) fixed in time but spatially disordered; this results from a so-called "mechanism of frustration', in which the position of the spins and the nature of their interactions prevent the appearance of a stable ferromagnetic or antiferromagnetic order. The renormalization techniques already considered in chapter 4 can be applied to the study of the transition from the disordered and fluctuating paramagnetic phase to this "quenched" phase observed at low temperatures. Spin glasses are a good introduction to the physics of disordered media, of which they are a typical example.

4B.1

Paramagnetism, ferromagnetism and spin glasses

A glass is an amorphous solid in which the arrangement of the molecules presents a local order on the microscopic scale. It differs from a crystal, whose ordered aspect is perceptible even on the macroscopic scale; it also differs from a liquid, since its molecules have "nearly" fixed positions (at least over short but nevertheless macroscopic time lengths): its structure is disordered, but fixed on the temporal scale of the observation. By analogy, a spin glass is a magnetic system with no long range order. The spins have fixed orientations over sufficiently short durations 83 but they are spatially random, unlike the situation observed in ferromagnetic or antiferromagnetic substances; the typical cause is the disordered distribution of the position of the spins, which is transmitted to their couplings and thus to their orientations. This is the case of alloys made of magnetic atoms of iron or of manganese dispersed in copper or in silver where the ferromagnetic or antiferromagnetic nature of the effective interaction between the spins depends on their mutual distance. -

The role of the temperature in a magnetic medium Thermal agitation of atoms modifies their positions and thus also their mutual distances, hence the interaction between the spins they carry. Without acting directly on the spins, it nevertheless influences their orientation. At high temperatures, thermal agitation is the dominating mechanism. The (statistically independent) temporal fluctuations of the orientations of the spins orientations are in fact slowly moving, which is associated to an aging, that is, to a relaxational evolution of the spin glass over macroscopic time lengths. 83 These

130

Spin glasses

induced by this mechanism rapidly disorganize the couplings of the spins: the total magnetization as well as the (local) magnetization at a site vanish on the temporal scale of the observation. The independence of these individual fluctuations ensures that the system visits all its microscopic configurations over time, with a probability related to the energy E of the configuration". If a constant external uniform field is applied, its influence dominates the thermal agitation and lines up the spins, so that a magnetization parallel to its direction appears. This paramagnetic situation, analogous to the liquid phase, is observed in every magnetic substance when the temperature T is greater than some threshold 710 . At low temperatures, thermal agitation does not involve enough energy to qualitatively modify the couplings. These no longer vary with time, so that the spinglass phase can appear Of the structure of the magnetic substance allows such a phase). Since the orientations of the spins remain random, without spatial organization, the spontaneous global magnetization is zero. However, these orientations are fixed, and the (local) magnetization at a site has a non-zero temporal average. DETAILS AND COMPLEMENTS: MODELING THE SPIN GLASSES

The determination of the macroscopic properties of a magnetic substance, for example the total magnetization or the response to an applied magnetic field, involves only the statistical distribution of the spin confi gurations of its magnetic atoms (Fe, Ni, Co or Mn); these spins, carried by the electrons of incomplete electronic shells, ca,n be treated like classical (non-quantum) spins in the study of the transition to the spin-glass phase. Although their interactions are not direct but relayed by the electrons of conduction of the non-magnetic metal, we can describe their consequences by replacing them with effective short-range pair interactions, involving only the spins. A model for spin glasses is given by replacing the coupling constant J of the Ising model (§ 2.1.1) by a family [Jai,) ) of random coupling constants, each associated to a pair of spins (ti, j). The consequences on the interaction potential of the random nature of the real distance between the spins i and j located at the nodes of a regular lattice are reproduced in the random and local value of Jii.

4B.2

Frustrated systems

The essential characteristic of spin glasses is the presence of "quenched disorder" and of "frustration". As illustrated in figure 4B.1(b), it can appear even if the spins are placed on a regular lattice. We speak of frustration when the nature of the interactions and the geometry of the lattice make it impossible to simultaneously minimize the energy of all the pair couplings between the spins: modifying the orientation of one spin will lower some of the energies of interaction and increase others. As a result, the orientations of the spins are still random and spatially disordered, as they realize any configuration among those corresponding to the energetically optimal trade-off. Nevertheless, at the opposite of what occurs at high temperature, these orientations are so slowly evolving (only over long macroscopic times) that we may speak of quenched 84

More precisely, this probability is given by the Boltzmann factor e —E14 B T up to a normalization factor.

4 B.2 Frustrated systems

131

disorder. This term "quenched disorder" also applies to the continuous models of spin glasses in which the magnetic atoms, hence the spins, occupy spatially disordered but fixed positions (over any time). As a result, the interactions J i .§i between the spins are also "quenched" (or "frozen") since the effective coupling constant /id between the spins i and j only depends on their relative distance ri,j. Typical examples of frustrated systems are fluoride CsNiFeF 6 , the basis of whose crystalline structure is a polyhedron with triangular faces and whose spin-spin interactions are antiferromagnetic, and the amorphous composition (Al203)0,1(Mn0)0,6(Si02)0,2, in which there is a competition between the ferromagnetic interactions between nearestneighbor spins and the antiferromagnetic interactions between next-nearest neighbors. —

Figure 4B.1 Antiferromagnetic system a.nd frustrated system The pair interaction between the spins is antiferromagnetic and reduced to nearest neighbors. It has potential V(; , §:1 ) = ../gai with J > 0 and its energy is minimal when the spins are aligned in alternating directions. (a): on a square lattice, the phase is antiferromagnetic. Once the orientation of one of the spins is fixed, there exists only one con fi guration minimizing gi ) of nearestthe total energy. This minimum is absolute since each pair configuration —J. neighbor spins is in the energetically optimal (b, c): the system obtained on a triangular lattice is called frustrated since it is impossible to simultaneously minimize the interaction potential of the different pairs; several configurations realize the optimal trade-off (namely the relative minima of the total energy).

DETAILS AND COMPLEMENTS: CONFIGURATIONS AT EQUILIBRIUM

In a spin-glass phase, the spin system has many configurations with very different symmetries realizing relative minima of the total energy, with values independent of the temperature. At high temperatures, the dominating effect of thermal agitation and the temporal fluctuations it induces prevents the system from stabilizing, even temporarily, in. one of these energetically favorable configurations; thus these configurations do not play any particular role. At low temperatures, on the other hand, they correspond in the phase space

132

Spin glasses

to locally stable (metastable) states at short macroscopic times: tlie spin-glass phase appears since the con fi guration of the system is then fixed at small time scales. Nevertheless, the spin glass still slowly evolves according to a thermally activated relaxational dynamics, which induces aging effects; the spin glass will ultimately leaves any locally stable configuration to reach another metastable state, typically of lower energy. The higher the energy barriers isolating a configuration and the lower the temperature, the longer is the lifetime of this configuration: indeed, the temperature controls the spontaneous fluctuations of the spin system thus the fluctuations of its energy, which allows it to cross over the energy barriers. Although macroscopic, this lifetime is finite; the system ends up relaxing, in a random arid experimentally unpredictable way, since the sequence of states visited and the lifetime of each of them cannot be determined a priori. We can, however, predict that they will depend on: — the initial configuration and the a.ssociated symmetries; — the way in which the system was cooled to the spin-glass phase; — the presence of an external magnetic field during the cooling; — the temperature T reached, since it indirectly controls the stability, so also the lifetime of the configurations realizing local minima of the total energy; the quantum fluctuations and the possible tunnel effect which will have the same consequences as the thermal fluctuations. —

(a)

Figure 413.2 -

(

(c)

3)

Frustrated systems: symmetries and energy barriers

The coupling between the spins is antiferromagnetic: the energy reaches a relative minimum Vm i n = —(3/2)1 in the three cases represented here. The configurations (a) and (b) have the same symmetries, so they can be deduced from each other by coordinated and continuous rotations of the spins at constant energy. The configurations (a) and (c) have different symmetries and are separated by an energy barrier AV = V* Vmjn = J/2 since (a) without passing through energy states at least cannot be transformed into (c) equal to V* = — J. —

4B.3

Transition to the spin-glass phase

4B.3

Transition to the spin-glass phase

133

Spin glasses present a phase transition at a freezing temperature Ti, revealed, as the temperature is lowered, by the properties of electrical conductivity and of response to an external magnetic field. Above Ti, the magnetic susceptibility (for weak fields) approximately obeys Curie's law ( 4.1.2), the more so as the temperature is higher. On the other hand, it presents a marked peak at 77. The calorimetric properties undergo a gradual transition, comparable to the liquid/glass transition. On the other hand, the magnetic properties undergo a sharp transition at Ti, comparable to the antiferromagnetic transition; this behavior cannot be explained by any reasoning based on the progressive freezing of the orientations of the spins as the temperature decreases. Although it does not correspond to a clearcut transition", the passage through Ti appears as a critical phenomenon.

<>

DETAILS AND COMPLEMENTS: DEFINITION OF

Ti

The so-called "alternative magnetic susceptibility" x(T, H, 4.)) with respect to an applied alternative magnetic field H oscillating with frequency w presents a peak at the temperature T = Tf(w) when considered as a function of T; this peak becomes more marked as the field H is weaker, but its position is independent of the intensity of H. It decreases very rapidly below Tf(co) since all the spins simultaneously cease to follow the oscillations of the field; this collective behavior is characteristic of a critical transition. We observe that T = T f (w) decreases as w increases, Its (intrinsic) lower bound is called the freezing teraperattire DI of the spin glass. Let us note that below T1 (w), the alternative susceptibility x(T, H ,w) also depends on the time t, which reflects the specific slow dynamic behavior of the spin glass, besides being reminiscent of the "critical slowing down" associated with critical phenomena. This time dependence implies a breakdown of the temporal translational invariance (invariance through a change of the origin of time) in the spin-glass behavior; this phenomenon of aging is one of the most prominent features of the spin-glass phase.

The particular properties of the transition can be explained by the multiplicity of the microscopic configurations realizing relative minima of the total energy and by the diversity of their geometric symmetries. Ti appears as a threshold temperature below which the system remains fi xed at short time scales since the thermal fluctuations are too weak to allow the system to cross the energy barriers isolating the configuration over microscopic time lengths. A possible order parameter is the spatial average =< mu > of the modulus IrhI of the local magnetization 86 fit; in some cases, it is possible to show, using the so-called method of replicas, that the second derivative 0 diverges at T = Ti and obeys a scaling of p with respect to the field H at H /71. The drawback with p is that it is not experimentally law with respect to IT —

'Indeed, there are by now doubts on whether the passage through Tik is really a phase transition in the classical thermodynamic sense. 86 is a temporal average, because of the macroscopic resolution of the observation, which explains the fact that it vanishes in the paramagnetic phase. Note that the order parameter cannot be the global magnetization since it vanishes both in the paramagnetic phase and in spin glasses.

134

Spin glasses

accessible. Since the state of a spin glass (T < Tic) is frozen, it is more relevant to describe the local geometric symmetries of its configuration, which are absent in the paramagnetic phase. The local configuration (b) of figure 4B.1 is, for example, invariant under the three rotations of angles 0, 27r/3 and 47r/3 whose center is the center of mass of the three spins. The order parameter is then a field AW, where A ( i) denotes the infinitesimal generator(s) 87 describing the local symmetry of the configuration in the so we can use the tools of group theory. neighborhood of

4B.4 Renormalization methods According to the outline given in chapter 4, the partition function Z(T, H) of the spin system can be computed via renormalization techniques, and we can study its behavior in the neighborhood of the transition temperature Ti for a weak field, The thermodynamic quantities, mainly the free energy per spin f(T, H), and the exponents of the transition can be deduced from it. The full computations are difficult and cumbersome and a lot of questions are still open; in consequence, we shall limit ourselves to pointing out some specific facts. — The dimension d of the underlying space is crucial in determining the universality classes; it is a simple numerical parameter when the analysis is performed in conjugate space, so it can be extended to any real (non-integral) values. The resolution in the neighborhood of a value do for which the solution is known can be done by perturbation, setting c = d — do. — It is interesting to apply the tools of group theory to take profit of the local properties of symmetry of the configurations; the geometry of the lattice is thus very important. — It may be necessary to take quantum effects into account. — The determination of the order of the phase transition corresponding to the apparition of a spin glass is still an open problem: it seems that this order depends crucially on the underlying dimension d, the transition being of order 1 for d = 4, of order 2 (critical) for d = 2 and of still unknown order for d = 3.

Extensions A spin glass is a typical example of a disordered medium, which is of great theoretical interest because of the small number of its parameters and variables: in some sense it is a minimal model of a disordered medium. It lends itself well to numerical simulations, for example via the Monte—Carlo method ( 3.3.4). This model is also useful in the study of neural networks since the evolution to-wards thermal equilibrium of a spin glass and its possible stabilization in a large number of different "frozen" configurations is remarkably analogous to the mechanisms of memory and recognition of patterns.

87

algebra of the group of symmetries

S

=

n E Z}, or an element of the Lie if it is a Lie group (see §3.4.1).

A is either the generator of a discrete group

SA

4B.4

Renormalization methods

135

REMARKS AND BIBLIOGRAPHICAL NOTES An introduction to the properties of spin glasses and to their possible explanations can be found in the survey articles by Binder and Young [1986] or Weissman [1993 ], A complete presentation of the theory describing the "spin-glass" phase is given in the books by Mezard, Parisi and Virasoro [1986] and by Fischer and Hertz [1991]. The links between the models of spin glasses and neural networks are exposed in Dotsenko [1994]. For a detailed survey of the "spin-glass" phases observed in metallic alloys, see Rammal and Souletie [1982]. We refer to the collection of papers edited by Young [1996] for the most recent advances on both the theoretical (with the associated numerical simulations) and the experimental modeling of spin glasses. We also mention the recent book by Luck [1992] for a general introduction to the physics of disordered media.

Dynamical systems and chaos In the framework of differentiable dynamical systems (already considered in § 2,2), renormalization is used to analyze the appearance of deterministic chaos; it is the translation into temporal terms of the spatial renormalization developed for/critical phase transitions. We introduce the notions describing asymptotic dynamics and their possible chaotic aspects: attractors, invariant measures, Lyapunov exponents and entropies. These aspects concern discrete evolutions as well as continuous flows, whose asymptotic study can be performed on a discrete interpolation given, for example, by Poincar6's method, In supplement 5A we explain the physical meaning and the operational interest of the results of ergodic theory, which is the basis of the statistical description of asymptotic evolutions. We explain in three sections the renormalization adapted to each of the following three generic scenarios towards chaos in dissipative systems (excited so as to stabilize in a non-trivial

state): — the period-doubling scenario. In supplement 5B we introduce a fibered renormalization designed to describe the in fl uence of noise on this scenario.

— the scenario of Pomeau and Manneville, via temporal intermittency, — the scenario of Ruelle and Takens via destabilization of a quasi-periodic

attract OT. In conservative (Hamiltonian) systems, the appearance of chaos is described by the KAM theorem supplemented by the study via renormalization of the diffeomorphisms of the circle, which is described in detail in supplement 5C. In supplement 5D, we explain renormalization applied to developed hydrodynamic turbulence. The basic notions of the theory of dynamical systems are collected in appendix

5.1 5.1.1

Deterministic chaos Asymptotic dynamics

The term deterministic chaos refers to erratic and irregular behavior observed in physical systems whose evolution is differentiable and does not involve any random mechanism. A first answer to this apparent contradiction is that the possible chaotic features and the irregularities of the motion appear only in the limit t co: thus they are compatible with the predictability of the evolutions of finite duration and

138

Dynamical systems and chaos

with their regularity with respect to initial conditions and to parameters ( 3.2.4). Dynamical chaos thus concerns the asymptotic dynamics, restricted to an attractor A defined as a closed, invariant, undecornposable set consisting of the accumulation points of all the trajectories coming out of an open set containing it; the statistical properties of this asymptotic evolution are stationary and are described by an invariant measure rn,, of support A.

DETAILS AND COMPLEMENTS: ASYMPTOTIC SINGULARITIES

The asymptotic limit t oc is the temporal analog of the thermodynamic limit encountered in spatially extended systems. The possible singularities with respect to the parameters p of the statistical quantities X,,,,(p) describing the global dynamics are due to the presence of this limit X(p) = lim t_,,„ X(t, p) in their definition: the existence of these singularities does not contradict the regularity of p —+ X(t, p) when t is finite. It is impossible to exchange the limit t oo with operations on p (such as differentiation) when p takes a critical value p c , so the dependence on t and on p must be analyzed simultaneously; renormalization is exactly the right tool to accomplish this. The procedure consists of bringing out an invariance (t t I k, p pc — b(p c p), X V' X), which reveals the scaling law:

X[t c,

p)]=ka X(t, p, p)

X(t, p c — p) -4-. t -aï[qp, - p)fr]

where

= log k log

The corrections to scaling occurring in the asymptotic scaling laws when the duration T of the observation is finite, as well as the differences between the asymptotic quantities X and the observed quantities X(T), can be analyzed in the same way as spatial fi nite-size effects (§ 3.2.4).

Conservative and dissipative dynamical systems Conservative systems leave the volume element rehraL of the phase space X invariant. The typical example is a system having constant Hamiltonian H 5, q)= Ho, where dm, L 05, = Ponq is preserved by the flow according to Liouville's theorem. In this case A = X and mc,„ = rrIL. In dissipative systems, the evolution contracts the "natural" measure rni, on the phase space X, which is invariant when the system is conservative". The asymptotic study of these systems involves only the attractor A on which the trajectories stabilize, an invariant measure m on it and the evolution f restricted to A. The dissipation causes f to contract the natural measure rni, at each iteration; consequently, raL (A) = O. The measure m oo has support A, so disjoint from the support of rraL. The initial conditions influence the transient regime but have no further asymptotic effect than specifying the attractor (according to the basin of attraction it belongs to), and the (

"This statement shows that the physical notion of a dissipative system is difficult to

formulate, apart from the tautology: "a system is dissipative if it is not conservative". A more rigorous but less tangible mathematical notion of dissipativity is given in § 5A.1.

5,1

Deterministic chaos

139

one among all the invariant measures of f which describes the permanent regime. From a physical point of view, such systems describe an evolution presenting a mechanism which dissipates energy, matter or more generally information. This dissipation is balanced by external inputs or by a mechanism of excitation, adapted for the system to dissipate exactly as much as it receives, so that the asymptotic state is non-trivial. The description (f, X, raL) of a conservative system and the description ( fl A, A, rracc ) of a dissipative system restricted to its attractor are formally very close; there is actually no need to distinguish dissipative and conservative systems in the asymptotic study, in the sense that the same characteristics are defined and analyzed on (A, and on (X, mL) respectively,

5.1.2 Discretization methods We can show that the study of a continuous dynamical system whose physical interpretation is clear can be reduced to the study of a discrete evolution, whose analysis is simpler; the physical relevance of discrete systems follows. Let qS be a continuous flow whose phase space X is d-dimensional: 0(41, t) G X is the position at t on the trajectory coming out of x o E X at t o .

Stroboscopic discretization A first discretization method works via stroboscopy: fixing an initial instant to and a step At = r, we extract the discrete trajectories [xn (to, so, 7') 0(to, xo, to + nT)1,1> o • This method is only useful if the flow is autonomous, in which case the study of the flow is reduced to that of the map Or . However, the discrete evolution [On, (x0 )]fi >0 does not in general have any particular properties: we must consider trajectories which are indeed discrete but which evolve in the whole space X, and then it does not simplify anything to study O r . This failure is due to the artificial nature of this type of discretization: the step r, chosen by the observer and equal for all trajectories, does not match the properties of the flow.

Discretization via the Poincaré section (Poincaré [1880], [1892 ]) Another method, known as Poincaré's method, can be profitable if the autonomous flow 89 has a periodic orbit 0 of period T. We choose a transverse section E to 0 at zo E 0, i.e. a hypersurface of dimension d 1 passing through zo but whose tangent space does not contain the tangent vector to 0 at zo . This choice is the only arbitrary step in the method; it is based on particular physical properties, for example the passage through a given state, the disappearance of a phase or the cancellation of a variable. The description keeps only the successive points of the passage of the continuous orbit through E, restricting itself to orbits near the periodic orbit to ensure that they intersect E: a trajectory x(t) is thus replaced by the sequence (x„)„>0 of its intersections with E. The autonomous character of the flow ensures that the interval of —

"The restriction to autonomous systems is not too serious, since as shown in appendix II, a non-autonomous flow can always be brought to an autonomous flow by adding an additional, "pseudo-temporal» dimension to the phase space.

140

Dynamical systems and chaos

time t1 -t n separating two passages x n =x(L n ) and x„ + 1= x(t,i+ i) depends only on x n (and not on n): we have t1 -t = r(x n ). The function T is called Poincares return time in the Poincar6 section E. The correspondence between x n and x n + i also depends only on x n and induces a map fE called the Poincaré map, defined and differentiable in a neighborhood Uzo of 2.0 and with values in E. By construction, r(z0)-=T, fE(z 0 )= zo and fE(x) 0 7.(x )(x). The advantage of this method is that it gives an intrinsic reduction adapted to the system since it is specific to each trajectory (7- (x) depends on x); it reduces the system to an autonomous discrete dynamical system of smaller dimension (at most d -1) since its phase space is Uza E E. The asymptotic properties of the discrete evolution h reveal those of the continuous flow it interpolates, with the gain of eliminating some specific details such as the time law. This procedure thus constitutes a first step in the search for universal properties. Moreover, it does not describe the motion along the trajectories, which is not interesting because the dynamics are invariant under translation along 0; this property is equivalent to the fact that the tangent to 0 at 2.0 has an eigendirection of eigenvalue 1 of D0(20 , r0 ), and one advantage of Poincaréss method is that it eliminates this marginal direction. Thanks to the existence of these discretizations of the continuous flows, we need to describe only the chaotic characteristics of discrete dynamical systems.

5.1.3

Chaotic properties

The appearance of the notion of deterministic chaos was the signal for the abandonment of the basic (idea established by Landau (Landau [1944]), according to which deterministic and regular evolution laws generate chaotic asymptotic behavior only after destabilization of an infinite number of degrees of freedom. The inadequacy of this theory was revealed when evidence was obtained that chaotic characteristics can occur in systems with only a small number of degrees of freedom, or even in infinitedimensional systems involving only a finite-dimensional subspace of the phase space. An example of this second case is given by a spatio-temporal system whose dynamics can be spanned onto a finite number of given spatio-temporal functions, which reduces the study to that of the dynamical system describing the purely temporal evolution of the finite number of coefficients occurring in this decomposition. The possibility of chaos is not a priori excluded° except for continuous autonomous evolutions in dimension 1 or 2.

<>

d OF THE PHASE SPACE X The role of d with respect to chaos can be discussed by constructing the phase portrait91 ,by which we visualize the flow in the phase space. The different trajectories do not intersect and are invariant under change in the origin of time since the flow is autonomous (or was DETAILS AND COMPLEMENTS: THE DIMENSION

assertion is different from the theorem of Ruelle and Takens (§ 5.4), which uses the number of excited (unstable) modes and not the dimension of the space (Li and Yorke [1975]). 91 See App. 11.2. The advantage of the phase portrait is that it can be obtained without solving the evolution equations; thus it gives no information on the motion along trajectories but still enables us to qualitatively follow the evolution of every point. 90 This

5.1

141

Deterministic chaos

made autonomous, cf. App. 11.2). In dimension I or 2, they behave like solid walls impossible to cross, so that the history of a. point is completely predictable over any length of time; no chaotic characteristic can appear if d < 2. If d > 3, no trajectory actually realizes a partition of X: two arbitrarily near, experimentally indiscernible points may actually end up having completely different evolutions, so these evolutions are unpredictable at the initial instant and are in this sense chaotic, The constraint d > 3 only seems to disappear in discrete systems, since they are obtained by projection and section of continuous flows in higher dimensions.

A Bi C1 A0

fito

Co

A1

A 0 Be

Figure 5.1 -

Co

Cl

A.1

Chaos and distortion of the volume elements

The fi gure represents an elementary step

On = 1) of three generic evolutions.

(a) The evolution contracts the lengths in every direction: the attractor is a fixed point. This case exists only for dissipative systems. (D) The evolution dilates certain lengths but without mixing the volume elements: the attractor is non-trivial but not chaotic; observing the points A, B and C at any given instant n gives information about their initial position.

(e)

The evolution dilates certain lengths and mixes the volume elements: the

attractor is chaotic since the dynamics "forget" initial conditions: observing A, B and C at an instant n oo no longer gives any information about their initial position.

142

Dynamical systems and chaos

Quali tative criteria for chaos A crucial necessary condition for the appearance of chaos is the presence of nonlinearities, In support of this assertion, we note that the transition towards chaos in the family [z t—P 1 — px 2 ]0 <m<2 is controlled by the parameter IA (§ 2.2). Non-linearities are necessary to ensure not only that there is an amplification of the distances but also that this amplification varies according to the direction and the position in the phase space, so as to ensure a mixing of the volume elements, as illustrated in figure 5.1. This mixing property, formalized in the framework of ergodic theory (App. 11.4), causes a phenomenon of "forgetting the initial conditions" and thus constitutes a criterion for chaos. The erratic character of the asymptotic trajectories is in general confirmed by the existence of statistical laws (as in the case of a stochastic evolution) relating almost all of them (in a measure-theoretic sense) to the same average quantities, which can be computed via the invariant measure of the evolution (§5A.2). The dissipation turns out to be a property independent of possible chaotic characteristics: the contraction of the volumes implies only that the dynamics restricted to the attractor control the asymptotic behavior t oo of the system. The term "chaotic" used to describe a deterministic motion was adopted essentially to describe its unpredictability. This unpredictability, illustrated in figure 5.2, is due to the amplification of the distances by the flow, which means that the slightest uncertainty about the initial state is crucial to the asymptotic behavior, so that we have no information about the longterm behavior of the system. The butterfly effect, introduced by Lorenz (Lorenz [1963]), is a (somewhat exaggerated) image intended to illustrate the consequences of this sensitivity with respect to initial conditions, stating that a single wingbeat of a butterfly, exponentially amplified by chaotic dynamics, can completely modify the climate. More pragmatically, this type of argument is used to limit the reliability of weather forecasting to about five days.

DETAILS AND COMPLEMENTS: EXAMPLES OF CHAOTIC SYSTEMS 92

— The logistic map x rx(1— x) models the evolution of populations" and reproduces its possible erratic and unpredictable behavior. Libchaber arid Maurer [1980] give evidence of various types of chaotic behavior, for instance an accumulation of period-doubling and of temporal intermittency phenomena in Rayleigh Benard experiments where one observes convection in a viscous liquid heated from -

–

below. A compass placed in two magnetic fields, one fixed and the other rotating, gives an example of Hamiltonian chaos: the physical system made up of the compass and of the rotating field is described in a phase space of dimension 4, and its evolution preserves its total energy, but is chaotic. —

Certain chemical systems present macroscopic chaotic behavior, for example the

'See also Gollub and Swinney [1975], D'Humieres et a. [1982], Lichtenberg and Libchaber [1984 'See May [1976]; one thinks naturally of lemmings and crickets.

5.1

Deterministic chaos

143

Belousov-Zhabotinsky reaction modeled" by a system of three kinetic chemical equations called the "oregonator". — The model introduced by Lorenz [1963] to describe the convection of an incompressible fluid gave one of the first known strange attractors. Obtained by truncating the spatio-

temporal equations of convection and seeking parametrized solutions of a given form, it has the form of a continuous dynamical system:

I

dX dt - X) dY dt = rX - Y - X Z dZ(t)Idt = XY bZ

Topological criteria for chaos A first feature to consider is the nature of the attractor; the asymptotic dynamics can be foreseen if the attractor is a fixed point, or a periodic or quasi-periodic orbit, i.e. a superposition of independent periodic motions (§5.4.1). In more complex cases, the attractor can be 'fractal" (lacunary and with non-differentiable structure), in which case the invariant measure on it is singular with respect to the Lebesgue measure; it can be "chaotic" in the sense that the restricted dynamics amplify the distances. These two properties are independent; when they occur together, the attractor is said to be strange. A second criterion is obtained by quantifying the unpredictability of the evolution. The duration T over which it can be predicted increases if we refine the initial resolution eo and if we relax the precision e„, required for the observed states. The relation coo c o eTstoP defines the topological entropy Stop (figure 5.2), which increases with the unpredictability. A chaotic flow is characterized by Stop > O.

40, DETAILS AND COMPLEMENTS: EXPRESSION OF THE TOPOLOGICAL ENTROPY

Initial conditions so and yo are called (n, e) -separated by the evolution f if


d[fi(xo), f.i(Y0)] >

i.e. if an observation performed over a duration n and with resolution c distinguishes the trajectories coming out of ro from those coming out of yo. If Ar(n,c) is the maximal number of (n, 0-separated initial conditions, the topological entropy Stop is defined by: Stop = lim lim n -1 log [N(n, e)] o n oo

.Ar(n , c)

exp(nSt op ) Ar(O, e)

Since the resolution c is fixed, each step of the evolution multiplies the number of e—separated initial conditions by e st°P, so that a "finite structure" of e s'°P trajectories appears in each of the trajectories distinguished in the preceding step.

"See Tyson [1976] or Lemarchand a,nd Vidal [1988].

Dynamical systems and chaos

144

at' 2

Figure 5.2

-

Exponential divergence of neighboring trajectories

ctio ) A flow is said to be chaotic if two initially very close trajectories (ao generic, liaoct o i l I. If ao and a are er t diverge exponentially fast: Ilat c411 exponent of the flow. This amplification of Lyapunov is the maximal 7>0 the distances makes the dynamics unpredictable over a long time since details neglected at the initial instant (because of the finite resolution co or of the simplified nature of the model) end up playing a determining role. Conversely, observing a trajectory over a time T with the precision co actually increases the precision E of its point of departure, which becomes c = where St op is called the topological entropy of the flow (one can show that St op

> E positive exponents

> 7).

Measure-theoretic criteria for chaos Measure-theoretic criteria describe characteristics which are generic in the measure-theoretic sense, which means that they describe the flow from which have been removed trajectories coming out of a subset of measure zero for the invariant measure m oo . In particular, we are concerned only with dynamics restricted to the attractor. The measure rn,, spontaneously selected by the evolution is assumed (if not shown) to be ergodic, since this is the case where the measure-theoretic description of deterministic chaos is relevant. Like rn, this description gives no information on the transient regime, nor on particular trajectories. Rather, it gives a global statistical overview of the asymptotic regime. The two principal indicators of chaos are the Lyapunov exponents, which generalize to arbitrary dimension d the definition given in appendix II for d = 1, and the statistical entropy relative to the measure m oo , called the metric entropy and written s(rrl). As shown in figure 5.2, a characteristic feature of deterministic chaos is the existence of a positive exponent, implying the amplification of the distances in the associated direction.

Deterministic chaos

5.1

0.

145

DETAILS AND COMPLEMENTS: LYAPUNOV EXPONENTS

(Lyapunov [1906])

In the case of a dynamical system (f, X) in dimension d> 1, one constructs: A = lim n—pco

(x)Dfn(x)]

1/2ra

XEXCR4

One shows that the limit exists and defines a real d d matrix A(x) which is rn-almost everywhere constant for every invariant positive measure m which is ergodic under the action of f. The Lyapunov exponents of the flow generated by f are defined to be the logarithms of its eigenvalues. There exists a manifold (invariant under f) of codimension greater than or equal to 1 outside of which the trajectories separate exponentially fast with divergence rate equal to the maximal Lyapunov exponent. On the manifold, the separation is slower and it is described by the other Lyapunov exponents, and the trajectories which lie on it are called non-generic. By definition, the Lyapunov exponents are global asymptotic characteristics of the flow, The advantage of these exponents lies in the fact that they provide a simple criterion for chaos but also a quantitative and analytically computable estimate of its intensity (Arnold and Wihstutz [19861),

5.1.4

Scenarios towards chaos

and

adapted

renormalization

methods A scenario towards chaos is a sequence of bifurcations leading from a system So with no chaotic characteristics (for example having a single stable fixed point) to a chaotic system Sc . A remarkable result of the theory of dynamical systems was the discovery of generic universal scenarios, induced by the continuous variation of a control parameter p. Genericity is physically essential for the scenario to be observable. The universality of such a scenario gives it a predictive role: whenever a certain set of qualitative characteristics is present, the scenario can occur and it then leads to quantitative

properties.

0 DETAILS AND COMPLEMENTS: THE GENERIC SCENARIOS

p is called the stochasticity parameter because it measures the importance of the factor inducing the chaos. In dissipative systems, chaos is due to the simultaneous presence of an excitation and of dissipation, taking place on different scales and at different instants; p is related to the ratio of their characteristic scales or of their intensities, or to the amplitude of the non-linearities which reflect the simultaneous presence of these two mechanisms. Three generic scenarios are possible: — the period-doubling scenario, describing accumulation of period-doublings towards an aperiodic attractor which is fractal at p = p c ; the chaotic characteristics appear for certain values of the parameter p beyond p, (§ 5.2); — the scenario of Pomeau and Manneville, via temporal intermittency. This local scenario describes the persistence of stable phases in the evolution just after the disappearance of certain stable fixed points at p = p, (§ 5.3);

146

Dynamical systems and chaos

— the scenario of Ruelle, Takens and Newhouse, via destabilization of a quasi-periodic motion with three incommensurable frequencies depending on p, which can lead to a strange attractor for p > pc (§ 5.4). For Hamiltonian systems, the principal (generic) result is — the theorem of Kolmogorov, Arnold and Moser; here the parameter p is related to the amplitude of a non-integrable perturbation (§ 5.5).

0 Renormalizati on and deterministic chaos The use of temporal renormalization to study chaotic asymptotic behavior is based on arguments analogous to those used in the study of critical thermodynamic behavior, the spatial dependencies being replaced by their temporal equivalents, Chaos results from an organization of the elementary temporal steps which is perceptible in the average asymptotic properties but not in an evolution of finite duration (a "sample"). Renormalization works well to selectively eliminate the degrees of freedom with no chaotic consequences. Its purpose is to reveal the scaling laws of the asymptotic dynamics and explicitly determine their universality; it fulfills this second aim by replacing the study of the trajectories in the phase space X by that of the evolution laws in a function space. The renormalization of a discrete dynamical system of law f is based on the scale change [t + t/k, x kax]. It preserves the topological nature of the attractor of l and also that of the invariant measure mf on it (mf is absolutely continuous or singular with respect to the natural measure). Its action on the Lyapunov exponents and on the entropies can be expressed by: —

—

Stop(Rkf) = k

y(Ril) = k 7(i)

-

St op (f)

s(rnRk f) =

k s(mf)

Once Rk is constructed, the analysis continues according to the usual steps, detailed in §3. 1 . To study a scenario towards chaos, it suffices to transfer the action of Rk to the parameter p: Rk 4, fro, in a neighborhood of its critical value p c . The possible universal properties of the asymptotic dynamics, in particular their dependence on p —

(revealing the transition towards chaos at p c ) are then related to the characteristics of Rk and rk •

5.1

Deterministic chaos

147

-

exponent '7(11)

-

1

0.5

1

1.5

2

control parameter p

Figure 5.3 - The Lyapunov exponent of fg (x) =-- 1 — px 2 The Lyapunov exponent 7(4) of L4 is plotted as a function of the parameter p E [0, 2]; it is numerically computed via the approximate relation N7(p) N -1 E i _o log .,fi,' (f4(s0))1 (here xo = 0 and N -. 103 ). In the zone 1,401, the graph shows the doubling bifurcations (7(pi) = 0). In the zone y > pc , the positive and negative values are intertwined; the drawing is not particularly reliable, and very sensitive to changing the resolution, to the choice of N and to numerical noise. The envelope ry(p) (drawn in boldface) is however quite robust. The analogy between its graph and that of the spontaneous magnetization of a ferromagnetic material (figure 4.1) leads to the interpretation of -y(p) as the order parameter of the transition towards chaos at tt,.

148

0

Dynamical systems and chaos

DETAILS AND COMPLEMENTS: PROOFS

The action of renormalization on the their qualitative definition Is — ylen't( Rki)

Lyapunov

exponents

-y(f)

I[Rkfnx) — [Rk f]'(yY

- ylek"(f) which implies that -y(Rk f) = kW),

fkn(ylc - ")!

spatial scaling factor k' .

Renormalization

is obtained from

f k r'(x

-

independently of the

accentuates the divergence of the trajectories in

the unstable directions -y( f) > 0 and the rate at which they get closer in the stable directions ry(f) < O. The action of renormalization on the topological, entropy can be computed by noting that:

{ x.,0 and yo (Rk f, n, 0-separated xo and yo (f , kn, e)-separated

k'xo and k - 'yo (f,kii, k - '0-separated 10 - " fi (x 0 ) and k - ' fj (yo) (Ric f , n, k" c) -separated for a value j < k

€). It follows that so .Ar(f, kn,k - "c) > Ar(Rk f ,n,c) and Af(f, kra, < k Af(Rk f , n, St op (Rk f) = kS(f) for every spatial scaling factor V' Indeed, the spatial scale change behaves like a magnifying glass and does not modify the chaotic nature of the flow. We finally note that if the measure mf is invariant under f, then the measure rriR, defined through rn f [B] = rn [k- " B] is invariant under Rkf- Thus, renormalization does not modify the nature of mf. In particular, we check from this relation that the metric entropy is transformed according to s(mR h f)

5.1.5

= ks(nf).

Transition towards chaos and critical phase transitions

The steps of the temporal renormalization of dynamical systems clarify the analogy between transitions towards chaos and critical phase transitions encountered in statistical mechanics, This analogy is supported by the existence of an order parameter revealing the transition towards chaos (figure 5.3), It can be the envelope°5 L(p) of the maximal Lyapunov exponent or any other global indicator of the asymptotic evolution which vanishes in the zone p < p c and becomes non-zero if p > pc . The existence of a positive Lyapunov exponent in the chaotic regime implies a symmetryFoo is breaking (t t) of the dynamics restricted to the attractor: the direction t is thus different from the opposite associated to the amplification of the distances and direction, which is contracting. In contrast, the dynamics on a cyc1e have no favored temporal direction, This irreversibility of chaotic evolutions is reflected in the positivity of their topological entropy. The chaotic phase is the analog of the (low-temperature) ordered phase: the amplification of the distances associated to chaos appears only if the successive steps contribute constructively to the evolution of the distances separating the endpoints of initially close trajectories; the temporal statistical correlation between the successive steps is then long range. The analogy extends to transient regimes since the phenomenon of critical slowing down, observed in the relaxation towards thermal -

-

-

"The irregularity of L(p), due to its asymptotic character L(p) = lim n - 0.3 (1/n) E7:01 log It(x .,)1, makes it necessary to consider its envelope L(ta).

5.1

Deterministic chaos

149

equilibrium at a temperature T near the critical temperature 71,, appears near the bifurcations in the stabilization 96 of the trajectories on the attractor (figure 5.4).

Number of steps 1000 -

00CP-

400 -

200 -

1-5

Figure 5.4 - Time needed to reach the attractor of f(x) = 1 — px 2 For p E [0,p, 1.401 ], the duration of the transient regime is estimated by the number of steps after which a generic trajectory stabilizes in a domain whose points are distant by at most Ax = 10 -3 from the attractor of fo . It diverges at the bifurcations, which gives a criterion for locating them. By reducing Ax and the numerical noise, we would obtain a reliable result for the next bifurcations, 96 Estimating the time needed to reach the attractor is a preliminary step in numerically constructing the bifurcation diagram, which makes it possible to eliminate the influence of the transient regime (figure 2,3), Note that the divergence of the time needed for reaching the attractor in the neighborhood of the bifurcation points makes the bifurcation diagram less reliable at these points.

Dynamical systems and chaos

150

In the table below, we summarize the analogies which justify the inclusion of the transition towards deterministic chaos in the family of critical phenomena.

Transition towards chaos

Phase transitions

Discrete formalism: global phase space 0 = 0 10v

discrete dynamical system temporal extent T N instants, N steps (j = 1...N) instantaneous position xi E (150 increment vj trajectory in -= asymptotic limit (T -> co or N 0o) forgetting the initial conditions permanent stationary regime

lattice system spatial extent L N subsystems (j = 1. . N) state variable s individual state si E 0 0 configuration in 0 = 06'7 thermodynamic limit (L co or N co) no boundary effects homogeneous thermodynamic equilibrium

Statistical description invariant measure (§5A.1, §5A.2) Boltzmann-Gibbs distribution statistical laws due to ergodicity ... due to molecular chaos (§5A.4) Control parameter 9 an

evolution law fm : the non-linearities increase as 6 = p decreases. M = envelope L(p) of the Lyapunov exponent -

order parameter M(0)

Effective Hamiltonian 'HT; thermal motion decreases as 0 = T decreases. M = for example, magnetization m(T) or jump discontinuity of density

Non-critical transition at 0* bifurcation not leading to chaos first-order transition dL I dp discontinuous at p = p* but dMIdT discontinuous at T =Te but with finite right and left limits with finite right and left limits Critical transition at 0e ; (M = 0 if 6 > 0,, M 0 if 6 < 0,) scaling laws and critical exponents v > 0, 71 5 2 divergence of the range of the statistical correlations at 0, non-summable correlations, collective structure, order on a large scale

0 < 0, (p > p c ): chaotic phase Pc)? >0 if /1 > pc characteristic time r(p)dp - p e ry temporal correlations C(t) or 0(r.,./) diverging at p c if t -+ oo or w 0

L(P)

C(t, pc ) C(t, p) e' (w , p)

RP

et(w, p c) - 0 -2 t 1-9 4:0[tIr(p)] 2 [wr(p)]

< Oc : magnetic or dense phase M(T) (T TOO 0 0 if T < -

length of correl. e(T) ,--, 171 spatial correlations C(i) or 0( diverging at 7', if 2; oo or q X 2—(1-1)

eu, T, )

C(i;, T) x 2-61 " CD[Sg(T)] T) q7)-2 [q(11)] e•-■

)

el -2

5.1

Deterministic chaos

151

Universality • exponent a of the singularity ot the • dimension d of the real space F(Xi a ) laws f(x) e) (expansion a = 2 - c, a = 1 + c) (expansion d = 4 - c, d = • dimension n of the phase space • "n-vector" model (n Lyapunov exponents) (n components in M) universality classes (a, n) = universality classes (d, n) External perturbation A, susceptibility x0(0)_[(9M/aA]4 =0(0)(9, 9) -7 A = amplitude Cr of the noise A amplitude H of the applied field L(ki, a) regular at p c if CT 0 O M(T, H) regular at IT, if H 0 -

EaLlacr](p,g =0)— 7

-

[OMI9H](71,11 = 0) IT-TI e(71 H) 11/1 - P

Pci -7

(11 c, cr)

Renormalization

having a hyperbolic fixed point of stable manifold V'

action rk (0, A) (of scaling factor exponents

rk(0, A)

temporal renormalization

k)

with a fixed point

r k(0 c , 0) = (9, 0)

k ll'(0 - 9 ) + 10 1 /PA . spatial renormalization

"macro-steps" , iteration of f Rnfpc = (fixed point) V' = evolutions at the onset of chaos

"macro-sites", partial trace W (fixed point) V 8 = critical systems

r(ri,p) = T(p)/k C(kt,p) k"qt,r k p)

Ork71) = e(T)/k C(ki,T) k 2-

5.1.6

rk T)

Power spectrum and correlation time

Up to now, we have considered only quantities defined as temporal averages along trajectories or, what is equivalent if the flow has an ergodic invariant measure rn, as statistical averages with respect to m. A description in the conjugate space is possible; it is based on the spectral analysis of the flow and of its temporal correlation function. The power spectrum of a temporal scalar signal A(t) which is locally integrable is defined to be the function: 1

5(w) = pracc,

ft.

2

Ti-to

e i" A(t) dt

If the limit 5(w) exists and is finite, we can check that it does not depend on t o . In the study of the evolution of a system S, described by the trajectory x(t) of S in its phase space X, we analyse A(t) F(s(t)) where F is a state function of S, i.e. a real observable quantity defined on X. The quantity 5(w) is only interesting for a stationary signal or for a sequence of pulses, so that the limit is non-zero; indeed, if A(t) is integrable then 5(w) 0. If on the other hand the signal is uniform, i.e. A(t) Ao = const, the spectrum reduces' to the peak 5(w) = 2.7r4 Ow).

"We use limT- c, T 1 foT e

t

dtI2 2rS(w).

Dynamical systems and chaos

152

If the signal is a periodic sequence of pulses A(t) S(w) = 27rc 2 E% 45(w 27n/0).

= Enœ_ o c6(t 1

-

n), then"

-

Power spectrum and temporal correlations If the temporal correlation function 99 C(t) of the signal A(t) decreases fast enough at infinity, it admits a Fourier transform a(w) (App. III.1):

1 G1 (1 ) Tlinico T

ft.,

A ( 9) A 0

+

C(w) =

+on e""C(t)dt - 00

The two quantities 0 and S have the same dimension, and they give complementary information on the structure of the signal:

— S describes how the "energy" is distributed among the different oscillating components whose superposition reconstructs the signal A(t); thus it reveals the normal modes of the signal and their possible hierarchical structure, the peaks of S corresponding to the energetically dominating components. We note that the "total energy" of the signal is equal (up to a constant, in order for the result to have the dimension of an energy) to: limT, LT0 44° A 2 (t)dt/T = f S(w)dw 127r = C(0) = f ef(w)dw127r,

— Via its Fourier transform, a describes the correlation function C(t) of the signal and its different characteristic times, called correlation times. The peaks of a(w), their heights and their respective positions in the space of the frequencies reveal the hierarchical structure of the different temporal correlations present in the signal.

The theorem of Wiener and Khinchin asserts the equality of S(w) and C(w), if

C(t) decreases fast enough at infinity. From an experimental point of view, it suffices to determine just one of these two functions; this enables us to avoid the difficulties of measuring one of them directly, and puts several methods of measurement at our disposal. From a theoretical point of view, this theorem proves the identity of the frequencies corresponding to the peaks of S and of the characteristic frequencies defined as the inverse of the correlation times. From the behavior C(t) e - t 17- if t -+ oc, one can even define the typical correlation time r directly from the spectrum S(w), as follows:

r 2 = lim [2 co S(w)] -1 [dS dcd(w) Intuitively, this time is the average time taken by the trajectory x(t) underlying the observed signal il(t) = F[x(t)] to visit a representative domain of the attractor.

We use N — 2rn)=E,7 0 8(42r n). e'"1 2 .27c "Usually, it is the centered signal 21-(t)=A(t)— A or A = lim T-1 faT A (t)di which appears in the definition of C and of S; here we take A(t) so as to keep track in S and 0(c,.7 = 0) of the stationary component A of the signal,

5.1

Deterministic chaos

153

Chaotic spectra The chaotic nature of the deterministic evolution observed via the signal A(t) is reflected on the spectrum S(w) in the following clues. — There is no lower bound on the existing frequencies, this property, analogous to the singularity q = 0 of critical phenomena, reflects the divergence of the autocorrelation time rcon. = I/ inf1w — It is a broad-band spectrum: there is almost a continuum of relevant spectral components, rather than some isolated spectral lines. — It is often self-similar; exactly at the transition towards chaos, we observe S(ow) = a'S(w) for an infinity of scaling factors a.

7

4

ao 6.23

B28

- 1 ,38.

(b)

(a)

S(A,w) of the map 44 (x) 1 — Figure 5.5 The spectrum w The numerical plot of the power spectrum (for fixed p) uses the approximation: -

S (p , w) = K -1

E einw fn (xo) 1 2

(here

K =104 , xo = 0)

1
The abscissa co is restricted to [O, 27r1 since S(p, w) = S(p,w 27r) . (a) is the typical spectrum of an evolution lying on a periodic attractor (here a 4-cycle for p = 1.35); the width of the peaks is due only to numerical noise. (b) obtained for p = 1.9, is a chaotic spectrum; the details of its graph are, however, not very reliable because of the finite value of K, and mainly because of the numerical noise, which is amplified by the chaotic dynamics.

Dynamical systems and chaos

154

A chaotic spectrum is thus qualitatively comparable to that of a "noisy" version of a deterministic evolution; we find that a chaotic deterministic motion cannot be distinguished via its observable consequences from a stochastic evolution, Note that a real system is unavoidably perturbed by external noise, whose continuous spectrum, of average height A but with no particular structure, is superimposed on the "deterministic" spectra and thus overwhelms all the spectral components of amplitude less than A. The power spectrum analysis turns out to be very efficient for experimentally evidencing the presence of deterministic chaos. This advantage appears convincingly in the case of the period-doubling scenario (§ 2.2) detailed below, It allows us to move the study of dynamical systems into (temporal) conjugate space. The guiding principles of the renormalization analysis are thus the same as in the case of the critical phenomena encountered in statistical mechanics and described in (spatial) conjugate space.

0. DETAILS AND COMPLEMENTS: SPECTRAL ANALYSIS OF

THE PERIOD-DOUBLING

SCENARIO

To determine a spectrum describing the discrete flow generated by each of the unimodal maps of a family [fiA L presenting an accumulation of period-doublings at bifurcation values [finin>o, we consider an observable signal A, of amplitude A(z) when the physical system is in the state x. Let us explicitly compute the spectrum if p Pni, in which case the attra.ctor is a 2 5 cycle [xi(p), x2(p) x2.(p)]. By the periodicity of the asymptotic dynamics, we find that if the evolution law is ft,, the component S(p, A,00) at the frequency -

Ro is equal to:

1 S(p, A,rt o ) = hm — N—co 2n N

E

A(x)

E

e2i2rfloV4,2

12

0
0<j <2n

S(p, A, Q0) is thus zero unless 2 n R0 is integral and its amplitude is related to:

l(p, A,

R0) -----

1

E e 2 air..01

A

f . N1 2

(ki <2 4 Besides the peak at the fundamental frequency Ro= 1, the spectrum presents subharmonic peaks at fl o = 2 -1 q where q is an integer and j < n. The minimal frequency present in the spectrum is Ro = 2 – n if and only if p n -1 < j < it, Writing 2ir/w o for the average time interval separating two observations, we obtain the following properties in the frequency space FA; = — the amplitude of the subharmonic frequencies wn = wo2

decreases as n increases;

— the spectra for p a and pit +i are similar: SOin-f-i, 4,0,1+0 = KS(Ii n ,w,,) where the scaling factor K does not depend on n, Setting K = 2' (a > 0) and using 12, — w Cl[w(1.1, — p )' ] where y = log 2/ log 5 and 6(Pc — Pn—i), we deduce that S(p,w) the function 0 depends on a single argument; — At p = p c , the spectrum presents an accumulation of spectral components at u; = 0, revealing the divergence of the characteristic time at the onset of chaos. Its structure is selfsimilar: by taking the limit as n oc) in the preceding relations, we show that the ratio of

5.1

Deterministic chaos

155

two successive harmonic components is constant: 100 8(.002 -5 ) = The typical spectrum just described gives a feature of the period-doubling scenario which is so characteristic that it is sometimes called a subharmonic cascade. This description is useful for experimental evidence: the spectral analysis of an experimental signal is a classical procedure 101 . The external noise affects the spectrum S(w) only by superimposing its own spectrum Sb(w) onto it; this spectrum is continuous but of bounded amplitude, and thus it does not perturb the appearance of the first doubling bifurcations.

5.1.7

Dynamical systems with noise

The term noise refers to any stochastic influence applying on a physical system S described on a macroscopic or mesoscopic scale. The stochastic character of noise is a consequence of the microscopic scale of the phenomena generating it: the description of S on a higher scale can only include statistical information on these phenomena, so that their consequences are perceived as random and taken into account in the evolution equations by a stochastic process b(t).

C> DETAILS AND COMPLEMENTS: STATISTICAL HYPOTHESES ON NOISE

,

— The statistical average of noise is zero since < b>, which appears as a deterministic term, can always be included in the deterministic part of the evolution. — If the noise results from numerous weakly correlated microscopic interactions, the central limit theorem asserts that it has a Gaussian distribution, which is entirely specified by the temporal correlations C(t, s) = . — If the noise is given independently of S (i.e. if it is an "external noise"), its statistics are not influenced by the state of S, so that we can suppose this state to be uniform in the phase space X of S. Otherwise, we must consider a process qx, t) where s E X, and the statistical characteristics of the noise are functions of s. — Noise can be assumed to be statistically stationary if the overall evolution of its source is negligible during the period of observation of S. — When the timescale on which we describe the evolution of S is large compared to the correlation time of the microscopic phenomena which generate the noise, we can ignore the temporal autocorrelation of the noise and suppose that the the processes 1)(0 and b(s) are s; thus C(t,$) = C06(t — s). statistically independent if t

Dynamical systems with noise The presence of external noise, if it is weak, does not invalidate the description of the evolution of the physical system. S via the evolution of a small number of global quantities, whose variable is the time t. Thus, by introducing some effective stochastic 100 See Nauenberg and Rudnick [1981], Huberman and Zisook [1981], Feigenbaum [19794

[19804 [1987], Kai [1981] for an analytic approach. m See Oppenheim and Schafer [1989], Charbit [1990 ] or Delmas [1991].

Dynamical systems and chaos

156

terms in the equations, we can remain in the framework of the theory of dynamical systems. If the noise is very weak, a linearization around the initial equation reduces its influence to that of an additive stochastic perturbation of amplitude c <1, which depends on time (respectively t or n in the continuous or discrete descriptions) and possibly on the instantaneous state of 8 (see Arnold [1971]):

[dX / dt](t) = F [X tl + 0[Xt, b1 , t]

where

xn+i = f (x.) + eb y,(s r„n)

Chaos and noise Noise is not necessary for the appearance of deterministic chaos, quite the contrary: deterministic chaos is due to a very complex, very delicate and highly sensitive organization of the trajectories, so that it may even be destroyed rather than amplified by a stochastic perturbation . Since noise is unavoidable in experimental or numerical observations, the study of the influence of noise on the scenarios towards chaos is essential to identify these scenarios with reliability in an observed phenomenon (see figure 5.6 and supplement 5B),

attractor in t the phase space

(

control parameter p

p*

without noise

with noise

superposition

Figure 5.6 - The influence of noise on the doubling bifurcation Without noise, the pitchfork bifurcation (figure 2.2) occurs for a well-defined value p* where the branches of the 2-cycle come out of the branch of the fixed point with a vertical tangent, Noise destroys these two characteristics: the asymptotic regime varies continuously with p. In the third diagram the first two are superimposed for a better visual comparison.

The period-doubling scenario

5.2

5.2

157

The period-doubling scenario

With the notation introduced in § 2.2, we return once again to the study via renormalization of the period-doubling scenario associated with one-parameter families of the set .T of unimodal maps. The goal is to explicitly describe the self-similarity (f properties of the bifurcation scheme and of the attractor observed at p = p, Following this, we will consider the extension of the renormalization analysis to universality classes of different regularity (Collet and Eckmann [1984 Collet and Lesne [ 1989]).

5.2.1

Self similarity of the bifurcation scheme -

A remarkable property of the period-doubling scenario is the self similar tree-structure -

of the associated bifurcation scheme. It is obvious from the graph (figure 5.7), and analytically it is revealed by the action of the renormalization operator R which shifts the different "levels" 1 tin-}-11n >01 each of which is associated to a bifurcation at p = p, Indeed, if f has a 2's-cycle [xj] o <j <2 ., then Rf has a 2" -1 -cycle [x2j/Af] o 1. —

<> DETAILS

—

AND COMPLEMENTS: EXPRESSION OF THE SELF-SIMILARITY OF THE

SCHEME Let (ft,) 1, be a family of unimodal maps presenting an accumulation of doubling bifurcations at the values (pi)j>0. For j_.i < < ji,, let 4n) (p) denote the element nearest to O within the stable r-cycle; the labeling j s j( n) (..t) is further constructed according to x j(n)(p)-4 r[ .ro(TO(p)]. The function ft, belongs to the domain of definition Do of R if and only if A t, a- ft,(1)< 0, fi,()i 1,) > 0 and ft2t (A)< —Ap ; in this case we show that its fixed point x i,* is stable if Ai, > s*m . As fpn E D0 and s'hit.. is unstable whenever n > 1, it follows that etty, > A, (1); using the parity of ft, and the fact that it is decreasing on [0, 1], we also show that f() n )= for, (—) n ) >xi% > A, > (An ). If —A n , its image under the action of ft,„ satisfies fi (x)> fi (A y,)> —An . The result to keep in mind is that the elements of the 2n-cycle of fp , lie alternatively on each side of A n , those of even label lying in the interval [An , —An] and those of odd label in [fl,„ (AO, 1] C [m n , 1 1. Since the renormalized evolution law Rft,„ involves ft! rather than fon , the renormalization decimates the 2' cycle . of ft, n , preserving only the elements of even label. This is graphically reflected on the bifurcation scheme in a truncation of the picture limiting the field of observation to the interval [A n , —A n ] in which the subcycle (4 ) )0 <2<2 .-1 lies. The two scale changes completing the expression Rf(x) = (A n s)/An amount to transforming it by 2: (since An < 0) and then bring its extent back to the initial interval [-1, 1 ] . The result is a 2n -1-cycle of Rfi,„ which is almost identical to the (unique) stable 2n -1 -cycle of f_,: the two relations —

—

—

—

-

[Rfp

k

n

5 2j (An ) ( (71) A,

fi2 k ,[s (n)(AO] 2i An

S (27j+k )(pn ) and

Rf„„

158

Dynamical systems and chaos

show the approximate relation s (;; ) (An ) An4n-1) (pn- ), which becomes more and more precise as n increases. This approximate relation expresses the asymptotically exact similarity of the tree structure formed by successive bifurcations; the subtree coming out of the point [pn , x2(7 ) (p n )], dilated by a factor of ..\; 1 , coincides with the subtree coming (n)

out of [p n -1,41-1) (Pn--1)]; moreover it shows that the sign of s o (p n ) alternates:

(-1)ns oe2) (pn ) > 0. Setting dr, = x(0" ) (tin ) - x (272_,(14,), we obtain the relation dt, dn _i, which leads to limn _400 dn idn-i = 9(1) = 0,3095.,. -

The preceding results can be transposed to the tree whose vertices are the elements of the cycles containing x=0 so also x=1. To prove the existence of such a 2-cycle for all n> 1, we introduce the manifold So = If E 1, Ai = 0} consisting of the functions whose stable 2-cycle is given by the pair {0, 1}. In the generic one-parameter family (fp ) p , there exists

ES

ER-i (So); the

vanishing of

ensures that the 2j+ 1-cycle of

[Ri fp](1)= (Ri fm ) 2 (0) =

2 3÷i

(0)/ f p (0) at

f1 i is the orbit of xo =0. As xl =1 and x2 = A m , , this cycle

reaches the endpoints of the image [, n of

It is stable since Pm (0) = 0 implies that

the derivative of f2 ,3 vanishes at each of its points. Such a cycle containing xo = 0 is called Pi which superstable. Since the attractor of fp, is a stable 2j+ I -cycle, we have Afi Jpi

_ 1 , [,

(tei )j>0 converges to pc at the universal rate 8. The sequence of [d'i =x(1 ) (A"i )-x (2)_., -1,(gri )li>1 obeys the same scaling law as the sequence [d1], >1.

proves that the sequence

distances The proof is based on the correspondence via renormalization of the cycles of expressed by the relation

fm ji, and fp _ ,

Rfpii

Beyond the critical value p

pc

The structure of the bifurcation scheme above the critical value A c , determined by the condition fil c E V', is extremely intricate, as can be seen from figure 2.3. Let us indicate some remarkable results. — The envelope L(p) of the Lyapunov exponent L(p) of fp is positive above p c , but we observe uncountably many values p, associated to stable cycles, a fortiori such that L(p) < 0 (figure 5.3). — There exists a sequence (iii)j >1 decreasing towards p c at the universal rate > 1 and such that the attractor of f is a cycle of periodl°2 3.2; f describes a -

-

manifold

Y of the form Y =

S6 so that i" RI-

— There exists another sequence and such that the invariant measure of

Lebesgue measure; one can show that f so that RI-- •••=, f-Pi Pi-1

Pi

.

f;.

also decreasing towards pc at the rate is absolutely continuous with respect to

WO where U = (f, f 3 (1) 1 f (1) = 01, - -

102similarly, for every integer k, there exists a sequence of parameters, decreasing towards tt. at the universal rate 15 and associated to cycles of periods [k. 2 13>0103 Approximate relations such as Rfi4 are asymptotically exact after an adequate projection onto the unstable direction of

DR().

5.2

The period-doubling scenario

159

Cl

Al

1

5. C7

B3. 03

d1 0.5

C4' BOP

CO 0

cer CZ

-0.5

,

0.5

,

P0

1

I

Pi

P2 P3 15

control parameter f_L

Figure 5.7 - Self-similarity in the period-doubling scenario The bifurcation scheme of the family [f/A(s) = 1 — P:c 2 j00 is observed. The numerical error has the effect of a noise and allows us to observe only the four first bifurcations. The associated periodic attractors are labeled from their element nearest to 0, alternatively positive (0 and Bo) and negative (A0 and Co). We construct d1= A0 A1, d2= B0B2, d3= C0C4 and generally dri= 2:0(P0-5 2n - 1(pri); this distance is algebraic and has the sign (-1r Its asymptotic behavior is given by limn ,,,„ d n id n _1=z.(1)= —0, 3995 ... The self-similarity of the bifurcation scheme is expressed on the subtrees [MNP] relating the vertex M to the points N and P: [A0 C2 C6] — A i x [0.130 .132] and [B0C0C4] ,--, .A2 X [A0B2)0B2] where .

160

Dynamical systems and chaos

— If p < pc , then fo has a stable cycle whose period T(fi,), which is equal to 2n if pri_ i < p< p,,„ satisfies the scaling law T(f) (p e — fer" with v= log 2/ log S. Above pc , the attractor of ft, is chaotic for infinitely many values of it, at which it has a structure of T(fti ) bands, We observe an inverse cascade, associated to a sequence (A,,), >no which decreases towards pc , such that at fi n an attractor with 2m -1 bands (for p>11,•,) becomes an attractor with 2' bands. Using the relations L(R f0 )=2444 ) and T(R4 4 ) = P(44 )/2, we can show that ran converges towards pc at the rate 6; we can also analytically prove the validity of the scaling laws L(p.) ,.., (p p c )' and (p — p e r and compute v' = log 2/ log 6 = v. —

<> DETAILS AND COMPLEMENTS: PROPERTIES' " OF THE FIXED POINT ic" OF

R

so is the only non-trivial analytic map whose graph and the graphs of whose iterates (ç0 2n ) n).0 are exactly self-similar near so = 0, with 9(0) = 1; this expresses the invariance Rso= 9o. Using (R90) k = sok with k = r_1, we can show by induction that 4„o2n (0) = An where A = 2 (0)=9(1)< 0. The fixed-point equation R'9o=9:7 can thus be explicitly written c)2f1 (..\ 11 .z)= A" so(x) for all n > 1; it ensures that the graph of SO2n , with abscissa restricted in each direction, coincides with the graph to [HAI", IA11 and dilated by a factor of A of 9p. We do not know the exact expression for 40 but only the first terms of its analytic expansion at s = 0 which, plugged into DR(,), allow us to compute the unstable eigenvalue > 1 and the associated eigenvector, giving the tangent at ça to the unstable manifold V' of R at so. This is how we compute the value 6 = 4,66920 ... The trajectory [so k (0)]k>0 presents remarkable properties of similarity, deduced from the identities (F09a) k = 9o k , from lies in the fact that which we see that 92nk (0) = sok (0). The importance of this function the asymptotic properties of every element f E V ((p) are similar to those of ça due to its convergence towards ça under the action of R.

5.2.2

Self-similarity of the critical attractor

Figure 5.8 explains the symbolic construction of the attractor A, of fpc at the onset of chaos; it displays the structure of a Cantor set, universal since it is identical for all of the stable manifold V (so). Its lacunary structure means that A c the elements has Lebesgue measure zero. The ergodic invariant measure tn, of support A, is thus singular with respect to the Lebesgue measure. A, is not chaotic since the Lyapunov exponent L(f) is zero.

0.

DETAILS AND COMPLEMENTS: CONSTRUCTION OF THE ATTRACTOR

A,

The two numerical methods used to construct the attractor A, are presented in supplement 5A (figures 5A.1 and 5A.2); we give here the analytic proof of its fractal structure. By definition, A c is invariant, closed and indecomposable, it is thus a union of dense trajectories. By passing to the limit p. pc in a subfamily [fp ,ji >0 whose elements have the orbit of so = 0 as their limit cycle, we show that A, contains so = 0, so it also contains its trajectory (sk)k>0 under the action of fpc . A, is constructed by taking the 104 See

Campanino and Epstein [1981], Epstein and Lascoux [1981] and Epstein [1986 ] .

5.2

The period-doubling scenario

161

closure of this trajectory. In particular, it contains the endpoints 5 1 = 1 and s2 = ..1 1,„ of the image f,([ -1 , l])=24 0„ 1]. lf x > 0 denotes its fixed point, foc exchanges [x* , 1 ] and [A, x*], so the iterates of even label are smaller than s * and those of odd label are greater than z * . The critical function foc (at the onset of chaos) belongs to the stable manifold of the renormalization operator R, and it is in the domain of definition of all the iterates and so also in that of [Rn fil ](1) < 0. Consequently, 52 r has the same sign as (-1) 1 . The step-by-step construction of A, is sketched in figure 5.8.

n=

1

2

zo

4

x• 3

4

x* 3

1

thl

n=2

2

6

8 zo

2 10 14

6

8 zo 1612 4

7

5

1

n=3

Figure

x °3 11 16 7

5 13 9

1

5.8 - Symbolic construction of the critical attractor A,

The attractor A, of fp, c and the associated ergodic invariant measure m c are constructed by recursion. The critical point 50 = 0 and thus its trajectory [5k = ft, c (x0)]k>0 belong to A c• At the step n, we construct (in boldface)

r disjoint intervals

1(1»=(erk,x k+2.). They are exchanged by the evolution

4n) ,

since foc [4.°) ] = 421 and f?2,:[.4n) ] = so they are visited with the sa.me frequency by the trajectories and thus they must have the same measure

rn, (4,n) ) = 2—n . the limit A

=

.4n) is replaced by ri(cri+1) U Iin++2 1). A,

At the next step,

is

Un i /In) of this recursive procedure.

=

In figure 5.8 we perceive the lacunary and self-similar structure of A, analogous to the shape of a Cantor set. The analytic determination of the ergodic invariant measure m, describing the asymptotic dynamics of foç is based on a corollary of Birkhoff's ergodic theorem: the measure m(B) of a subset B is equal to the visiting frequency of a trajectory in B. The different intervals [4(n) ] 1
,

and valid for every element of the stable manifold V (9o).

162

Dynamical systems and chaos

5.2.3 R,enormalization analysis

renormalization procedure presented in §2.2.4 is not the only possible one. A first variant 105 uses the operator Ro acting on the elements f of the same space T via [Roflis) = a[f o Asia)] where a = —1/9o(1). This operator Ro has the same fixed points f(x) E- 0, f(x) s and so as R. Its disadvantage is that it depends on the

The

—

unknown coefficient a: a preliminary step must be to prove that there exists a solution of the equation ça= R(p, i.e. a fixed point, and to determine the value = —1/a. The advantage of Rg is that its action on f is simpler, which facilitates its linear analysis and makes it better adapted to the study of the self-similarity properties of the bifurcation scheme. A second variant 106 is based on the similarity of the branches of the bifurcation scheme located closest to x=1. It introduces the operator (Tf)(x)=a7 2 [f o f(a.2f x b1 ) — b 1] where bf >0 is the positive pre-image of 0 (f(bf)= 0) and a3 =1 — b1 . The operator T has fixed points,f(s)E 1, f(X)E x and a hyperbolic fixed point 7,b (different from 0). As above, we could also consider an operator T o where a and b have the (fixed) values associated to the fixed point t,b of T. This renormalization depends on two coefficients a and b, making, it technically more difficult, which is why it was not pursued.

Other universality classes A transition towards chaos by accumulation of period-doublings can also be observed for families which do not satisfy the same regularity properties at x = 0. It is possible to apply the previous analysis by letting the same operator R act in other classes [F,],>0 defined as follows:

{f(x)

F(t) where t =12:11 +' , (f, F) satisfies (i), (ii) and (iii)}

(i) fa 1,1D C [ Li.] with f(0) = I -

—

and 1(1)

E—

(ii) 0 is the only critical point f in [-1, 1] (such that r(0) = 0) ; (iii) f has a negative Schwartzian derivative in [-1,1] — 0. These three properties ensure that, generically speaking, the one-parameter families follow the period-doubling scenario. A typical family of .7; is as 1 plx1 1+ED1
(iv) F is analytic in a complex neighborhood B = B(0, r) of [0,1] and F restricted to [0,1] takes real values; F(0) = 1 and F(1) E] — 1, 0[ ; (y) P(t) < 0 if E [0,1]; (iv) and (v) imply (i) and (ii);

(vi) F (B[ F(1)] 1 +9 c B(0, ro)---] oo, 0] where ro < r, which ensures l°7 that Rf induces an analytic function in B and satisfies (ii); —

variant actually was introduced before the procedure described previously, by Feigenbaum [1977], [1980a). 1°e This is the renormalization introduced by Coullet and Tresser [1978a,b], [1980]. 1°7 B is the topological closure of B, i.e.. the smallest closed set containing B. 105 This

5.2

The period-doubling scenario

163

(vii) O 1 describing the accumulation of bifurcation values (pi)j >0 at p c . This value b e is identical for all the one-parameter families transverse to in this sense, the transition towards chaos is universal in FE . The sets [.7-4,>0 appear as universality classes of the period-doubling scenario. The point to keep in mind is that in the case of dynamical systems, the universality class is determined by the regularity of the functions. The Holder exponent (1+c) here plays the same rote as the dimension of the space in critical phase transitions (§4.3.4). The various universal characteristics, such as 8, > 1, the associated eigenvector or the critical manifold V:, can be (approximately) computed by numerical means. When 0 < 1 — e < 1, a perturbative procedure is possible starting from the results obtained for E = 1. Similarly, an analytic perturbative computation works well if e oo or if 0 < e < 1; in this last case we can show the convergence of the expansions obtained 108 .

DETAILS AND COMPLEMENTS: PERTURBATION IN

E

IF

0
The first step is to solve 16,o, = soc ; the existence and the uniqueness (ignoring the constant function equal to 1) of a solution and the fact that it belongs to V, are proved by topological arguments. Under the hypothesis 0 < e < 1, its expression is obtained in the form of an analytic expansion converging normally in the ball B(0, r) of the complex plane whenever rkoE (1)I < 1. The computations lead first to the (real and negative) value of = $o,(1), related to c by:

c = A,

[1

log

Rt,1] -1- [1+ 00 ()]

=

= doge + 0(e) < 0

„N E ) + E A 1 t(1 0/2 0(41 t)r) 3,) follows. The linear The expansion $0, (t) = 1 analysis of Dii(io„) establishes the hyperbolicity of the fixed point: all the eigenvalues are stable, of modulus bounded by !Ad < 1, except for one of them, given by the expansion: —

—

be = 2

—

—

—

e/A, e/2+0()i); ; 2 -

This is the eigenvalue be which is the universal number describing the accumulation of the bifurcation values of the period-doubling scenario in J. The linear analysis can be pursued by determining the expansion of the eigenfunctions. The analysis is then extended to the nonlinear action of the renormalization operator by determining the equation (in the approximate form of a perturbative expansion) of the stable and unstable local manifolds of R at pc .

'See Collet and Eckmann [1980] or Collet and Lesne [1989].

Dynamical systems and chaos

164 5.3 5.3.1

The intermittency scenario Description of the scenario

This paragraph is devoted to the renormalization analysis of another scenario towards chaos, namely the intermittency scenario. This term designates a disordered alternation of regular and predictable phases i°9 and of chaotic phases. The explanation Ax 2), we propose here uses the family of discrete evolution laws [x f o (x)=—p+ (with A> 0) as the typical situation.

<> DETAILS AND COMPLEMENTS: REDUCTION TO THE NORMAL FORM

+ x Ax 2

The first step is to take a Poincaré section of the continuous flow, lying in a space of dimension at least 3 and controlled by a parameter Y. If the Poincaré map F, has astable fixed point for y < 0 which loses its stability at y = 0 when its maximal eigenvalue passes through +1, then for v>0 near enough to 0, the other eigenvalues remain of modulus less than 1 by continuity. It is thus legitimate to project F2, in order to eliminate its contracting components, which are inessential in the chaotic properties since they asymptotically disappear; this = in R. Using projection reduces the study to that of an effective evolution p as well as an adequate conjugacy variable change z —÷ s and parameter change = °m o 0; 1 , we eliminate the terms of degree higher than 2 and reduce to the case — Ax 2 with A> O. f = —p .

0 The saddle-node bifurcation The explanation of intermittency uses the following type of bifurcation. For p < 0, fp has two fixed points x = TV pl A; 0+ is stable (if p> 1/A), x; is unstable and they coalesce at p 0; for p > 0, there is no fixed point. This bifurcation is generic and is known as a saddle-node bifurcation (App. 11.1). In the neighborhood of s = 0 and for p > 0, the graph of fo remains very close to the first diagonal y = x: these two curves determine a channel in which the dynamics are very slow because of the weak values taken by the variation Ax = f(s) — x over a time step. Figure 5.9 shows that the discrete trajectory x,+1 = fo (x,,) is trapped in the channel: the evolution is slow, monotone (xn+ i < xn, if the graph of f is under the bissector) and thus predictable. —

—

Intermittence and chaos The major deficiency of this scenario is that it describes only the regular phases of the evolution. The chaotic characteristics are exactly those of the parts of trajectories located outside the zone of the phase space described above. To observe the phenomenon of intermittency effectively:

— the dynamics must re-inject the system into the channel so that several regular successive phases can be observed; this property can be ensured by non-linear terms whose influence is negligible in the channel but dominates at infinity; it can also 109 Called

iarninary, with reference to non turbulent flows in hydrodynamics. -

5.3

The intermittency scenario

165

follow from the influence of components eliminated from the reduced one-dimensional description (which is only valid in the neighborhood of the channel) but which become essential once the trajectory leaves the channel; — the evolution must present marked chaotic properties outside of the channel to clearly distinguish between two types of behavior; the flow must have no other attractors, otherwise we could observe an irreversible convergence towards one of them rather than the expected intermittency. —

This local and incomplete scenario thus describes, more the persistence of the slow monotone phases in a globally chaotic evolution than the actual appearance of chaos in a regular and well-controlled evolution. DETAILS AND COMPLEMENTS: INTERMITTENCE OF TYPES I, II AND III

Several scenarios can lead to (temporally) intermittent phenomena: three different types of intermittency have been introduced. Here we only study type I, the earliest and most studied of the three, whose physical relevance is well-established. Type II has never yet been observed and is thus to date only a theoretical scenario. Type III makes use of the normal form f(s) = (1 + p)x ax2 bx 3 or a 2 b < O. A renormalization analysis analogous to the one designed for type I intermittency (of quadratic normal form) is possible, but the presence of the cubic term, which becomes essential, makes it technically more difficult (see Schuster [1984]) . —

0

gp(x)

Figure 5.9 - Temporal intermittency The boldface curve is the graph of a map g 0 parametrized by p and conjugate to the normal form fo (x) = —p-l-s—s 2 . The diagram shows that the discrete evolution 2:,,+/ = Mx-0 is monotone and slow if p is slightly greater than a value Po associated to a saddle-node bifurcation (40 (0) +1).

166

Dynamical systems and chaos

5.3.2 Renormalization analysis We suppose henceforth that there exists a non-linear mechanism of re-injection into the channel, which ensures that a temporal intermittency phenomenon is observable. The point of re-injection, depending on the chaotic evolution outside of the channel, is very variable and unpredictable. Since it determines the number of steps performed by the system in the channel, or equivalently the time which the system passes in the channel, these are random variables. The goal of the renormalization analysis is to determine explicitly the scaling law obeyed by the average duration r(fo ) of the regular phases (or the average number N(f o ) of steps spent in the channel) as a function of the distance between the control parameter p and its bifurcation value po =0, then to prove its universal character. The basic idea is to consider double "macro-steps" since the time spent in the channel is twice as short if the trajectory is generated by n. The adequate renormalization operator is thus:

(ftxf)(x) =

f o f(xA)

The scaling factor À is obtained by identifying the coefficient of the monomial s in f and R.,xf in order to make RAf as similar as possible to f. A qualitative result is obtained with the typical family [f(x)= -1 + x sl o , the factor A taking the value lead to the estimation 1/2. The two relations N(fo ) = 2N(R 1 12fo ) and R1124,

N(fm) 'T(P)"11 /VFA,

Renormahzation equations and fixed points To prove rigorously the universality of the result obtained in this way, we must find the fixed points of R E Rip. We check that [0a (s) = s/(1 + ax)] a>0 gives a family of solutions. The associated linear operators are:

[DR(04.v1(s) = 2 v Their

x

2 + as

1 v (r)

2 2

(2+ ax + ax )

eigenvalues are [-yr, = 2 2- 'jri > 0 , with eigenvectors given by: =

x'

1 (1 f.

ax) -2 [(1 + ax) 3- n

1]

A parametrized family [go], of evolution laws has po as a bifurcation value if gp 0 belongs to the stable manifold of a fixed point in which case we can expand

=

E.D cn(p) e a,n G

n

with the condition that ci (g o ) = 0 where j labels the unstable (j = 0, 1) or marginal (j = 2) directions of DR(0a ) and where Go appears as a negligible term with respect to the action of R. The condition of transversality is that (dco/dp)(po) 0 0. Keeping only the leading linear terms, the action of R on gp is given by:

Rg A = Oa+

E n=0

cn (p)e„,„ + • •

The intermittency scenario

5.3

167

Ri(g00 )=0a , which is what we obtain here By hypothesis on po , we must have 1im since c( o ) = 0 if 17,1 > 1. One can show"' that g, has a saddle-node bifurcation at the value po. Expanding co (po) around po gives the relation Rg g r (B ) or r(p) p o = (p • • to the leading order. Note that the value of (dc o ldp)(11, 0 ) is not involved here. The relation 2N (Rg)= N (g A is rewritten 1V(g oo+1) 2N(g p0+1,n ), which proves the scaling law: —

—

',t o)»

)

1V(gm« (P ,— Por'

with u

= log2/ log

-

= 1/2

(70 = 4)

Renormalization gives the value of this exponent I) and proves its universality since the value obtained does not depend either on the transverse family considered nor on its parametrization. On the other hand, renormalization does not give the probability distribution of the time spent in the channel, which depends on the complete dynamics of the system and on its specific details.

<>

DETAILS AND COMPLEMENTS: OTHER UNIVERSALITY CLASSES

As in the case of the period-doubling scenario, it is possible to use the same renormalization analysis in other classes corresponding to the different regularity properties of the evolution laws. The typical family becomes:

fo.cW= z — p — (4 1 1 +

e>0

The adequate scaling factor is A, = 2 -1 /f. In each class, the suitable renormalization operator R.), has fixed points (0,, a ). ).0. The functional sets indexed by the Header exponent 1 e thus appear as universality classes of the intermittency scenario. The fixed points are given explicitly by: çt = sEl — fa sgn,WWei- i f'. As in the case e = 1 above, the linear analysis of the renormalization in the neighborhood of a fixed point gives the average duration r(, JA) of the regular phases for a transverse family having u = O as bifurcation value : fr e/i+E . r ( f,A )

5.3.3

The influence of noise

The influence of noiselil on the intermittency scenario is modeled by adding an effective stochastic term c- bu of amplitude a to the evolution law:

xn+ i =

f p , c1 (X. rg) = f p ( XII)

ab n

Purely random noise The first case is that of noise whose temporal autocorrelation, of range much smaller than the duration of a step, can be neglected in the discrete description. 11 0 A simplified proof consists in plugging the values 0. (0 ) = 0 and q51.(0) = 0 into the expansion of goo (ensuring that the fixed point z = 0 of O. is the location of a saddle-node bifurcation) then the relations e,,(0) = 0 (for n > I) and e,„(0) = 0 (for n > 2); we can then conclude by looking at the lowest order terms since co(po)= ci (R) = O. "See 5.1.7 for a general description of noise in dynamical systems.

Dynamical systems and chaos

168

The noise, of external origin, is often statistically stationary and uniform in the phase space on the scales of the description. The terms (b„)„ are then independent random variables; they are identical, centered and of variance normalized to 1, so that the amplitude of the noise is a. Formally replacing the discrete evolution law by a stochastic differential equation, one can show' that intermittency appears for a413 with respect to the value in the a value /4 of the parameter shifted by Apt absence of noise. The average time r spent in the channel obeys a scaling law of the form:

Ang -413] where the function 0 behaves like e(z) 1/ N5 at infinity, so that we recover in the limit as a- —4 0 the scaling law r 1/ N/i/-•,/*' in the absence of noise.

Quasi-periodic noise In this case, the temporal autocorrelation is such that the random character of the perturbation can be modeled by an additional dependence, whose (angular) variable 0 evolves in a deterministic manner:

= fm (x) crb(s „, = 0, + w o-g(x, , 6+7 )

w

27Q

measures the amplitude of the noise. The periodicity 27r/co of the noise is adapted to the discrete evolution only if w E 27rCt; in the case u.) 27rQ considered here, the noise is said to be quasi-periodic and the sequence of its values appears random over quite a long duration since it is uniformly distributed on S I . The functions b and g are 2r-periodic in 9. One can show that intermittency appears for a value of the parameter /4 which is shifted by Ap2 cr2 with respect to the value in the absence of noise; the average duration r of the regular phases obeys a scaling law of the form:

(P,

a - ARP

P;) /172]

where the universal function A behaves like A(z) 1/Vi at infinity, so that we recover the scaling law in the absence of noise in the limit as a —) O. The different approaches, each limited to one type of noise, can be unified by adapting the formalism for period-doubling explained in supplement 5B to the situation of intermittency.

5.4

The scenario of Ruelle, Takens and Newhouse

5.4.1 quasi-periodic attractors and strange attractors Quasi-periodic motion The observation of a pendulum makes the notion of periodically oscillating motion intuitively clear; similarly, a typical quasi-periodic motion is seen in a pair of independent harmonic oscillators with incommensurable frequenciesi 13 . The term 112 "3

See Eckinann et al. [1981] and Hirsch et a/. [1982]. i .e., whose proportion wi /w2 is irrational; w i Z + co2 Z is then dense in R.

The scenario of Ruelle, Tokens and Newhouse

5.4

169

"quasi-periodic" describes any motion resulting from the superposition of k periodic motions (or "modes") of frequencies wi, w2 .. wk. Replacing the possible pairs (wi = pc, ', w5 gui, p and q integers) by their common subharmonic w' reduces the situation to that of rationally independent frequencies. As the evolution of the angular variable ei(t) 9i(0) 1 wjt (modulo 27) suffices to represent the j-th mode, the quasi-periodic motion is described by a function 0[0 1 (t), 62 (t). 9k (t)] which is 2ar-periodic in each of its arguments, and its invariant measure is drn(0) = d91 d92 . . d9. Its trajectories can be expressed in angular coordinates by (9k, 92,..., Ok), forming a torus Tk=(s1)xk round which they densely wind. lithe frequencies become rationally dependent, the trajectories become dense trajectories on a torus of dimension smaller than k. To complete the trilogy of scenarios towards chaos in dissipative dynamical systems, we will study here the stability of a quasi-periodic attractor. Since it is obtained by superposition and not by coupling of periodic motions of pairwise incommensurable frequencies (wi)i, we can already see that it will be destroyed by the slightest non-linear coupling. The quasi-periodic motions can also appear in conservative systems; the k angular variables are then those of a set of action-angle coordinates adapted to the Hamiltonian system of dimension 2k, The k invariants of the system are the k frequencies. The study of the destabilization of these motions, based on the KAM theorem and the renormalization of circle mappings, is the object of § 5.5. -

- -

Strange attractors The rise of the notion of deterministic chaos, distinct from both stochastic evolutions and from turbulent behavior involving an infinity of unstable modes, is historically related to the experimental and numerical discovery of more complex attractors than the quasi-periodic ones 114 , called strange attractors on account of their particular properties. The notion of a strange attractor developed into a precise definition: firstly, it is an attractor, i.e. a closed, invariant, indecomposable set containing the closure of the trajectories coming out of an open set containing it; secondly, the adjective "strange" applies if the restricted dynamics are chaotic, in the sense that they have the property that neighboring trajectories diverge exponentially fast and the ensuing property of unpredictability. Thus it is a tautology to say that a dynamical system is chaotic if it has a strange attractor!

5.4.2

The theorem of Ruelle and Takens

A theorem proved in 1971 by Ruelle and Takens and completed in 1978 by Newhouse, Ruelle and Takens, states that the destabilization of a quasi-periodic motion is typically accompanied by the appearance of a strange attractor (Ott [1981], Eckmann and Ruelle [1985]):

Every neighborhood (an a sense which we will make precise) of a quasi-periodic motion on a torus Tk of dimension k > 3 contains a strange attractor. 'For example, the attractor observed in the chemica1 reaction of Belousov-Zhabotinskii

and that resulting from the numerical study of the Lorenz model ( 5.1.3).

Dynamical systems and chaos

170

We emphasize that this theorem does not give a condition on the dimension d of the phase space but on the number k of "excited modes", i.e. of periodic motions of incommensurable frequencies present in the asymptotic dynamics (so not damped). It states the non-generic character of the quasi-periodic attractors and their sensitivity to the slightest non-linear perturbation. This sensitivity can be foreseen since the definition of a quasi-periodic motion involves superposition and not coupling of periodic motion: a non-linear coupling between the components of frequencies wi and wi generates harmonics pwi + qwi (with p and q integral) which form a dense set in R and destroys the quasi-periodicity of the attractor. An equivalent statement of the theorem is:

The sel of vector fields on the torus Tk (k > 3) having a strange attractor is an open set of the C 2 topology (the CD° topology if k > 4), whose closure contains the constant vector fields having a quasi-periodic attractor. This shows that the presence of a strange attractor is a robust, and thus a physically observable property. The weakness of the theorem of Ruelle and Takens is that it states a possible, but not necessary, result: it does not give sufficient conditions for a transition from quasi-periodic to chaotic dynamics on the strange attractor to actually occur.

The scenario of Ruelle and Takens The preceding theorem is the basis of the scenario of Ruelle and Takens, which extends it by describing how to pass from a stationary state (fixed point) to a quasiperiodic attractor to which the theorem applies. The typical mechanism, induced by a variation of the control parameter, is a succession of three Hopf bifurcationsi 15 , each of which causes an oscillating mode to appear in the asymptotic dynamics. After the third one, the asymptotic motion is quasi-periodic with three frequencies; if these frequencies are incommensurable, the attractor can destabilize and become replaced by one of the strange attractors whose abundance and proximity are described by the theorem of Ruelle and Takens. Otherwise, a fourth Hopf bifurcation is necessary to lead to the same result. Although the destabilization is generic, it is not actually necessary, which limits the predictive power of this scenario (Grebogi et al. [1983]). <> DETAILS AND COMPLEMENTS: THE APPROACH VIA RENORMALIZATION

In this situation, the generic situation preceding the appearance of chaos is a quasiperiodic motion with three incommensurable frequencies (col, co2, w3) of proportions:

C1 1 =

(4.01/4.03) 0 Q

= (W2/W3)

Q

The continuous flow t (91,o +t wl , 92,o -Ft W2, 93,0 -Ft W3) (of dimension d = 3) is replaced in the Poincaré section 03 = const by the discrete fl ow:

( 9 1, 9 2) " (91 +2iri, 92 + 2702) 115 See App. 11.2; this scenario thus requires that d > d, = 6, which confirms that the threshold equal to 3 concerns, not the dimension d of the phase space, but the dimension (k < d) of the attractor.

5.5

KAM theory and Hamiltonian chaos

171

The destabilization of the continuous asymptotic motion can be studied on the perturbed discrete evolutions of the form:

( 01 + 27rni + Ft.h.(9 1, 02) 92 + 271- n2 + Pf2 ( 9 i 92) where the parameter it controls the amplitude of the perturbation. To obtain universal characteristics of the scenario, one can develop renormalization arguments generalizing those used in the study of the destabilization of circle mappings (§5C.3). A delicate extension is however necessary here, because two diffeomorphisms (the two components of F4 ) are involved. We refer the reader to the research of Braalcsma et al. [1990] in this way.

5.5

KAM theory and Hamiltonian chaos

Let us now consider the application of renormalization to the study of the appearance of chaos in conservative dynamical systems, whose evolution is described by a Hamiltonian. Via various reduction procedures, it is possible to reduce their study to that of area-preserving maps (in the plane). The transition towards chaos is described by looking at the discrete evolutions generated by these maps via Moser's "twist" theorem, which extends the results on the stability of integrable systems given by the "KAM theorem". We then use the analysis via renormalization of circle mappings, given in detail in supplement 5C, to obtain universal results on the transition.

5.5.1

The Hamiltonian formalism

A physical system S is said to be Hamiltonian if it has a system of 2N coordinates (,p), pi ] i
{(dq i ldt)(t) = (8H1Opin,2:51 ,t) j

1,...,N

(dpi I cli)(t) = — (81110qin,pt ,i) pj is called the momentum conjugate of the coordinate qj , and Liouville 's theorem states that the evolution preserves the volume element di dN (i of the phase space of S. Note that this space must be of even dimension 2N; N is called the number of mechanical degrees of freedom 116 of S. The function H(qt , pt , t) is the Hamiltonian of S; it can be physically interpreted as its total energy and it is easy to check that dmdt. The system S is called conservative if its energy remains constant throughout the evolution, which is equivalent to the fact that the Hamiltonian does not explicitly depend on time (i.e. mat=o); the trajectories then lie on a manifold {14,.25)= fro } of dimension 2N — 1.

mot=

With reference to the harmonic oscillator of Hamiltonian II(g,p) = p2 /2m. mq2 /2, which has a single degree of freedom since the motion takes place in a single direction. 116

172

Dynamical systems and chaos

Canonical transformations and action-angle coordinates

A canonical transformation is a diffeomorphism which sends the pair /5) adapted the Hamiltonian H to another pair (e), P) adapted to another Hamiltonian H' and

to leading to the same equations of motion. A necessary and sufficient condition is that the change of coordinates is given via a generating function cD(4, P, t) according to:

= 1904, Qi = aom,

= 1, . . . , N)

H = +Not

canonical transformation preserves the volume: dvi5dN4 = dN PdN O. In conservative systems, we restrict ourselves to generating functions .1 independent of the time t, so as to have H' = const. A particular class of canonical transformations exists for systems having k 1 > invariants of the motion besides H. The system is called integrable if k = N; in this hv which, case one can show that there exists N constants of the motion II. , via a canonical transformation (preserving H), may be taken as new impulsions. Let çoi , soiv denote their conjugate variables. The fact that (15)1<j
A

—

= all

in.;

wj (1-1,..., IN )

const

=1,...,N

Integrating them gives (pi (t) = 9oi (0) + wi t. The coordinates (II. , ..., IN ) are called the action variables , .. ,9oiv ) the associated angular variables. For a partially integrable system (with only k < N invariants), the transformation involves only a subfamily of 2k coordinates. If N > 1, the trajectories of a partially integrable system lie on a surface of dimension 2N — k < 2N — 1; thus these systems are non-generic in the set of conservative systems with N degrees of freedom.

5.5.2

The KAM theorem and Moser's "twist" theorem

The notion of Hamiltonian chaos lies within the notion of deterministic chaos presented in §5.1.3; its specific property is that it involves the entire phase space X=1(4,15)) of dimension 2N, since the evolution generated by the Hamiltonian H(q,p) preserves the volume element dyn,(, 13) = dNpd. The invariant measure which occurs in certain criteria for the existence of chaos is in this case just the natural measure mL. The conservative systems of dimension 2 (so that N = 1) are all integrable since their Hamiltonian H is the required invariant; their trajectories are described by 0(t) = 00 w(H)t (modulo 1), so no chaos is possible. Thus in order to observe chaos, we must be in dimension at least 4 (N > 2). Starting with the type of system farthest from chaos, namely an integrable system So in dimension 4, we will study the appearance of Hamiltonian chaos when we add a generic perturbation of amplitude c; this destroys the non-generic integrable character of the system. The trajectories lie on the surface 11 of constant energy Ho , so that the dimension of the "effective" phase space is equal to 3. An adequate Poincaré section of this three-dimensional flow brings us back to dynamics which are discrete and restricted to a cylinder S 1 x R, designed so that the integrable system we started

5.5

KAM theory and Hamiltonian chaos

173

with has evolution law:

F0 :

sO

( + w(x) (mod 1) ) x

("twist map")

The cylinder decomposes into circles z = const which are globally invariant under Fo . The energy 1/0 being definitively fixed, each of them is characterized by the rotation number (x) of the discrete trajectories supported on it.

The KAM theorem (Kolmogorov [1954], Moser [1962], Arnold [1963a,b]) The first result on the stability of the invariant curves of 110 is qualitative; since it is due successively to Kolmogorov, Moser and Arnold, it is called the _KAM theorem (KMA being unpronounceable). It concerns the maps F, constructed on a compact set K =S1 x [a, 1)] by perturbing Fo :

Fe :

( 9 ± w(x) ch(O , z) (mod 1) ) g(9, )

The functions g and h are one-periodic with respect to the variable 0, and they are assumed to be regular and bounded on K. The statement of the KAM theorem is as follows:

If F, preserves the area drnc10 and if does not vanish on [a, b], then for c 0, these curves A have a one-periodic parametrization on R.; on each of them, the evolution is generated by a diffeomorphism of the circle (or circle mapping) fA, i.e. a regular increasing map, specific to the curve A and such that fA(z + 1) = fA(z) + 1. The trajectories [frdti (z o )]„>0 are dense on A and the rotation number p(f) lirn 4 , n —l [f;i (z o )— z o ] (independent of z o ) is irrational. The notions of circle mapping f and of the associated rotation number p(f) are detailed in §5C.1; p(f) appears as the asymptotic average angular velocity of the evolution. An intuitive justification of this theorem is that the invariant curves of rational rotation number are composed of discrete finite trajectories, which reconstruct a continuous curve when juxtaposed; this situation requires a respective adjustment of the different finite trajectories, and is destroyed by the slightest perturbation. Conversely, if the rotation number is irrational, each trajectory lying on the invariant curve is dense on it and alone suffices to reconstitute it, even if a perturbation of the evolution slightly modi fi es its successive elements. The numerical study of the destabilization of the invariant circles of F0 when we add a non-integrable perturbation of amplitude c reveals a typical scenario, whose qualitative description is presented in figure 5.10: as c increases, the invariant -curves become deformed and some of them disappear. Analytically, the KAM theorem must be completed by explicitly determining the invariant curves which disappear and those which are preserved. A partial answer is given by Moser's theorem detailed below. It quantitatively defines a class of "very irrational" real numbers and states that the robustness of an invariant curve is determined by its belonging to this class, and not by the form of the perturbation.

174

Dynamical systems and chaos

(b)

(a)

e = e2 >

X

el

--------....„-C).../----

._.

CD

0 = 2ir

0=0

B =0

(c)

Figure 5.10 - Moser's "twist" theorem and Hamiltonian chaos The boldface curves represent some of the invariant curves of an areapreserving map F E in a compact set S1 x [a, b], sufficiently regular (C5 ) and of the form: F(9, x) = [0 + w(x) + ch(0 , r) (mod 1), s ± eg( 0 , r)].

Moser's twist theorem describes the scenario observed if infkbj- I w'( 0)1 > 0: (a) For e = 0, the invariant curves are circles x = xo = const, on which the restricted dynamics reduces to a rotation of angle w(x 0 ). (b) Whenever e > 0, the circles of rotation number w(x0) are destroyed if co(so) is rational, and differentiably deformed if co(x o ) is irrationa/. (c) As c increases, regular invariant curves disappear when the rotation number of the trajectories lying 011 them. is near a rational number. Chaotic zones appear but they are trapped between the remaining invariant closed curves. (d) Above a value e*, all the invariant curves are destabilized; the last ones to disappear are those whose rotation number is the golden number cr = (1,/ — 1)/2. The evolution is chaotic (i.e. highly sensitive to initial conditions) and the trajectories can cross the phase space from one end to the other in the direction of s; this property, known as Arnold's daffusion, is illustrated by the dotted trajectory.

KAM theory and Hamiltonian chaos

5.5

175

<> DETAILS AND COMPLEMENTS: MOSER'S THEOREM (Moser [1973)) Moser's "twist" theorem extends the KAM theorem under the additional hypothesis that w(x) is C 5 and that Ico l (x)1 > c > 0 on the compact set [a, I]. This theorem states that: There exists E » such that if e < e' if p E waa, satisfy1P—PN> for two real numbers C > 0 and a > 0 and for all integers p and q, then there exists a curve which is invariant under F,, diffeomorphic to an invariant circle of Fo and on which the restricted dynamics are conjugate to a rotation of angle p. The interpretation. of Moser's theorem in terms of chaos requires us to return to continuous systems of dimension 4. An invariant curve A, = { x A 6 (0),0 E S 1 } of F, corresponds to the Poincaré section {92 = const} of a torus parametrized by two angles 0 E S 1 and 02 E Si and lying on the hypersurface of constant energy fl (of dimension 3). These invariant surfaces of dimension 2 are called KAM tori when the trajectories which lie on them are dense. For era- 0, the dynamics on a KAM torus are generated by (0 , 02)-4. (0 ± f1 i (x), 02 ± n2(X)) where w(x) =.. RI M/S/2W is irrational. Even if > 0, one can still represent H using the variables (s,0,02)ERxS 1 x51 adapted to Fo. The RAM tori then divide up H into topologically compact zones, not correlated with each other, which bounds the length crossed in the direction of x. As e increases, certain KAM tori disappear = E * of and the average shift L, in the direction of s increases. The disappearance at the last torus corresponds to the transition towards chaos; arbitrarily distant points of the space can now become linked by a deterministic trajectory, and the ensuing divergence of L, gives this transition a critical aspect. The existence of such trajectories, of arbitrary extent in the direction x, makes the evolution unpredictable since it becomes impossible to locate the system at a given instant if we know only a neighborhood of its starting point, whereas when the extent of the trajectories remains finite, even an approximate initial information allows us to explicitly determine at least a bounded domain in which the trajectories lie.

5.5.3 Renormalization and universal properties Although it is more explicit than the KAM theorem, Moser's theorem describes only of F,, diffeomorphic to the invariant the persistence of the invariant curve circle Cw of F0 of rotation number w, as a function of the amplitude c of the nonintegrable perturbation and of its own rotation number, specific to the curve A,,„, and

denoted by Let us note that the parameter w is the frequency imposed while the rotation number p(e,w) is the actually observed frequency. Understanding the transition towards chaos requires rather the description of how and in what order the invariant curves A,,,„ become unstable. The deficiency of the first results motivates carrying on the analysis using renormalization tools to describe the universality of the transition towards chaos and the self-sintilarity of the dynamics at the onset of chaos. As long as the invariant curve A,,,„ remains diffeomorphic to C„ the discrete but dense trajectories lying on it are entirely described by the evolution of the angular component 0 E Si:

= A,,(0), 0 E SI} This function satisfies f e ,4,(9 + 1) = fe ,(0) + 1; it is thus a diffeomorphism of the circle Si when c is small enough, of rotation number p(e,w). Studying the 0

= 0 +w + eh(A 2 (0), 0) where A,,„, =

176

Dynamical systems and chaos

"breakdown" of A Fw , i.e. how it stops being diffeomorphic to C,„ is thus equivalent to studying the destabilization of the asymptotic angular dynamics generated by f,. Denjoy's theorem implies that if p(w,, e) is irrational and if infs f(0) >0, then fa, ,, is differentiably conjugate to the rotation 0 ■ • 9 + p(w,c); a possible destabilization can occur only if the diffeomorphism is critical, i.e. such that infs , t, e (9) = 0. The renormalization analysis of the asymptotic properties of critical circle mappings, detailed in supplement 5C, then gives the quantitative and complete description of the appearance of chaos. —

<> DETAILS AND COMPLEMENTS: THE "DEVIL'S STAIRCASES" The situation where p , is rational and equal to p I q is observed along a whole interval of values of w. It is called a (p, q) phase-locking, and corresponds to periodic, therefore regular and non-chaotic behavior of a cycle with q elements and p turns. The bigger the associated interval of values of w, the more robust it is (and so observable). The intervals of values of w for which the rotation number pt,„ is rational form the complement of the values of w for which each trajectory of f is dense on the invariant curve A. Phase-lockings and persistence of the KAM tori are thus two complementary asymptotic situations, which occur (for fixed e) irregularly and totally discontinuously with respect to the parameter w. The entanglement of the values of w which lead to irrational rotation numbers () cm with those leading respectively to irrational and rational rotation numbers is particularly complex in the case of a family of critical circle mappings, obtained for example when e =_- E. The graph of the rational values plq of p(co, co) as a function of w is called a devil's staircase: the steps, each associated to a rational value p q of p, have unequal height and depth (given by Aw(p/q)); the restriction of p to rational values can make it lacunary, in which case we

EPoi

have L1w(plq) < 1. In the contrary case, the staircase is called complete. It often has self-similarity properties.

Results of the renormahzation analysis For fixed c > 0, the invariant curves (all having a discrete, dense trajectory on them, with irrational rotation number according to the KAM theorem) can be grouped into sheaves 117 structured around curves I(c) with particular rotation numbers p, whose expansion as continued fractions "8 are periodic, i.e. p = [0; po, pi, P2 • • • Pk 'PO, pi .1. Among these, the ones whose expansion is of the form play a particular role, and especially the golden number cr = [0;1,1 [0; p,p, As f increases, the invariant curves become deformed, first •in a way which leaves them diffeomorphic to the circles {x = xo } they come from (at c = 0), i.e. they are still sections of a KAM torus. The restricted evolution generated by f,,„ remains monotonous and asymptotically identical to the rotation of angle p(w,, e) . They eventually disappear and the breakdown turns out to be a function of c and of the degree of irrationality of p(w,, c), quantified using its expansion as a continued fraction and the rational approximations given by this. At the moment when A,,,„ disappears, 1 ""Sheaves 118 The

of hoops' would be a better term. definition and properties of these expansions are detailed in §5C.2.

5.5

KAM theory and Hamiltonian chaos

177

= 0) is no longer monotonous; the notion of critical circle mapping (inf s., corresponds to this transition. Beyond it, we observe that the order and the orientation under the action of f, are not preserved, nor is the appearance of mixing mechanisms and of separation of neighboring trajectories. Consequently, the dynamics become unpredictable. This transition thus corresponds to the onset of chaotic characteristics which, however, remain localized in the phase space as long as other invariant curves associated to KAM tori still remain, since they behave like solid barriers. The sheaves [4:0pb, become thinner, until each of them contains only the particular curves 1; these then disappear and with them the last track of the sheaf Op. The curves Ii,, and consequently the associated sheaves (1)/) , disappear in a well-determined order, related to the irrationality of p. The last sheaf to disappear is the one associated with the golden number a = (1,/ — 1)/2, which appears as the most irrational real number of 10, 1[, i.e. the one most slowly approximated by a sequence of rationals. <> DETAILS AND COMPLEMENTS: EXPLICIT PROCEDURE Renormalization allows us to describe the universal properties of each invariant curve at the moment of its disappearance, then those of the scenario according to which the sheaf O p becomes progressively thinner until it contains only Ii,. The basic idea is that an adequate combination of iterations and of translations brings a circle mapping to the identity transformation. In practice, it is done in the space of critical circle mappings by constructing a renormalization operator for each periodic rotation number p = rn P2 • • • Pk pi, P2 • , o Rp.i . This has R p adapted to the arithmetic self-similarity of p: R ,, Rp • • o R a hyperbolic fixed point Op, appearing as a universal function related to the critical circle mappings of rotation number p: it describes the typical characteristics of 1f, when it breaks down. This fixed point Op has a single unstable direction associated to an eigenvalue 8p > 1 and a stable manifold 11,. One can then describe the universal behavior of families [fij i, of critical circle mappings transverse to The element corresponding to the onset of chaos is f G11,, which satisfies R pn [fpj - Op. One then shows that there exists a sequence [pi]j >0 such that:

..J

— limp= p; pc), f o belongs to the domain of definition V , = Dip ; belongs to the boundary 5V — for all

f,

p E

[D pi ] of Rip ;

,

Rp[f 03 ] E 87)-jp -1 ; if j is large enough, we write R p [ftki ] oo; — pi p (p, pi ) jf j this the asymptotic relation

we deduce from

—

— if p > pi, the rotation number p(p) of fo lies between qi and ql +1 where qi is the rational number whose expansion as a continued fraction consists of the first j integers of the expansion of p: 112i -1 G q2 j +1 < P < q2i+2 < q2j ; — consequently, lirni—ioo

p(p) P_ (p c

—

p)

= p; where cic

—21ogpl log

> 0.

178

Dynamical systems and chaos REMARKS AND BIBLIOGRAPHICAL NOTES

A general bibliography on dynamical systems is proposed at the end of appendix II; the following selection concentrates more specifically on the notion of deterministic chaos formalized in this framework. The first suggestion of an explanation of the appearance of chaos goes back to Landau (Landau and Lifschitz [1959]); its main defect is to use the progressive destabilization of an infinite number of degrees of freedom., whereas one of the most remarkable characteristics of deterministic chaos is that it appears in systems whose number of degrees of freedom is very small, The first advance towards a solution of this problem. was made by Ruelle and Takens [1971] and completed by Newhouse, Ruelle and Takens [1978], for continuous dissipative dynamical systems: their result states that the destabilization of three degrees of freedom can induce the appearance of a strange attractor. Somewhat later, and concerning discrete evolutions, the period-doubling scenario was numerically displayed by Feigenbaum A third [1977], [1979a],[1980a], who evidenced the universal constant = 4,66962 scenario, via intermittency (of type I), was numerically observed by Manneville and Pomeau [1979], [1980], [1981] on the Lorenz model (Lorenz [1963]); it was experimentally confirmed by Berge et al. [1980] in a convection experiment of Rayleigh—Benard and by Pomeau et ai. [1981] in a chemical oscillator. For details on the two other types of intermittency (II and II1), we refer to Schuster [1984]. An accessible yet detailed survey of these three scenarios can be found in Eckmann [1981].

Moser 11973] or Lichtenberg and Libchaber [1982] give general presentations of stability and the stochastic aspects of an evolution. Some "technical" books are Hellemann and boss [1983], Schuster [1984], Stewart [1986], Gleick [1987], Baker and Gollub [1990], Wiggins [1990], or Peitgen et al. [1992], in relation with fractal structures. Let us also cite the collections of articles edited by Berge, Pomeau and Vidal [1984] for experimental aspects and Cvitanovie [1989] or Hao [1984], [1988], [1990] for re-editions of important articles. More specifically, the reader can consult Vidal and Pacault [1981] or Lergarchand and Vidal [1988] for details on chaos in chemical systems and Haken [1981], [1982] on the link between chaos and the notion of order; Ruelle [1989a] discusses the notion of predictability. Eckmann and Ruelle [1985] detail the tools, in particular of ergodic theory, necessary for the quantitative description of chaos (5.1,3). The notion of a strange attractor, which appears in Curry and Yorke [1977], is detailed in Ruelle [1980], [1989], Ott [1981] or Eckmann and Ruelle [1985]; now classical examples are the Lorenz attractors [1963] and those of Henon [1976] or of Roessler (Wegmann and Roessler [1978]). The analogy between phase transitions and the transition towards chaos (5.1.5) or more generally bifurcations, is developed in Kozak [1979], Hu [1982] or Hu and Mao [1982]. The spectral characterization of chaos (5.1.6) is dealt with in Eckmann and Ruelle [1985], Nauenberg and Rudnick [1981] or Huberman and Zisook [1981]. The tools used in spectral analysis are presented for example in Champeney [1973] (for a deterministic evolution) and Hannan [1960] (for stochastic dynamics). The influence of noise on scenarios towards chaos ( 5.1.7) is summarized in Eckmann [1981]; its analysis via renorrnalization in the case of the period-doubling scenario is detailed in supplement 5B. Complete proofs concerning the period-doubling scenario (§5.2) and its extensions to less regular functions and to dimension greater than 1 can be found in the articles by Collet et al. [1980], [1981], Feigenbaum [1980b], [1983], Hu and Mao [1982], Lanford [1982] and in the monograph by Collet and Eckmann [1980]. The renormalization analysis of intermittency (5.3) is presented in Hu and Rudnick [1982] or in Guckenheimer and Holmes [1983]. The

5.5

KAM theory and Hamiltonian chaos

179

application of the renormalization of the diffeomorphisms of the circle to the scenario of Ruelle and Takens (PA), which is not fully concluded even now, is considered in Braaksma et al. [10901 the genericity of this scenario is discussed in Li and Yorke [1973] and Grebogi et al. [19831. Chaotic behavior can also appear in conservative dynamical systems. The Hamiltonian formalism used to describe them is presented for example in the course by Guckenheimer and Holmes [1983]; let us also cite the very useful book by Nayfeh [1973] for the technical aspects of canonical transformations and the perturbative methods used in this context. The study of Hamiltonian chaos has its roots in the work of Poincaré [1892], then of Birkhoff [1927]. Recent research is based on the RAM theorem (Kolmogorov [1954 ], Arnold [1963a,b] and Moser [1962], [1967 ] ) on the stability under perturbations of integrable systems. More recent articles on this topic are those of Benettin et al. [1980a,b] and those collected by Dell'Antonio and D'Onofrio [1987]. The renormalization analysis of the appearance of Hamiltonian chaos uses the analysis of the destabilization of circle mappings, discussed in supplement 5C. For further details on the onset of chaos in Hamiltonian systems, consult Escande and Doveil [1981), Kadanoff [1981], Feigenbaum et al. [19821, Mac Kay [1983] or Shenker and Kadanoff [1982]; the necessary results of Diophantine arithmetic can be found in Falconer [1990]. Devil's staircases are studied for example in Jensen et al. [1083] or in Bak [1989]. For the sake of completeness, we mention the monograph of Mac Mullen [1994] about renormalization in the framewok of dynamics in the complex plane, a vast topic not approached in this chapter. To conclude, we point out that a bibliography on chaos was recently published by Zhang [1991 ].

Supplement 5A

Ergodic theory and related statistical results The statistical description of an evolution is based on ergodic theory, which describes the conditions for purely deterministic trajectories to satisfy statistical laws. We will em:phasize the operational aspects: reconstruction of an invariant measure, choice of adequate ensemble averages. The basic results are applied in various contexts: dynamical systems, stochastic processes, statistical mechanics. We compare the probabilistic and ergodic analyses of a time series and discuss Boltzmann's ergodic hypothesis. The notions we use are defined in appendices I (measure and probability theory), II (dynamical systems) and III (statistical mechanics).

5A.1

Invariant measures

Let us consider a discrete dynamical system with evolution law f, in a phase space X. Experimentally, the impossibility of controlling or precisely measuring the initial state makes it random: its relevant description is the probability of its realization rather than a point of X Thus we endow X with the structure of a measurable space (X, B) and instead of studying trajectories coming out of specified points x o E X, we study the evolution of the initial distribution mo (defined on the a-algebra B) under the action of f; taking m o to be equal to the Dirac measure bzo at 2',0 links these two approaches. This evolution is described by the sequence of measures (rn„)„ > ,3 where mt, is the distribution of probability in X of the point xt, representing the system at the time n:

r (so) EAJ = mo(r n [A ]) mn _i (f - j [A])

VA G B, m r,(A) Prob[x„=

Instead of representing the discrete flow as a sheaf of trajectories each describing the whole history [x„= fn(x0 )]„>0 of a single point 2; 0 of X, we consider it "transversally" as a sequence of instantaneous images [m]> o , each representing the whole phase space at a given instant n. The support of mn is the domain of X accessible at the instant n. mn is linear and can be obtained by iteration of The correspondence mo (mo ). mo —).mi; we introduce the Perron Frobenius operator P1 such that mn = This operator is linear, injective, positive and continuous for weak convergence lig of measures; it preserves the normalization of m since pf (m)[x] = m (X ), and the support of Pf(m) is contained in the image f (Supp m) c f (X) of the support of m. -

,

'The sequence (222) )3 > o of measures is said to converge weakly to the measure ni if VB E 8, Ern )

c.,,m)(B)=sit(B).

182

Ergodic theory and related statistical results

114111"-.1

-4■11

0.5

71,

.1■11

Ala

Figure 5A.1 - The ten-thousandth iterate of fi,c (x) = 1 — associated image-measure

pc s2 and the

In this figure we simultaneously see the graph of fANc (where p,, = 1.401...) and its histogram j cri = ,60; card{ k, f oNc (kA's) E LiAY, + 1 )AY[}, giving the N-th image PN (rraL) by foc of the Lebesgue measure

crifllJ WeN LiAy, (j+ i)AyD

rni, of [-1, 1]:

pN(rnd ([jAy, (j

1)Ay [)

Choosing Ax < Ay reduces the statistical distance between the histogram and its theoretical limit (Ax 0). Decreasing Ay gives a finer knowledge of PN (mL). Modifying N tests the asymptotic character of Ply (m.0: we may thus identify PN (mL) with the invariant measure rn, of f whose support corresponds to the attractor A c of fpc . (Here N=104 and Ax = Ay =10-3)

If the sequence of measures (m a, =P7 (rno))„>0 converges weakly to a measure the continuity of pt ensures that m,, is a fixed point of pf , so it is invariant under f: mc,,, describes the statistically stationary state in which the evolution generated by f stabilizes, The measure-theoretic analysis of the asymptotic dynamics is intended to describe the measures invariant under f (there exist several of these in general); thus it must determine the fixed points m* of P1 . A basin of attraction is associated to each fixed point; this is the set of initial distributions evolving to the fixed point under the action of f. If the basin of m* is not trivially reduced to the single measure the support of m* is an attractor for the dynamical system (X, f). A second step is the spectral analysis of the linear operator Pf (in a suitable space of measures), which enables us to describe the robustness of the invariant measures and the way in which an initial distribution mo approaches or drifts away from them during the evolution induced by f. In an experimental context, the relevant invariant measure m will be the one whose basin contains the initial measure m a constructed from physical data, It can be written mc,,, =lim,,_,, P7 (m0) . Figure 5A.1 uses this property of convergence to construct mc,„„ numerically.

5A.2 Ergodicity and the law of large numbers

183

(> DETAILS AND COMPLEMENTS: DISSIPATIVE OR CONSERVATIVE SYSTEMS There exists a measure-theoretic notion of dissipative or, on the contrary, of conservative evolution; it is less vague than the corresponding physical notion and can be stated as follows: (f, X, B, m) is conservative if there exists no set y iv of non-zero measure whose inverse images [f , > 0 are pairwise disjoint. A particular case is when m is finite and invariant under f. This notion is equivalent to the incompressibility of ( f, X, B, in) expressing the fact that there exists no set Z c X such that Z C f -1 (Z) and rn[f- I (Z) 2] > O. This property of incompressibility suffices to ensure that every measurable subset B E B satisfies 120 the recurrence propertv (Birkhoff [1912 ] ) stating that almost all trajectories coming out of B return there, i.e. that there exists a subset Bo C B such that m(B — Bo) = 0 and for all x E Bo, there exists n > 0 such that f' (x) E Bo .

c

(n r

—

5A.2 Ergodicity and the law of large numbers The notion of ergodicity is a characteristic of the triple [(X, B), m, f] consisting of a measurable space, a measure and a transformation. It makes sense only when in is invariant under the action of f: f is m-ergodic if every measurable subset invariant under f has measure zero or full measure; equivalently, if every din-integrable function F invariant under f is constant almost everywhere. The importance of this notion comes from Birkhoff's ergodic theorem (Birkhoff [1931]): If f is m-ergodic, then for every drn iniegrabie function F on X, the sequence [n-1- F(fi (x))]„>1 converges rn-almost surely to a constant F* . If rn(X)
The physical meaning of this theorem appears when X is the phase space of a physical system S, f its evolution law (n is then time) and in its distribution of probability in X; the invariance of rn under the action of f expresses the fact that S is in a stationary statistical state. If f is rn ergodic, applying the theorem to the characteristic function of a measurable subset B E B (i.e. to the function equal to 1 inside B and to 0 outside) ensures that the visiting frequency of S in B is asymptotically equal to m(B) whenever it starts at an m-generic point of X. As a corollary, if B E B and na (B ) > 0, there exists a subset of full measure (rn(X — = 0) such that every trajectory coming out of it passes through B infinitely many times and at arbitrarily large instants, since their frequency is rra(B) > O. Figure 5A.2 illustrates the method of reconstruction of the invariant measure and thus of the attractor (which coincides with its support) given by this statement. We must compare figure 5A.1 with 5A.2: the second method (figure SA.2) is preferable since it converges faster to the desired invariant measure; it is much less expensive in terms of computation time'. More generally, the theorem shows the equality of the temporal average A = nA(x 1 ) and of the statistical average < A >=- f Actin for -

xs

E0,i ,„

.y =

"It suffices to consider {x E BVn > 0, f n (x) B}: its inverse images are pairwise disjoint so that m(Y) = O if f is conservative. 121 The first method needs Ni N iterations to obtain N1 2/Ax images under fN , distributed into No = 2/Ay boxes. The second requires only Ni iterations (for the same number N1 of exploitable points), which reduces the computation time by a factor of N.

Ergodic theory and related statistical results

184

every dm-integrable state function A of S. Saying that a single trajectory suffices to reconstruct the invariant measure m of f and the average quantities < A > in fact means that there exists a set of trajectories, called generic (in the sense of measures), having the same statistical temporal properties. Ergodicity is the property which ensures an adequate organization of the temporal steps in these typical trajectories, giving to each of them the same global temporal behavior. Ergodicity gives sense to the statistical description of a deterministic evolution, since it enables us to replace the notions of initial condition and of temporal trajectory with probabilistic statements in which time no longer explicitly intervenes, Suppose the invariant measure na of f is normalized by rn(X) = 1. If A is a real dm-integrable state function, we can consider [A o fi = A 3 ]i >0 as a sequence of random variables on the probabilized space (X, B, na), identically distributed because of the invariance of m: Prob[Ai G I C R.] = rn(A -1 [1]). Then the ergodic theorem takes the form of a strong law of large numbers: limN,,, Nal EN. -01 A = < X> almost surely. The m-ergodicity of the evolution f is the condition or this statistical law to apply to the sequence of observables [Ai ] >0 . It is complementary to the condition of statistical independence usually attached to this law since here the random variables Ai +i and Ai are totally correlated: Ai+i A i o f.

3000 "'

1003

0.5

Figure 5A.2 - Attractor of the critical map fmc (s)=1—te, x 2 T, E [- 1 1] A discrete approximation (of step Ax) of the ergodic invariant measure m, of fp . is drawn with the help of the ergodic theorem, enabling us to identify mc ajAx, (j+1)AxD with the visiting frequency in this interval of a trajectory of N steps coming out of xo. We test the reliability of the result (it is exact if N oc) by increasing N, and the genericity of the trajectory by modifying xo. The support of m, is the attractor A, of fmc , to be compared with that of figure 5A.1 (here Ax =10 -3 and N = 5104).

185

5A.2 Ergodicity and the law of large numbers

In the following table we compare the two possible approaches to the analysis of a sequence OA > 0 of observations of a system S; these two approaches are valid under exactly opposite conditions, namely the absence of temporal correlations versus the existence of a deterministic relation between the successive states of S. This innocentlooking table actually contains the foundations of the study of dynamical systems (• and in particular of their (asymptotic) chaotic properties, of statistics ( • 2 ) and of statistical mechanics (• 3 ).

Data

Sequence

(a)>/ of observations (n

=

"time series"

Context Deterministic evolution law f

Sequence of random variables

(An )„ > 0 defined on (12, T,P) : on = An (w), w

is time)

X

X z n = f(x n _ i ) = A(x y,) = A o fn(x o )

G

Hypotheses

f autonomous, asymptotic regime (normalized) invariant measure rra (m, f) ergodic

identically distributed variables probability law rnA independent variables correlation time = 0

correlation time

= oo

p associated to the evolution = 6(x — f(g)) xn-i = b) = P(An = a) J P(xn. =

Conditional density

p(A n = a

An-1 =

Statistical law

J Birkhoff's ergodic theorem

strong law of large numbers

Type of convergence

1

almost sure convergence

rn almost -

everywhere convergence

Convergence of the empirical average to the statistical average

iim N-1 E a• = <

N -ec

i<j

>

lim N -1 ri aj = f

< INT

A (x )drra( x

i <j < N

Experimental reconstruction of the distribution from the visiting frequencies

1

lim — N-+co N where

XA(ai) = niA(A)

mA (A)

Prob[A E A]

-E x A (ai )zat[A -1 A]=rnA (A) so m A (A) ----- fraction of points visiting A c A(X) (•1)

Access to all the average quantities

urn N-, 00

1 — N

F(ai= )

empirical average = = statistical average

< F(A) >

1

F(o)dmA (o) i=1

(• 2)

temporal average = = ensemble average (

•3)

Ergodic theory and related statistical results

186

DETAILS AND COMPLEMENTS: ERGODICITY AND CRITICAL ASPECTS

The notion of critical phenomena, referring to the existence of a collective behavior of the elementary constituents and to an organized structure relating the microscopic scales of the physical interactions to the macroscopic scales of their observable manifestations, can be applied to ergodic dynamical systems. The evolution law xn-Fi = f(x) describes the shortrange physical coupling: the state xn at an instant t n depends directly only on the state xfi—i at the preceding instant t n_ i and suffices to determine the state xi at the following instant t n +.1. This model is comparable to a Markov chain, and replacing the temporal variable n by a spatial variable, to the one-dimensional Ising model (Gallavotti [1973]). Thus ergodicity is a property describing an organization of the successive steps such that globally, almost all trajectories have the same statistical properties. Just as all the thermodynamic properties of a critical phenomenon are contained in the global statistics, all the asymptotic properties of the evolution can be deduced from the ergodic invariant measure associated to it.

5A.3

Temporal correlations and the mixing property

A transformation f on the probabilized space (X, 8,m) has the mixing property (with respect to the probability rn) if:

VA, B Es,

rim rn[A

r n (B)] = m(A) m(B)

(m[X] =1)

n--+ 00

or, in an equivalent functional form, if for two square-integrable functions F and G;

lim

n—p

x

(F

G dm = f Fdm f Gdm X

(fx dm =

X

This property expresses an asymptotic decorrelation of the evolution generated by f, since it states that the conditional probability rn[A fl f- n(B)]/m(A) that a trajectory pass through B at the instant n, knowing that it came out of A, tends as n oo to the probability m(B) of belonging to B. The probability of presence in B is thus asymptotically independent of the starting point. This property of forgetting the initial conditions is typical of chaotic systems, so the mixing property can actually be taken as a criterion of deterministic chaos (Ottino [1989]). The mixing property implies the invariance but also the ergodicity of (f,X,B,m), so all the results obtained in the preceding paragraphs are valid. DETAILS AND COMPLEMENTS: MIXING AND STATISTICAL LAWS Since A is real in X, the correlations of the random variables [A,-4 = A o fl n>0 are

2 Cn = < A n +jAi >

2 =

(A o fn+i)(A o fi)dm

Adm] ;

they are independent of j because of the statistical stationarity expressed by the invariance of m. The mixing property states that Cn tends to 0 as n tends to infinity. Introducing stronger mixing properties makes it possible to describe the asymptotic form of the dependence on n

5A.3

Temporal correlations and the mixing property

187

of Cn precisely, and to ensure the summability IJ 1C.1 0 and shows that the random variable SN defined on X: Siv N -1 / 2

E

[A

0 fn

[ Add a N -1 / 2 X

0<j
E EA„ < 0
converges in law to the centered Gaussian law of variance 172 = Co + 2 x[ar b] for the characteristic function of any interval [a, bj, we thus have lim N—P

Prob[a Siv
—

N.00

A >]

x[n ,b] (SN(s)) dm(x) =

Er , C. Writing

e` z2 / 26-2

dz

a

If the microscopic evolution of a physical system is mixing for an invariant measure rra defined on its phase space, its temporal correlations are rapidly decreasing and thus have finite range 7-0 : the successive states (a)>o observed on a scale r can be considered independent if r» 7-0: Thus they can be described as realizations of a sequence of random variables (An ) n >0 with values in X, independent and identically distributed according to in. This means that mesoscopic models without temporal correlations or with independent increments can be justified by a hypothesis of microscopic mixing. 0 DETAILS AND COMPLEMENTS: THE CASE OF A SPATIALLY EXTENDED SYSTEM

If the microscopic evolution of the extended system SN consisting of N constituents is mixing with respect to a measure on its phase space whose support is the set of all configurations, it decorrelates the states occupied by SN at the instants to and to + 7 for sufficiently large r. It follows that the spatial correlations present at to will be completely modified after a length of time T, the configuration at the instant to r no longer having any statistical relation to the configuration at to. Even if strong spatial correlations are present at a given instant, the mixing property of the microscopic evolution gives them a very short lifetime To: their temporal average over an interval of time T > ro is thus zero, so that on the scale T the constituents are spatially independent. The existence of long-ra,nge temporal correlations fixing the system and preserving the microscopic spatial correlations, i.e. the absence of microscopic mixing, is thus necessary for these microscopic spatial correlations to remain perceptible at the observation scales and to structure the system in a durable way; in particular, the joint presence of spatial and temporal correlations is necessary for a critical phenomenon to appear. A mixing situation is one where the microscopic evolutions of the different constituents are incoherent, individually mixing and strongly dominating the influence of the couplings; this is the case in the high-temperature phase, where the kinetic energy of the thermal motion of a constituent is much larger than its interaction energy. The opposite situation where the N constituents evolve in a collective manner corresponds to the ordered or quenched phases observed at low temperatures.

188

Ergodic theory and related statistical results

5A.4

The microcanonical hypothesis and ergodicity

The etymology 122 of the term "ergodacity", related to the notion of energy, recalls that it was introduced in 1884 by Boltzmann as part of the foundations of the statistical description of a system S of constant energy consisting of N »1 particles. Boltzmann proved that an equiprobable weighting of the set Co of accessible configurations gave average quantities identifiable with the classical thermodynamic quantities if S is isolated, so of constant energy Eo . To justify the physical relevance of such a weighting (called microcanonical), in the framework of a mechanistic description of the microscopical configurations of S and of their evolution, he formulated the ergodsc hypothesis in a bounded and discretized phase space E. Still known as the hypothesis of molecular chaos, it assumes that S occupies all possible microscopic configurations with the same frequency during its evolution. After a great deal of controversy over this theory of Boltzmann, the notion of ergodicity was developed in a purely mathematical context, until the recent progress of the theory of dynamical systems and deterministic chaos revealed its physical relevance. The equiprobable microcanonical weighting is precisely the measure mû which must be put on Co to recover the present notion of ergodic evolution under the Boltzmann ergodic hypothesis. This mo-ergodicity of the microscopic evolution of S confirms the validity of the statistical approach whenever the minimum time scale r of the observation is much larger than the microscopic time scale 7-0. Let A([s]) be a state function, defined on the space E0 of configurations [s] of S. On the scale r, we perceive the temporal average A = r--1 for A([s](t))dt, and we can apply the ergodic theorem to it since T > rip, In the limit Tiro --- Do, it is thus almost surely constant, identical to its statistical average < A >= f A([8])dm 0 as1). This identification shows that the microcanonical statistical quantities < A>, which are analytically computable, describe the observations made on the scale r. Note that the ergodicity implies that the system is in stationary statistical equilibrium, described by the invariant measure mo: on the scale r, it is described by quantities which are almost surely constant (and not random) and no longer time-dependent.

<> DETAILS AND COMPLEMENTS: PARTICULAR SITUATIONS If other macroscopic constraints of conservation V --= Vo =-- canst are added to the energetic constraint E = E0 = const, the ergodic hypothesis naturally applies to the hypersurface £(E0, Vo, ...). —

— When the system has symmetries, only the microscopic states compatible with the symmetries are actually realized; the ergodic hypothesis is thus exact only in a subspace of the hypersurface E(E0). — If the different configurations are separated by energetic barriers, E(E0 ) is separated into zones where the system remains trapped over macroscopic intervals of time: the ergodic hypothesis is only valid within each zone, during the time that the system passes there,

122 A

critical discussion of this etymology cari be found in Gallavotti [1989].

5A.4

The mieroeanonieal hypothesis and ergodicity

189

A detailed analysis of the statistical distribution of the values taken by a state function A([s1) of S (isolated) leads to the equivalence between: (i) a microscopic hypothesis of weak temporal correlations, valid if the microscopic evolution is strongly mixing (§ 5A.3) and implying weak spatial correlations on higher scales; (ii) (iii)

a mesoscopic hypothesis of additivity of the entropy and of the energy; a macroscopic situation where for each extensive quantity A:

We can identify the most probable value A m with the value A ob, observed on the macroscopic scale and with the statistical average < A >; The fluctuations V; moreover, the fluctuations A, are of order \fi— AA = A 0b, — < A > and 6A = A are quasi-Gaussian, and the order of magnitude of the irreducible moments (§I.2) is given by: —

<< Ak >>=< 45A2 > k/2 0(N i—k/2)

The notion of molecular chaos which justifies the microcanonical formalism is made precise by the condition that the 'molecular dynamics must be uniformly and rapidly mixing, in order to decorrelate the elementary constituents. The system S then stabilizes in a statistically stationary and homogeneous state, in which the statistical correlations are weak, which makes the additive quantities proportional to the number of particles. This mixing hypothesis ensures the ergodicity of the evolution, which makes it possible to determine the (a priori random) temporal averages as statistical averages in the microcanonical formalism, i.e. with respect to a uniform weighting in the phase space. It shows the non-critical nature of 5'; the microcanonical weighting also expresses the fact that no order appears spontaneously in the system: the equiprobability of the configurations is the very opposite of an organized situation.

REMARKS AND BIBLIOGRAPHICAL NOTES For further details on invariant measures with respect to transformations and on ergodic theory, the simplest reference, defining the notions discussed in this supplement, is Halmos [1959]. One should first consult the background of measure theory given in §I.1 and that of ergodic theory given in §11.4; a general bibliography on each of these theories is given at the end of the corresponding appendices. The statistical description of the asymptotic evolutions supported by ergodicity is detailed in Lasota and Mackey [1985]; its relation with chaos is described by Kuramoto [1984], For recent advances about mixing and the associated asymptotic results, we refer to Doukhan [1990]. The ergodic hypothesis, introduced by Boltzmann around 1880 (Boltzmann [1964]) to justify microcanonical statistics gave rise to a great deal of reflection, for example Maxwell. [1890], Ehrenfest and Ehrenfest [19591. As further support of this discussion, see Lanford [1973]; a general exposition of the connections between statistical mechanics and dynamical systems can be found in Ruelle and Sinai [1986].

Supplement 5B Fib ered renormalization The study of the onset of chaos in deterministic dynamical systems must be supplemented by the study of the influence of noise on the various scenarios, where here the term "noise" refers to a given, statistically stationary stochastic perturbation. The consequences of noise increase with its amplitude; this dependence can be made precise via renormalization techniques, which moreover show up its scaling and universality properties, if any. It is essential to do this in order to be able to compare theoretical predictions with experimental or numerical observations, unavoidably plagued by noise, and to identify with certainty a possible deterministic scenario in the latter. In this supplement we give a detailed analysis in the case of the period-doubling scenario. We refer to appendices I and II for the basic vocabulary.

5B.1

External influences and fibered formalism

In order to analyze the evolution of a physical system S, the first step is to make precise what covers S by specifying its phase space X. The second step is to model

the influence of the "external medium" .A4 on it. The external medium can consist of other physical systems distinct from S, but also of aspects of S which are not taken into account in its description by an element s E X, typically because they are on a scale smaller than the minimal one (which is called "resolution" throughout this book) fixed by the choice of X. In the simplest model, this influence appears via a control parameter y, whose value, well-defined and constant over time, appears in the deterministic evolution law f of S.

0.

DETAILS AND COMPLEMENTS: VALIDITY OF THE MODEL

In order to describe the influence of M on S via the single parameter p,, the characteristics of M which are involved must be in a stationary state and insensitive to the variations of the state of S, which is assumed too small to react appreciably on M. It is also necessary for the average time interval T separating two observations of S to be much larger than the characteristic time of the correlations between the microscopic configurations of M and S, and also larger than the characteristic time To of the temporal statistical fluctuations of M around its average state, so that the interaction of M on S does not create any implicit temporal correlation between the successive states of S.

Fibe red renorrnahzation

192 The deterministic semi-product model

One way to improve the preceding model is to include the undetermined and uncontrolled influence of M in a random component, added to the evolution law; the sequence of states of S is then a stochastic process. We do not use this approach here, but one which is closer to the deterministic formalism of dynamical systems, which will turn out to be technically very useful. Noise is introduced as an additional variable y, belonging to a probabilized space (y, 0. As S is assumed not to react on the state of M, the evolution of the noise is described by a transformation g of y into itself, independent of x, invertible and leaving i/ invariant (to reproduce the reversibility and the statistical stationarity of the influence of M on S). The evolution thus takes place in a new phase space X x Y; the law f describing it appears as a transformation of X x y into itself, fibered over g. Its first component is the transformation f defined on the "real" phase space X and depending on the realization y of the noise; it describes the evolution of S when a noise y is involved. The second is the evolution law g of the noise:

7:X x

—> X x Y

(f(s,y)

9(0 )

This gives the fibered iteration formula for the evolution law f of S:

fin)(

)

nfra-1]( x, y) , gn- 1 (y)1

It depends on the evolution g of the noise: the complete notation should be f[g;n , This formula shows that it is the evolution g of the noise which is essential in the asymptotic analysis of the evolution of S as n oo, and not the noise itself, which is a variable of the phase space. One advantage of this formalism is that the influence of the noise is not limited to a weak additive perturbation. ]

DETAILS AND COMPLEMENTS: Two EXTREME EXAMPLES [0, 1[ endowed with its Lebesgue measure For quasi periodic noise, y is the circle Si du(y) = dy, and the evolution law g is a rotation, characterized by its rotation number (.4) E S 1 : g(y) = y+4...) (mod 1); -

For noise not correlated with time, described by a sequence of independent random variables ("white" noise), y is an infinite product of measurable spaces (yo, up), which are identical because the noise is stationary, and the evolution g is the shift:

y = yoeZ , =

g = Cr where [cr(O]j = Yi÷i if 9 =

Ez

0 5B.2 Renormalizatiort in the presence of noise Starting with the results obtained in § 2.2 and § 5.2 for the transition to chaos by accumulation of period-doublings, we will study the changes in this scenario caused by external noise.

5B.2 Renormalization in the presence of noise

193

Numerical or experimental observations The influence of noise on the bifurcation scheme of this scenario makes the cycles of large period unobservable: one can sketch the effect of noise by drawing the diagram with a line whose thickness a(77) varies with the amplitude $7 of the noise (figure 5B.1). The points whose distance is less than a(q) are no longer separate and we now distinguish only a finite number N(q) of bifurcations, which decreases as 77 increases. The question is to determine, as a function of the universality class .7", in which the scenario is considered and also as a function of the type of noise perturbing the deterministic evolution laws belonging to F, in what proportion 7, > 1 the amplitude 71 of the noise must be reduced in order to observe an additional bifurcation: 1. N(-y c sq)= of unimodal maps and for "white" noise (i.e. Gaussian and In the class spatially uniform noise without temporal correlations), we numerically find the universal value 7,-i = 6.62 ... This universality motivates the use of renormalization tools in approaching the general analytical study. —

Renorrnahzation analysis The first 123 analytical investigations on this theme used a perturbative approach (with the relative amplitude ri <1 of the noise as parameter) around the deterministic scenario; the renormalization of the evolution law was then completed by a linearized renormalization of the stochastic perturbation added to reproduce the noise effects. We present a more general analysis here; it is adapted to all types of noise, with no restriction on the amplitude or on the temporal autocorrelation, so it enables us to discuss the universality of the conclusions with respect to the noise under consideration. It is based on the formally deterministic model given above, which makes it possible to transpose the renormalization used in §2.2.4 almost directly. The basic idea is to move the asymptotic limit n oc) taken along a trajectory of the extended dynamical system (in X x ))) onto the iteration of a renormalization operator The variables E X (representing the in the space of the fibered evolution laws configuration of S) and y E y (describing the instantaneous realization of the noise) are considered here on the same footing. Consequently, it is natural to renormalize the evolution laws f of S and g of the noise in a similar manner, which leads to the following definition of the renormalization operator R:

1.

f[f(A[2 - 1 (y)] where

[Rggs , y) =

y)

OM]

A[9(9)]

and A(y) = f(f(04 -1 (y)), y) = .0 21 (0,y -1 (y)). Just as in the absence of noise, 7Z contracts the apparent duration of the evolution by a factor of 2 and Rg preserves the normalization f(0, y) E 1. The spatial dilation of the graph of f[2] which completes the transformation Rg is, on the other hand, no longer isotropic in X x X, nor stationary since the scaling factor is equal to 1/ALq -1 (y)] for the abscissas and to 1/A[y(y)] for the ordinates and it varies in time with the state y of the noise. The composition rule 123

See

for example Crutchfield, Farmer and Huberman [1982] or

Schraiman et al. [1981].

Fibered renormalization

194 for

R9 follows from that of IZ:

(An] f) g

= Rg 2a-

0

0

R92

o R9 R=R

The relation between the trajectories of S in X and the iterates of

(y) f (2ft) (A„[g-2n (y)] x, y) fj(x , y) = .&

with

-1 o

g

R9 is given by:

A n (y) = f [21 (0 , y)

R9 is well-defined whatever the form of the dependence of f with respect to the noise y. If f is independent of y or if g = idy (in which case y is a constant parameter), then Rg coincides with the operator R constructed in the absence of noise (§ 2.2.4). Restricting ourselves to the case X = [ 1,1] considered in the absence of noise, we -

want to study the flow generated by R. in the space g, consisting of the functions f(x, y) whose dependence on z belongs to the universality class .7", introduced 124 in § 5.2.3 and whose dependence on y is dv-square-integrable. The space G, can be endowed with a Hilbertian structure extending that of L 2 (3), dv). The fixed point cp, of R, independent of the noise, is thus a fixed point of all the operators R9 , whatever the evolution g of the noise. We set (x, y) cp,(2;) to obtain a fixed point of R. in ge

5B.3

Linear analysis and spectral decomposition

The linear analysis of the renormalization R. around the fixed point 0, E GE is based on spectral decomposition of the Hilbert space g, adapted to the linear operator Ug induced by g on g, according to the relation [Ug gx, y) = f[x, g(y)]. Acting only on the y-dependence of the functions, Ug satisfies Ur. 0 (Jet = gn+k (Ug) +k ; the invariance of the measure v with respect to g and the invertibility of g imply that Ug is unitary in Ç. It is then shown 125 that there exists a126 spectral measure p.9 on the circle S 1 , and a family of Hilbert spaces (g,,g ,,) E s i , such that:

ge =

Is , G

dpg (w)

and

(Ug

= eiw Idg gç

com

The meaning of this expression is that every function f c g, is associated to a family [fg-,,,] WE s i of spectral components, where f9, G gE,g ,,„ according to:

F(z,y) is any fixed y E y, we have f(z, y) F(Izr", y) where the function z analytic in a complex neighborhood EL = B(0, r€ ) of [0,11, takes real values on 0, 1], satisfies F(0, y) E 1 and aiF(z,y)<0 on [0,1] and is such that Sf(z, y) is negative on X — {0}. Iii g,, we must use the fibered iteration rule; for example, property (vi) of § 5.2.3 now is written F 0[ F(1,g-1 (y))] 1 +',y) n] 00, o] = O. 125 See for example Hannan [1960] or Yosida [1971]. 126 /1 9 is defined up to equivalence since a.tag , defined by d(a.pg )(w) = a(w)dp, g (w), also works if the function a(w), defined on S I , is strictly positive and if we modify the norm lig,,,,,„ by the factor [a(w)] -112 , We speak of the class of spectral measures, which is entirely characterized by their common support according to the theorem of Radon-Nykodym (App. 124 For

—

1.1)

—

5B.3

Linear analysis and spectral decomposition

VI v(i),

E

i 2 eg,

gE,

=

195

is,

< f(1)if(2)>,c=

d(w)
The spectral components of f are unique once the measure ki g and the norms This decomposition should not be confused with Fourier bge,g,e.t)SI are fixed. analysis: the spectral components of f are not scalars but elements of Hilbert spaces (in general different spaces for different components). The expression is, jaig ,„dpg (w) thus has no meaning a priori; only the integral expression of the norms 11f H 2 and of the Hermitian products < f 112 >g is well-defined. Moreover, the decomposition depends explicitly on the operator Ug , and it realizes by construction a diagonal representation of this operator: the components of the function U9 f (I; f are given .

ft

by [eir'w fg ,„Lesi . Note that the invariance of the fixed point O E under the action of Ug , due to the fact that 4),(x, y) does not depend on y, implies that the spectral component is zero ifw O.

Lyapunov exponent of the renormalizat2on operator R The linearized operator can be explicitly written Leg E DR g (cp,)=Ug A,+ B, + Ug- 1 C where:

(A f v)(x) =

)4 1- [v(w,(A,$))

(B,v)(x) =

v(1)(p,(z)]

['P'e(WE(Afx))v(Acx)]

(C,v)(x) = [24'1E(s)v( 1 )] The diagonal character of Ug in the spectral decomposition carries over to L g . where = the component of L,, g acting in Qf,91W is of the form L e ,w idg. eiwA, B, C, is a linear operator independent of the evolution g of the noise and acting uniquely on the dependence of the functions on x; it is iterated according o o Lo w o L. The renormalization g —) g2 of the LE to the rule L[,71 on the spectral evolution law of the noise, replacing Ug by U: and its action eiwIdg 24.4 (modulo 1) on component by e 2i'idg, reduces to the transformation w the circle S i . By mathematical arguments, we deduce for a suitable c, for example c < 1 or c =0(1), that the family (L,,,,) e s , has a continuous field of eigendirections > 0). We ci)we s i where eE E gc,91,, satisfies e » = a e ,e,,2, (with

4,

show that the following limit exists:

= lim n -1 log 1 11, oo

II

(w E S i fixed)

0 <
The real number log p,,„, can be interpreted as a Lyapunov exponent of the flow generated by the renormalization in the space g,„. We can then show the existence, the differentiability and the convexity of the limit:

(30,(0) = lim n -1 log f co

S

exp [(3 log

11 0<j
p,,] di-t9(w)

196

Fibered renorrnaiization

If the spectral measure pg is invariant and ergodic with respect to the transformation w 2w (modulo 1) on S 1 , the application of Birkhoff's theorem (App. 11.4) gives the typical global Lyapunov exponent of the renormalization operator 7 in g,; for 'iv -almost all w E Si:

lim n -1 log II pc, 21,„ 0<j
(

ir log p,,„dpg (4.4;) a- log 7„ 29 ) SI

The limit depends implicitly on pg via the pg-full measure set of the values co for which it has the value log 7,(19 ); this dependence with respect to pg appears explicitly in the integral expression of The linear analysis of the flow generated by the renormalization operator 7Z ends with the determination of the stable linear space ec c g, such that for pg-almost all G S 1 , the asymptotic behavior of the spectral component under the action of (zero fl Ec, g or takes ig,„ to L,,„ is described by the exponent log( 9 ) if f — if c.,.) t 0) if f 0, E ges . The non-linear analysis of the action of R, in a neighborhood of 0, shows that the linear results remain valid for the real flow generated by 7"?. if 0, + £,,9 is replaced by the stable manifold of 7?, at the fixed point 0E .

L.,

—

5B.4

Critical exponents: results and conclusions

Once the parameter e > 0 (measuring the regularity of the evolutions) is fixed, the asymptotic behavior of C,,9 in the space g, depends only on the spectral measure pg , which proves the universality of the exponent log 7,(19 ) with respect to noises of a given type, defined to be noises with the same class of spectral measures (the phase spaces (y, or the evolutions g of such noises being possibly different). In the case of the period-doubling scenario perturbed by noise, the real number 7, can be interpreted as the number by which the amplitude of the noise must be divided in order to bring out an additional bifurcation. On the other hand, 7,(pg ) is not universal with respect to the type of noise: it is identical for Gaussian noise and noise with Lebesgue spectrum, but different for quasi-periodic noise. As in the absence of noise, the regularity of the functions, specified by the parameter e, has an essential influence on the universality class. It is actually possible to explicitly compute 7,(p g ) analytically in two situations. — For c =1, the functions of gE , with fixed point 0,, are analytic with respect to x 2 ; one proceeds by analytically expanding all the relations (the equation of the fixed point, the equation of the eigenvalues and so on). The computation is computerassisted. — For e <1, we take perturbative expansions of the equations to be solved with respect to small parameters E > 0 and A, = c log c 0(c) < O. The whole of the study of transition towards chaos via period-doublings can be transposed to other transitions, such as intermittency (§ 5.3).

513.4 •O•

Critical exponents: results and conclusions

197

DETAILS AND COMPLEMENTS: MAIN TYPES OF NOISE

— Spectral measure singular at co =0: dpg (w) = 6(w)du.). We recover the results obtained in the absence of noise: 70(c) = p=o = (h 5.2.3). For c 1, we have 70 4, 6692 ... 70 2— loge 0(). and for c <1, we have — Noise with Lebeague spectrum: dttg (w)= dw. The value of the exponent is log 7L( € )= e c (2)/2. It coincides with the exponent log 7G (c) obtained for noise described by a sequence of random Gaussian variables. For e= 1, we have 7G (e) = 6,619 ... and for c < 1, we have (€) ;:-2,

[

log Ei

•

— Quasi-periodic noise with irrational rotationnumber 0. The spectra] decomposition of G, comes down to expanding its elements into Fourier series with respect to the noise y along the basis (e2i7n1) rzE 7. The determination of the exponent log 7(c) is intricate y F. because it involves an infinite number of eigendirections of the linearized renormalization operator. In each Fourier sector (labeled by n E Z), except the sector n 0, this exponent is given by the maximal Lyapunov exponent log 7(c) of the family (L e , w )wEs„ but in order to describe the overall behavior of the flow generated by we must control the large deviations of the global quantities coming from the sum of the Fourier components. We can show that for a quasi-periodic regular noise, i.e. if we require the elements of to be C' y E S 1 , the global exponent is given by the Lyapunov exponent with respect to the noise observed in each of the sectors n O. For f = 1, the exponent is numerically close to the exponent observed in the case of a noise with Lebesgue spectrum, but we can show that it is actually different. If c< 1 one obtains: 7(c) = [cl log cl] -1

c

s1,

ge

REMARKS AND BIBLIOGRAPHICAL NOTES

Details and proofs about the spectral decomposition of a noise can be found in Hannan [19601 or Yosida [1971]. For reference books on the influence of noise on dynamical systems and in particular on their asymptotic behavior, see Freidlin and Wentzell [1984], Katok and Kifer [1986] and Kifer [1986]. More specifically, the period-doubling scenario is studied for example in Crutchfield, Nauenberg and Rudnick [1981], Crutchfield et al. [1982] or Schraiman et al. [1981]. The general approach presented in this supplement unifies these early works and makes it possible to investigate possible properties of universality with respect to noise (Argoul et al. [19871); the proofs are detailed in Collet and Lesne [1989].

Supplement 5C

Renormalization of circle mappings The asymptotic analysis of the evolution generated by diffeomorphisms of the circle (or "circle mappings") occurs in the scenario towards Hamiltonian chaos described by Moser's theorem (§ 5.5), and in the scenario of Ruelle and Takens (§ 5.4). It, is based on the expansion of their rotation number as a continued fraction (§ 50.2). Via renormalization, we can establish universal properties for critical diffeomorphisms with remarkable rotation numbers such as the golden ratio (§ 5C.3), and describe the transition towards chaos observed in families of diffeomorphisms (§ 5C.4).

5C.1

Circle mappings and rotation numbers

A diffeomorphism of the circle f, more usually called a circle mapping, is a regular, strictly increasing map of the real line R, to itself which satisfies the circle mapping identity As ± 1)=f(s)+1. Visually, R and its image under f can be rolled up in a coordinated manner on two circles of perimeter 1: the images of points s and x+n for an integer n which become superimposed on the first circle also become superimposed on the second, separated by the same number n of turns around the circle, This definition expresses the necessary and sufficient conditions for f to induce a diffeomorphism f on the circle SI, which associates the class f(s)={f(s)+n, n E Z} ki) to the class fs n, n E Z). If A = {z = A(s),s E Si} is a closed curve inscribed on a cylinder S 1 x Z and diffeomorphic to a circle z = cons, f can be interpreted as an evolution law describing a discrete and monotonous motion along A. The simplest example is X p of uniform rolling speed p in x. The definition implies that a rotation s As) — x is periodic of period 1, so for every integer n, we have As n) = f(s) ± n. If y E [s, s+1[, then since f is strictly increasing, we have f(s) < f (y) < f(s)+1. A circle mapping f is said to be critical if there exists xo E [0, 1] such that f(s 0 )= 0; the fact that f is monotonous excludes extrema, so s o is an inflexion point and r(so) =I 0. The derivative f is 1 periodic and regular, so it attains its bounds on [0, 1[; f is noncritical if and only if inf[o,ij f(s) > 0. The iterates of f are also circle mappings, and they are critical if and only if f is. -

40, DETAILS AND COMPLEMENTS: CRITICAL CIRCLE MAPPINGS

g m (s)L, where g, is regular, I-periodic and Let us consider a family [ft (s) -= depends regularly on a real parameter p. The function 4, is a non-critical circle mapping for the values i where inf[0,131;(s) >0, typically given by one or more conditions p > pi. The value p i for which there exists a point so where rpi vanishes is a borderline case beyond which fp is no longer monotonically increasing. This case corresponds to a critical

200

Renormalization of circle mappings

diffeomorphism and separates the situations where 4-1 is differentiable from those where this inverse is multivalued; in particular (4—,1 )`(4,, (s e )) = -Foo. Via the translation 71, fp (s) = fi„(s x 0 ) — s o , we can reduce (so) = 0 to the case (Ty° foz ) 1 (0) O. Beyond it, i.e. for p < p i , fp (x) 2: is still 1 periodic but Pi, vanishes an even number of times in [0, 1[ since it is 1-periodic and regular: the graph of ft, has at least a minimum and a maximum in [0, 1[. —

-

One defines the rotation number pi of a circle mapping f by: pf = lirn

(s)

One shows that this limit exists and does not depend on s; it describes the asymptotic speed of rotation around the circle of the evolution generated when iterating f If pf E [n, n + 1[ for an integer n, the asymptotic study of f is reduced to the study of f = f o of rotation number p— E [0, 1[. Indeed, f commutes with 9p_ n. so fk

fk çj9p_ n k and therefore p1 = pi — n E [0, 1[. As the observable quantities

A of

a system of phase space S / are 1-periodic, the sequences [A o fi] k>0 and [A o lki k>0 coincide and we cannot distinguish the evolutions generated by f from those generated by The rotation yoc, : s + a has rotation number equal to a and often we can asymptotically relate the evolution under the action of f to the evolution of the rotation s s pf For example, Denjoy's theorem (Denjoy [1932], [1946]) states that f and this rotation are differentiably conjugate if infzE [011 ] f(s) > 0 and if pj is irrational; one of the consequences of this theorem is that a diffeomorphism of irrational rotation number must be critical if it is to generate chaotic asymptotic behavior.

1.

0

DETAILS AND COMPLEMENTS: EXISTENCE AND PROPERTIES OF

p(f)

Since f is a circle mapping, we have for all 0 < x < 1 and all n > 0:

so

lim inf, 00 (fn (0)in,) = lim inf,„ (f n (x) — .r) (fn (0)/n) = lim sup„_ (fn (x) — x)

If the limit exists for s =0, it exists for all s and does not depend on s, which justifies calling it p( f) instead of p(f, , s). Let k y be the integral part of r (0), satisfying fn (0) -1 < k n < f(0). The fact that f is increasing and the relation f(k) = fP (0) + kn ensure that for all n,p > 1, we have k n + kp < < 1 + k +kp . The proof then uses this sub-ad.ditivity of n kn . For fixed n, every integer p can be written p = qn r, where 0 < r < n and so qkn
Letting n tend to infinity shows the existence of

kp ip = p(f) (kp depends on f).

5C.2

Continued fraction expansions and the golden ratio

p(f) has the following properties: every real value is attained since p(s) = a; p(f + a) = p(f) + a for every constant function a; p(Tx , f) = p( f) for every real number xo where [Tal(x)

201

This number

- P(fn) --= nP(f) ; — p(f 1 ) = —p(f) since 1 72 (0) — 1 < x

-

5 0 .2

f(x + x0)— xo;

f(x) < fn(0) +1 ;

P(fi o f2)=- p(fi)+ p(f2) if the diffeomorphisms /1 and 12 commute.

Continued fraction expansions and the golden ratio

Expansion as a continued fraction measures the irrationality of a number by measuring its distance from rational numbers approximating it. Moreover, it can reveal self-similar aspects in the irrationality of certain real numbers, in a form which can be

used as the basis of renormalization techniques. The continued fraction expansion [p](s) = (pi(x)Di >0 of the real number x associates to it a relative integer po and a sequence (pi)5>i of strictly positive integers

such that:

1 x([P1) = Po +

+ P2

+ p3

1

We have po = E(x) (the integral part of x) and pi = E((j(x)) > 1 where we introduce ((s) = 1/Frac(s) (here Frac(x) = s E(x) < 1 denotes the fractional part of x). We stop the expansion at the element p5 0 if pi o+1 = oo. By construction, the correspondence x [p](x) is a one-to-one mapping from R to the set S = Hp] = (p.i)o< j <)0 <00, po E Z, 1 < p5 < oo} and increasing for an adequate ordering on S. The sequence [p](x) is finite if and only if s is rational and r_ 1 (s) = [po(s); • - Pn— 1 (S)] is the best rational approximation of x using n integers. The greater the number p,(x), the nearer the rational number rn _ i (x) will be to the rational number r,,(s) and to x. The convergence of the sequence of rationals {r(x)1 >o to xapl) is thus slower when the integers (pi)i> i are near 1, so these integers provide a measure of the degree of irrationaltty of s.

0-

DETAILS AND COMPLEMENTS: PROOFS

sap]) is rational if the sequence [p] is finite; one shows the converse by constructing the expansion of a rational Q 1 N o , by successive Euclidean divisions of the numerator by the denominator. The correspondence x [pl(x) is increasing if we order S via the lexicographic ordering for the sequences [( 1)i > 0: this ordering is given by stipulating that [p]< [pl if there exists jo > 0 such that pi =p1'• for j< jo and (-1)i 0 pl0 <(-1)i°29.12.0 . We deduce from this the rational approximation of an irrational real number s: —

r25(x) =7 [p o; pi • . • p2i] < s < r25+1 (S)

[PO; P1 • • = P2j-1-1]

since ( -1 ) 25+1P25+1(r25) = —00 < —p2i 4.1(5) and P25+2(r2i+i) = cx) > P25+2(x). The rational ri(x) = [po (x);p 1 (s) .pi(x)] is the best rational approximation of x using j

202

Renorrnalization of circle mappings

integers since r = [q0; q 1 ...qk must satisfy k > j and q,„ = pi, for n < j in order to lie between x and ri(x). From the equality (1 (z) = pi(x)-+ 1/(j+ 1 (x) we deduce that: ]

(5 (x) — C i

(Y) = Cic+i1+ (IY()s--)C./C:+j+) 1 (1i

if

Pi (s) = Pi (Y)

This relation, iterated for y = rn (s) from j = 0 to j = n — 1 (with C( r ) = p(s) so OW - C(rii ) =11(n-"(x)) shows that: 1

0 (— On [2', — r(2)1=

(" 1 (s)

1

<

(n7=1 ci(x)(i(rn (s))) —

pn+1(x)

1ir;=1 p5 (5)2

It shows that the rational sequence [rn (x)] n > 1 converges alternatingly to s: ro = E(x) < .. • < r2i < r2j-}-2 < X < 7'25+1 < r2.. 1 < • ..=< ri < E(x)+ 1 The upper bound of Is — rn (x)i shows that the greater the integers [pi]j>0 in its expansion, the nearer it is to a rational number. 13y recursion, one shows that s and C k (21) are related by (k (s) = [Akx +Bid / [Cks 1 Dd. The coefficients appearing in the fraction are integers and depend on a; only via the integers [p0 (x)...pk_1(5)], according to the formula: - -

{

Ak+2 = Ck+1 = - Pk(X)Ck ± Ck -1 Bk+2 = Dk+1 = -Pk(S)Dk + Dk-1

Co = Ai = 0 Do = Bi = 1

Ci -= A2 = 1 D1 = B2 = - P0(X)

The integers ak = (-1) k +1 Ck and bk = (-1) k Dk are positive and satisfy the recursion formula Xk = Pk(S)Xk-1 + Xk-2, with 00 = 0, ai =150=1 and Di - =p0 (x).

o

The most irrational real number c, i.e. the real number most slowly approximated by a sequence of rationals, is the one where all the integers in its expansion are equal to 1:

a - [0; 1,1, 1...) <=> cr -= cr-1 - 1 = Frac(o 1 ) =#.. a = (■/ - 1)/2 Its inverse a -1 = ( 1,/ +1)/2, whose expansion [1;1,1 .. .] is invariant under a shift of the indices, is a fixed point of C. It is called the golden ratio and has been known since antiquity for its particular proportions; since Fibonacci the number c has been associated to the Fibonacci sequence of integers (an ) n>0 defined by recursion via a,24. 1 = an + a n _ i starting with a0 = 0 and al = 1. The explicit determination of the sequence (a n ) >o gives the relation can - an-1 = (-1)n+ l on. The sequence (anian+i) n> 0 gives the best rational approximations of a (Garland [1987]). The truncated and shifted expansion s'[pj = [Pn; Pn+1 ...] is associated to (n(s) if [p] is associated to x. This relation shows that the periodicity of the expansion [pl(x) is equivalent to that of the real sequence generated by the action of ( on s. The equality C (X) = s thus ensures that x has expansion [731(x) = [Po; Pi • • • Pra —1> Po pi • • -]; if ("k (x)=(k (x), the periodicity appears after the transient sequence [po, • Writing (k(s) as a rational function in x as above, we show that a real number ,

5C.3

Asymptotic stability and renormalization

203

s has an expansion which becomes periodic after a certain point if and only if it satisfies a quadratic equation with integral coefficients Px 2 =Qx R (P > 1, R> 1, Q E Z). These numbers, necessarily irrational since their expansion [p] is infinite, are the only ones for which we can define a suitable sequence with a structure analogous to the structure of the Fibonacci sequence. This property reveals a certain arithmetic self-similarity of these numbers s hence an associated self-similarity of the rotations 0 0 -Fs (mod 1) which they generate on the circle. They will play a particular role in the renormalization of circle mappings. DETAILS

AND

COMPLEMENTS: PERIODIC REAL NUMBERS

AND

ADAPTED

SEQUENCES

Given a number x with a periodic expansion as a continued fraction, let us show that there exist recursive relations between the integers [Pk(x)]k> 0 and [Qk (x)]k> 0 from which one can express the successive rational approximations [rk(x)]k> o , with r k _ Pk /(2 k . — A transient sequence [po ...pk_ i ] can be eliminated by replacing x by (k (x.). — The case of the golden ratio g leads to the Fibonacci sequence (a„), > 0. The sequence (bk)k >0 defined by bk.+1.=Pbk-l-bk—i, 4=0 and bi = 1 works since rk_ i (xp )= bk +i ibk and rk (1/x p )= bkibk + 1. it can be explicitly expressed by b k =cr[(s p ) k (—xp rk] where a = xp /(1 sp2 ), and it satisfies bk — The condition for the sequences [Pk (X)]k> 0 and [Qk(x)]k>o to verify a finite recursive relation with Integer coefficients is that x must be periodic, with pk +n (s) = p k (x) for all k > 0. The proof uses n sequences Rai,k)k>o,i = 0 ...n 11, all satisfying: ak +i = pkak ak_ i (but with different initial conditions) and such that: r,, q+ i(x) The main point is that there exists a real number K(s) of modulus > > Po ...Pn-i _ 1 and a constant C such that every solution (Sk)k>0 of the recursion satisfies the exact relation Sqn +1 XS qn = CK -7 . The remarkable point is that the elements of this sequence are integers whenever So and Si are and it depends only on the integers Po • • • Pr2. 1 (apart from So and Si ); the initial conditions are needed in C but not in K. —

-

5C.3

Asymptotic stability and renormalization

We showed in § 5.5.1 how the study of the transition towards four-dimensional Hamiltonian chaos comes down to 127 the study of the destabilization of the asymptotic movement generated by a circle mapping f having irrational rotation number; temporal renormalization methods were conceived to deal with this question. The relevance of such a method is supported by the existence of a universal number c associated to the most robust invariant surface (KAM torus) of the Hamiltonian system and by the critical character of the transition towards chaos observed when it disappears. The analysis should give the following results. 127 Far from hampering the asymptotic study, this reduction is the first step of the procedure which consists of eliminating the specific details of the behavior of each system in order to emphasize the universal characteristics.

Renorrnalization of circle mappings

204

It should relate all the critical diffeomorphisms f E .7-0 of rotation number a to a function f* describing the universal properties of the transition. Recall that f describes the evolution on an invariant curve of an area-preserving map obtained by projection and discretization of a four-dimensional Hamiltonian system. If f is not critical and if pf is irreducible, then Denjoy's theorem implies that the asymptotic behavior of f is that of the rotation x x+pf ; the invariant curve is thus not located at the bifurcation point, where it is no longer diffeomorphic to a circle and in this sense "breaks down''. If f is critical but of rotation number pi 0 a, a local cusp appears at the point where f' vanishes, but the breakdown of the invariant curve leads only to a very localized chaos, because of the persistence of invariant curves diffeomorphic to a circle on which the restricted evolution has rotation number which is more irrational than pi (for example a). We observe that the evolution of a situation inf E [0,1[ P(x) > 0 into a critical situation infvE [0 1 1[ (x) = 0 is specific to each family of diffeomorphisms. Since of critical diffeomorphisms we are seeking universal properties, we limit the class by bringing their critical point to so = O.

It should explicitly express the ordering, as a function of the rotation number (which is irrational by Moser's theorem), along which the diffeomorphisms become unstable. We thus need to construct a renormalization operator R such that the successive renormalizations of an element f E To converge to a fixed point f', and whose action on any other critical diffeomorphism g E.F moves it away from the transition towards chaos by adequately modifying its rotation number pg 0 cr. The fixed point f should thus be hyperbolic, with stable manifold 2F7 , which prescribes (at least partially) the choice of R. The search for similarity properties between critical diffeomorphisms is based on the properties of the asymptotic characteristic quantity given by the rotation number p. We will use the two properties: p(fn) = np(f) and p(f + a) = a + p(f) (for every real constant a), as well as the expansion of p as a continued fraction; this will play an essential role by describing in quantitative terms the proximity of p to its rational approximations and by translating the arithmetic properties of selfsimilarity of p into properties of periodicity. The idea of renormalization is to use an _ i (-1)" 1 an between a method of coincidences, based on the identity aa, expresses the fact that a, steps the successive integers of the Fibonacci sequence. It under the action of f c .7", correspond to a„_1 complete circles in the limit n oo: fan an _ , with rotation intuitively, the asymptotic action of the diffeomorphism f, number p(fr„)=( should tend to that of the identity. We have fi = f since a o =0 and a l = land = fn+i by the recursion formula an+ i an-Fan-1 defining (a„)„ >0 . We first construct 0(f) = (9 1 ,11 i ) with g i (s)= f ( x) and h 1 (r) = s + 1. Then, setting kn = fn (0), we associate to fn a pair (g„, h„): —

1)"+1crn

-

—

g,(x) =

fn(kn_ix)

—

hn (x) = kyV i fn _ i (k n _ i x)

(n > 2)

The renormalization operator is defined by its action on the pairs (g, h):

(kV g[h(kg x)], k; 1 g[kg x])

1Z(g,h)(x)

kg = g(0)

We check that 7 (g„,hri ), (g, +1, h, +i ). The renormalization Rf of f E 2F is implicitly obtained by O(Rf) = R.[O(f)]. Let D 1 denote the domain of definition of R and -

5C.4

Universal properties

205

v ) We show that p(f) = a is equivalent to f E this is the case, we have:

kn (-1 )' +1 an and

lim n—rco

= lirn n—ice

nn>ipri; if = —or

By not limiting ourselves to f E 2, we show that:

f E Etn-1

(-1) n+l an—i/an < (-1)n+4gf < 1r4anian+1

We deduce from this that the action of R modifies the rotation number p(f): if so (-1)na rja,,fi < (-1) 3 p(f) < (-1) n an +2/an+ 3, we have f E Vn .i — p(Rf)>(-1)na Rf ED,, Dn _i and (-1)na n _ i /a,,>(-1) 1 n + i Ian+2. DTI,

—

5C.4

Universal properties

One shows the existence of a hyperbolic fixed point (g 4 , h 4 ) of R. The linearized operator DR(g* ,11, 4 ) has a unique unstable direction, associated to an eigenvalue 6, >1. Its stable manifold V = 0(V5 ) coincides with Do. = O(D,.0 ): it is thus also the set of couples OM where the critical diffeomorphism f has rotation number cr. As desired, renormalization relates each one to the typical universal couple (g h 4 ). Transversally, renormalization modifies the rotation number and increases the distance from the stable manifold. Let [fi,L, be a family of critical diffeomorphisms transverse to Vs , with rotation numbers [po ] ,. By construction of V 5 , poc = a where fo, is the unique element of the family such that fm , E V'. We extract a sequence [44 ,1 j>0 such that foi E 6Di (where SD/ is the boundary of Di). If tti < < f E and (-1)jai/ai i.i < (—Wpm < (-1)jai ÷ dai+2. We deduce from these bounds that poi = a; this relation can actually be foreseen since the manifolds [673Pi]i>0 accumulate at V 3 . By construction, they can be deduced by renormalization, so that co) gives the Rfoi E 6V -1 . The linear analysis of the relation Rfoi f oi_, (if j 6;i and: scaling laws: ti c —

a—

pp ,

(p c

pi ) where cr = —21og al logb, > 0

Up to now we have restricted our study of the critical circle mappings to diffeomorphisms with rotation number Cr and to families which become unstable as the rotation number tends to a. In order to generalize the renormalization analysis to rotation numbers p cr, the expansion of p as a continued fraction must be periodic; this periodicity is associated to a self-similarity of the rolling around the circle of the trajectories generated by the diffeomorphisms of rotation number p, which reveals a self-similarity of their asymptotic dynamics.

<>

DETAILS AND COMPLEMENTS: OTHER ROTATION NUMBERS

. . 1, we use C°(p) =11a and the If f has rotation number p=[Po;Pi to obtain pX n, — fractional expression of = (-1)"1 or' Z, where Z is a constant and X„ and Yn are integers satisfying the same recursion formulas Xn +1 Xn Xn-1 and 37,2 +1 = Yn, +11;1 _1. The procedure described for f E.T0 can also be applied here by constructing fn =fX . — Yn , of rotation number p(fn ) = (_on -fiez tending to O.

206

Re normalization of circle mappings

If p = [0; pa • , Pn 1.) Po - • .1 E]0, 4, there exists a sequence (Sk )k >0 of positive integers and a real number K of absolute value IK I > 1 such that Sk+1 = PkSk + Sk_i and pSg,i+i — 8 g7, = Cp.K - q. We thus set fk = f S k+1 — Sk , which gives p(fq ) = CpK - g. This definition of fk corresponds to the method of coincidence on which renormalization is based. Let us first consider the case where po = pi = . .=-..p > 2. We construct Rp acting like "R. on the pairs (g, h) = 0(f) by setting: -

-

-

= (k;' gP[h(kg .$)],k;' g[kg .s])

kg

g(0)

Its domain of definition is D 1 (14) = {(9,h),gP+1 o h(s) < x < gP o h(x)}. If pf = [O; popi ...pn .... i ,p0 ...], we construct Rpop i = R.p n _ 1 0 .. .0 R.p i 0 R1, 0 . We then show that these operators each have a hyperbolic fixed point, around which we perform a linear analysis as usual. We also show that the expansion of pi begins with [0; PoP1 - • •Pnl if and only if ci)(f) is in the domain of definition of R.popi ,.,p„. We can obtain bounds for pi according to the number of iterates of '74 01, 1 . .13 „ which can be applied to (f), and prove universal properties of convergence of the rotation numbers of families (transverse to the stable manifold of Rpop ,.„pn ) to the periodic rotation number p.

o

REMARKS AND BIBLIOGRAPHICAL NOTES The study of circle mappings in the context of the KAM theorem goes back to Arnold [1965]. Properties of conjugacy with rotations are established in Denjoy [19321 and Herman [1980]. To complete our presentation of the essential tools, in particular of renormalization -, which was introduced to describe their destabilization and thus to understand Hamiltonian chaos better, we refer to Mac Kay [1983], Rand et a/. [1982], [1983], Jonker and Rand [1983], Lanford [1984] or Ostlund and Kim 119851.

Supplement 5D Fully developed turbulence Developed turbulence is different from the temporal chaos observed in dynamical systems (of finite-diMensional phase space) since it appears in spatially extended physical systems (modeled as continuous media) and involves an infinite number of degrees of freedom ( 5D.1). The explanation of turbulence proposed by Richardson in 1922 and completed by Kolmogorov in 1941 and then in 1962 is still the qualitative foundation of the more recent theories ( 5D.2). The scaling theory it introduces, added to the scale invariance of the hydrodynamic equations, lead naturally to the use of renormalization principles; the spatio—temporal method we present here illustrates a more general procedure for describing the large-scale behavior of solutions of stochastic partial differential equations ( 5D.3).

5D.1

Fully developed turbulence and deterministic chaos

Developed turbulence is observed in a spatially extended system 8 (for example a fluid in motion) when the parameters of S satisfy a certain threshold condition. More precisely, it designates a class of complex spatio-temporal types of behavior which are:

— unstable in the sense that arbitrarily weak disturbances can be amplified until they have observable qualitative consequences; — exhibiting a large range of spatio-temporal scales; — apparently random on the macroscopic scale of the description. Examples are atmospheric turbulence and the magneto-hydrodynamic turbulence observed in plasmas l28 (Laval and Gresillon [1980]). This turbulence is in some ways analogous to deterministic chaos: it concerns asymptotic dynamics; its elementary equations are deterministic but the observable evolution of S cannot be distinguished from a stochastic evolution; it is unpredictable in the long term and displays great sensitivity to initial conditions and to disturbancess. Thus, only a statistical description makes sense on the (macroscopic) observation scale. The non-/inearities play an essential role by providing amplifying mechanisms and by correlating the different scales. However, turbulence is fundamentally different from deterministic chaos in that the instantaneous state of S is not represented by a finite number n of global state 128

A plasma is

an extreme state of matter observed at very high temperatures (for example

in stars), in which the electrons and the nuclei of the atoms are totally dissociated; local inhomogeneities in these two populations of particles of opposite charges cause intense electromagnetic effects.

208

Fully developed turbulence

functions ioi ... yon.. S is spatially extended 129 , and has an infinite number of degrees of freedom, corresponding to the instantaneous values io i (±) ...90„(i) of the state functions at each point of the real space Rd and, by definition, developed turbulence cannot be described in a finite-dimensional subspace of the functional phase space {0(.) Rd i— R'}. Thus, in order to assimilate S to a continuous medium, the volume element ddi, which is infinitesimal on the macroscopic scale, must be large enough compared to the scale a of the fluctuations and of the microscopic inhomogeneities for a deterministic and continuous description to make sense on the mesoscopic scale dx > a associated to it. The field o(t,t) describing S at the instant t is then continuous and even differentiable. Its evolution no longer involves a finite number n of ordinary differential equations (of variable t), but rather partial differential equations (of variables i and t) (§5D.2).

5D.2

Hydrodynamic equations and Richardson's cascade

From now on we restrict ourselves to the case of hydrodynamic turbulence in Rd. Consider an electrically neutral and incompressible fluid, of constant and uniform mass density p; its evolution equations can be expressed using its field of velocities V (± , t) and its field of pressures P(t,t):

{

atv+(v.tw = --

p-1

(7 P) + v (AV) + §

The first equation is the continuity equation, which expresses mass conservation; here it has a simple form because that p is constant. The second equation is the NavierStokes equation describing the dynamics of an element of volume of the fluid. The system can possibly be supplemented by the addition of boundary conditions. § is a given field of accelerations which can be stochastic; it describes, for example, the gravitational acceleration or the mixing induced by an external agent. In order to quantify the ratio of the scale L on which energy is injected into the system to the scale on which the energy is dissipated by viscous effects, we introduce the Reynolds number Re = LV0 I v where Vo is the average velocity of the fluid observed on the scale L and u is its viscosity. Re is dimensionless and it is the only parameter of the fluid motion once the evolution equations have been standardized.

((>

DETAILS AND COMPLEMENTS: STANDARDIZED HYDRODYNAMIC EQUATIONS

introducing the following variables, functions and dimensionless parameters

u —*

t _, t , = tvo i L PIP P111 = PI PI7o2 —

(17.t7)1Tf ---, (V , .V)7' , L(f( .t)V IV? AV --+ A'17` -, L 2 Aillof7

reduces the equations to standardized hydrodynamic equations with a single parameter

129

Re:

The relations between extended systems and dynamical systems is considered in Ruelle

[19914

5D.2

Hydrodynamic equations and Richardson's cascade

v'

av v' +(v'

=0

209

— t`(P' 1,91 )+ Re -1 A l f('

0 Scale invariance of hydrodynamic equations Let [170 , Po JO, d PO VO be a solution of the hydrodynamic equations in dimension d. Let us see if there exists a solution [171 , Pi , i , p i , vi ] whose field of velocities is t) 170 (1('.i, Kt). Plugging Vi into the equations gives the relation = 1—a and shows the invariance of the hydrodynamic equations with respect to the scale transformation: ,

]

K - cl "io

Xo

-4

OI =

to

—,%, —,N. —,+

t1 = to/K

PO

Po

Pl = K"Po pi _...._ K2-i-a(ct-2)po

go VO

vo Reo

=

IC 2- a#0

= K l— crVo = K 1-2a vo Rei = Reo

The quantities K > 0 and a are arbitrary and this transformation preserves the Reynolds number whatever their values, showing that Re is a physical invariant of fluid motion. The value a = 2/(2 — d), which preserves the pressure, indicates that d=2 must be a special value, and indeed analytic and numerical analysis confirms the particular status of two-dimensional turbulence.

Qualitative explanation of turbulence The appearance of turbulent behavior results from a trade-off between a mechanism of dissipation of energy, here viscosity, and a mechanism of injection of energy, contained in and implicitly in the imposed boundary conditions 130 on V. The simultaneous presence of these two mechanisms, which have opposite consequences on the total energy, is necessary for the system to stabilize in a non-trivial stationary regime. When the two mechanisms take place on the same spatial and temporal scales, their mutual compensation is local and instantaneous, and the system stabilizes in a regular and predictable permanent regime, called iarninary. Conversely, when they take place on very different scales, the transfer of energy from the scale of injection to the scale of dissipation can feed a complex structure of eddies and generate a regime having disordered and unpredictable realizations, whose spectral characteristics are different from those of the laminary regime; this is called a turbulent regime (or fully developed turbulence). This regime displays critical aspects: the spatial range of the influence of a perturbation is generally macroscopic; the relaxation time towards the stationary state after a disturbance can also be macroscopic; we observe multiscale and even scale-invariant spatio—temporal structures. 130 These

can, for example, reflect the injection of fluid at a velocity Vo into the entrance of a pipe, or the effect on the fluid of a shifting of the boundaries.

Fully developed turbulence

210

The Reynolds number estimates the ratio of the scales of injection and of dissipation of energy, and it thus suffices to control the laminary or turbulent nature of the motion. Experimental observations confirm the qualitative reasoning by showing that developed turbulence appears only when these scales are very different, which expresses in a condition' Re > Re* 2000, where the threshold value Re* is not universal but depends on the experimental device. The properties of turbulence increase markedly with Re.

Richardson's cascade and the Kolmogorov model The basic idea of Richardson's cascade [1022 ] is the relation of turbulence to the transit of energy across a large range of spatial scales; this idea was formalized by Kolmogorov [1944 [1962]. In this model, energy is introduced on a large scale L, on which the dissipation by viscous efects is negligible; thus it concerns only systems with very large Reynolds numbers (which agrees with the experimental criteria of turbulence). The fluid then develops cascades of eddies, each feeding relative eddies on lower scales, and this descends to a level at which the viscous dissipation is actually effective, Thus apparently random turbulent behavior arises from the non-linear interaction of local and instantaneous structures on different scales. The quantitative analysis is based on the assumption that the transfer of energy is constant along the cascade: the quantity e of energy (per units of time and mass) given by the eddies on the scale 1i to the set of eddies on a lower scale 1i44 is independent of 1,. The cascade stops at eddies of size 1, and of relative speed v„ corresponding to a Reynolds number equal to /m v„iv 1; indeed, the energy of these eddies is entirely dissipated by viscous effects and thus is no longer available to feed further motion on lower scales. We obtain: LRe -3 / 4 , which is called the Kolmogorov scale, This model of Kolmogorov is still one of the qualitative bases of our understanding of turbulence. Its deficiencies and inadequacies stern from the fact that its hypotheses are too strong: experimentally encountered turbulence is neither homogeneous nor isotropic and depends heavily on the geometry and the constraints of the specific problem,

5D.3 Renormalizat ion for large scale dynamics -

In view of the unpredictability of fully developed turbulence and its sensitivity to perturbations (even localized in space or time), the issue is to determining to what extent the hydrodynamic equations which are valid in alaminary regime can reproduce the statistical characteristics of a turbulent flow, To see that they do so, we must prove that the external random forces or the microscopic internal fluctuations, taken into account in the noise term have no consequence on the turbulent dynamics on large spatial and temporal scales. Renormalization methods are necessary to solve this crucial problem; they work by comparing the universality class of the deterministic model with the classes to which belong models with noise. The relevance of this approach to fully developed turbulence via renormalization is increased by the scale invariance of the hydrodynamic equations. It can be compared to the methods developed to describe critical dynamic phenomena (§4.4), rather than to the purely temporal methods used in the study of dynamical systems: in particular, the analysis takes place in conjugate space-time rather than in real time,

5D.3 Renormalization for large-scale dynamics

211

After standardizing the hydrodynamic equations, the parameters are the Reynolds number Re and the forcing term of the Navier—Stokes equations. We will use a unified formalism where this term p is stochastic and can model a random external influence (extrinsic turbulence) or reproduce the effects of microscopic phenomena internal to the fluid, amplified by turbulent dynamics (intrinsic turbulence). The statistics of are given a priori; a Gaussian model is generally relevant, so that it is entirely specified by its moments of order 1 and 2 (§1.3). We will suppose it to be stationary, homogeneous, isotropic and not correlated in time at the mesoscopic scale of the description.

(> DETAILS AND COMPLEMENTS: MODELS FOR THE STOCHASTIC TERM g By hypothesis on the statistics of we have < > 0 and < M, co) > 0; the correlation functions of g. express (with g =

< ;ri(a , tg,j(i"

—

>=

= (2r)d+ 1 beV

6(t — t') 8 (w

f Ô1 ( q )

Under the hypothesis that the statistics of g are Gaussian, the different modes of g are statistically independent as soon as the covariances vanish for (q,(...)) # (q',(.4.0, We expect the flow to display universal properties with respect to the large-scale components of g, which we denote by gL. Indeed, the injection of energy described by the term g,r, induces a "Richardson's cascade' in the fluid, whose structure, independent of the specific details of gL , suffices to determine the dominating statistical behavior of the turbulent flow. We add the condition 5(q) = 0 if g > A, indicating that g acts only on scales larger than the minimal (mesoscopic) scale 2r/A of the hydrodynamic description, either because we are dealing with an external field described with the same minimum scale as the fluid, or by the very construction of a random term reproducing the effect of the microscopic fluctuations on 4 — i — we can scales greater than 27r/A. By replacing — 4p() p — i by _ip()p1±0.Goolq2 include the component of G(4i) parallel to

dii(fy) =

q -2

( q

26ii

in the pressure term; thus we consider: )

where

00(q) = Oif

q =:11q11 > A

The analysis via renormalization will confirm, a _posteriori, that only the generic characteristics of gL , i.e. of the behavior q 0 of G(q), are essential in the large-scale dynamics of the flow, Thus it suffices to consider the cases 50(0) = 0 and 50(0) 0 0, with the help of the typical models 6'0(0= out to be inessential.

K 0 q2 and G- -0 (g)= Ko;

the higher degree ternis turn

The statistical properties of determine those of the solution V, so that we can write <> for all the statistical averages, on g or on V. The fact that p and u are constant makes the hydrodynamic equations invariant under spatio—temporal translations, which ensures that the stationarity and are transmitted to V. It follows that the homogeneity of the process w) in < V(4, co) > 0 and < 1.?(Cw)V(e,') >= (27r)d+ 1 6d(q q1)6(u) + conjugate space. This second relation means that the Fourier modes ' 31 V(C co) are 131

See appendix IV on the Fourier transform.

212

Fully developed turbulence

well-defined physical entities, and motivates an analysis performed in conjugate space. The correlations C(.i, t) and their Fourier transforms C(i, w), called dynamic structure factors, entirely define the Gaussian approximation of the flow, limiting the essential part of the statistics to the part described by the moments of order 2 and neglecting the irreducible moments of higher order (§I.2). These are El X d matrices, starting from which we define the spectral density of energy E(q) of the flow by (writing Tr for the trace of a matrix): -1-00 2E(g) = p q d-1

(27r) -(d÷ 1) 11511=1 dd-i fif

Tr [a(qft,w)] du,

-00

Since the trace is linear, the relation Tr[C( = 0,t = 0] =< 11 17 (i,t)11 2 > (where the variation is independent of and of t) ensures that E 0 f E(g)dg is the 2> density of kinetic energy (1/2)p < f((,t)1 of the fluid in motion, If Tr[a(q, w)] is in a neighborhood of q= 0 by an integrable uniformly bounded continuous at 4=0 and function of w, then Lebesgue's theorem (of dominated convergence) shows that we can exchange the limit g —> 0 and the integration over w; thus we prove that E(g)g_(d -1 ) is continuous at g = 0, i.e. that E(q) ,- q d- as g tends to O.

Large scale turbulent dynamics -

Restricting the study to asymptotic scales corresponds in conjugate space to focusing the study on 0); the renormalization should thus be designed to describe possible divergences at the origin (g 0 , w —> 0) of the structure factors C(,4.4.)), reflecting the presence of long-range spatio-temporal correlations. The goal of renormalization is to study the behavior as q --+ 0 of the spectral density of energy E(g); more specifically, it is to investigate whether the stochasticity of the turbulent regime induces differences with respect to the relation E(g) gd-1 corresponding to an equipartition of the energy between the different temporal modes Vi (t) and obtained in the purely deterministic description or more generally whenever â(, w) is sufficiently regular at 0. Still obtaining the result E(g) q d-1 would prove the inessentiality of the stochastic term in the large-scale dynamics.

Outline of the renormalization analysis The goal of the renormalization analysis is to split the set of possible statistical models (assumed Gaussian) for into universality classes. It is done according to the typical procedure illustrated in figure 1.7 (see also §4.4.2):

— cutoff in g = A/bq of the spectral components of V where A is the spatial extent in conjugate space; — dilation of the wave vectors by a factor of bq >1, and of the frequencies by a factor of b, >1 to preserve the apparent extent A (we set bq = b and b„ = bP(b); these real numbers parametrize the renormalization operators); — the modes g > Alb must be taken into account in a transformation of the equations, or if possible only of their parameters in order to analyze the action of the renormalization in a finite-dimensional space.

5D.3 Renormaiization for large-scale dynamics

213

The iteration of the procedure concentrates the study on the behavior as q > 0 since after n renormalizations the apparent flow depends explicitly only on the initial only intervene via the influence they have on modes q< Ab"; the modes q> Ab spatial scales larger than 27rbn/A, which is taken into account in the parameters of the renormalized equations. -

Let us give the essential steps of the renormalization analysis. We set R d+i , dk = dd0).) and 6(k) =

k=

E

• One useful computational trick is to introduce _a "small" parameter Ao in front of )17 appearing in Navier-Stokes the pressure term tPlp and the advection term (17.t equations, in order to formally perform a perturbative analysis; we take A o = 1 at the end of the computation to recover the original problem. -

• In conjugate space, the hydrodynamic equations are given by:

.17(k) = 0 icoPn (k)+ Ao(27r) - (d+ 1 )f fdkidk2 6 (k - ki - k2)E dm.i. (ig2m(k 1 )) is/n(k2) (n = 1 ...d) = Ot o qn,p-1 P(k) Re - le VT,(k) g'„,(k) -

-

• The modes q > A of 177 are not explicitly described since their spatial scale is smaller than the minimal scale 27r/A below which the model of the continuous medium no longer makes sense: they are taken into account in a noise term included in g (whose components q > A are zero by construction). • In conjugate space, the separation of the large and small scales is straightforward, given by the decomposition: 170 (k) =

iT(k)

îi(k) = 0

if

q
if q < Alb,

1-70 (k)= 0

if A/b <

q

i5(k) = V(k)

if A/b <

q
• The homogeneity of the statistics plays an essential role by ensuring that:

< 170,j(k)Vi(k' >= O

Vk, k' E Rd+ i ,

j E 1 ...d

• The most delicate step of the renormalization analysis is to "eliminate" the spectral components A > q > Alb from the explicit description:

— the partial resolution of the hydrodynamic equations enables us to express the components of j as the sum of functions of Vo and g with terms still depending on but of higher order in A o . We plug the obtained expressions into the equation satisfied by each mode Vo; — the independence of the modes k and le k of g gives a meaning to a selective statistical average over its modes q > Alb. Thus we obtain a system of equations which, at the lowest order in A o , involves only the modes q< Alb of Vo and of g., This procedure can be iterated up to any order.

214

Fully developed turbulence

The result of this procedure is to take into account the large-scale influence the modes Rk) and "i(k) with q > Alb in effective parameters and in a renormalized noise

term. DETAILS AND COMPLEMENTS: COMPUTATION OF THE EFFECTIVE PARAMETERS

Let us give a more explicit description of the computations leading to the renormalized equations. — Taking the product of the Navier-Stokes equation with and using the continuity equation .V((y) = 0, we can express the pressure terni as a function of V and -4 , then eliminate it from the Navier-Stokes equations to obtain:

i7j(k)= r(k) -gi(k)

2(27r%4-1 A) r(k)

Qiim ( 4)

1-70 1 )17,n (k

-

r(k) is a complex scalar called a propagatorT(k) = is the Fourier transform of the linear differential operator

at -

k') dk'

We note that r(k) 1 Re -1 A. Q(4) is a projection

operator given by: Qi ttn (4) Qm [45i — Q5 Qi /Q2] + q [65 — Q5 Qm /Q 2 ]

E

=

— The natural procedure would be to expand V in powers of „No by iterating the preceding relation, schematically written V = Ot o rQf VV. This becomes at order 2 in Ao:

= — Di o rQ f(r-g)(r-4) - AF,N1(r3) [rcjof Vf7] [rQf 17id

± 0(4)

This perturbative computation of V(Ii0,‘.4.)) and of ef ( ,i0,w ) introduces integrals f q'd d qqo where the exponent a depends on the statistical model chosen for 4- . We find that a = 2 if 60(0 /Co e and a = 4 if 60 (g) = Ko . The perturbative approach is correct if d> a, since then these integrals converge at q = O. If d < a, the expansion converges only if qo is not too near O. It diverges if qo 0, and renormalization is then essential to compute the behavior go 0 of the spectral components. However, the advantage of renormalization is not restricted to the cases d < a since even if by this direct perturbative method we can obtain the solutions V, only renormalization analysis can determine their universal properties. — The computation shows that the quantity z = Ao-V.CoRe 3/ 2 is the true parameter of the expansion (rather than the artificial parameter )O. It is valid for I for all d and gives the expression of its components as functions of the components of Vo and of so that we can eliminate from the equation for Vo. — The scale change = 4 and . c4.)' = V( b )c.,.) transforms the velocities as follows: = Vi(b) V(b -I4,b - i8(b) w)

if q < A

so that by the coherence between i(k) and —ir.A2(k), Wis transformed as follows:

= b)+(b)

if q < A

5D.3 Renormalization for large-scale dynamics

215

The contributions coming from the selective statistical average over the modes q > Alb of g are included in a modification of the parameters (.A 0 and propagator F) and of the source term, or even in additional terms. The scale change above ends the construction of the renormalized equations, i.e. the equations for V' which involve only the components of the stochastic source. The exponents (3(b) and 7(b) are chosen so as to make these equations as similar as possible to the initial hydrodynamic equations. —

—

The two models detailed above, specified by the amplitude Go (q) = K0q 2 or

Go(q) Ko of the source, are prototypes of the two mainly encountered universality classes. Most of the modifications (weak temporal correlations, higher order or higher degree terms, departures from Gaussian. statistics) turn out to be inessential as it is possible to show that they are reduced by renormalization, so actually eliminated after enough iterations. The complete computation for the model 6'0 (q)= K0 q2 leads to the following transformation:

1?-0 (b) = &a -2 K 0 (1 ± A ) ( 2 ) -1 = bi-aoz (1+ bz4427

(b) = b2 P Re (1 + 1, •( d2 _Ad2)) -' 3:0 (b) = 30(b)

where Ad is a numerical coefficient depending only on the dimension d of the space. For any .A0V7CoRe3/ 2 < 1 can be choice of g(b), the renormalization -AO of the parameter z reduced to the ordinary differential equation:

(di I db)(b) = b-1 (1 - d/2) 2'(b ) b - O[(b)3]

-i(b) < 1

If d > 2, the renormalized flow has a perturbative parameter i-(b) smaller than that of the initial flow, with -i(b) = O. Thus, the validity of the perturbative approach is improved by renormalization; the iteration of renormalization, corresponding to letting b tend to infinity, brings the initial flow to a "hydrodynamic" fixed point of the transformation for which the scaling law E(q) ,--, q' 1 is valid. If 0 < 2 - d <1, renormalization brings z to z(b) = f = V871-(2 - d) + ([2 - dI3 / 2); the renormalized flow at infinity thus differs from the "hydrodynamic" fixed point, which means that the scaling law E(q) qd-1 is modified in an essential way by the presence of the stochastic term. As before, the value d, = 2 of the dimension d of the space plays a special role. Analogous results are obtained for the model do(q) = Ko, with a turning point in d = d, with d, = 4. The result of these different steps is that the renormalization 7Zb can be entirely expressed via the transformations:

c4) Re —0-

e (b)

z

A o,

) tre-3-

b0 (b)w 2' (b )

17 —4 0 ro(b)

where VITO- is related to the mean amplitude of the stochastic force term #. The structure factors are then transformed according to the formula

Ôb((b), .-E0(b), To. (b),

‘.,))

b27(b)4fi (b) +d 0(Re, K o ,

b— 0 (b)w)

(for q < A), since 170,w) = b7(b )V(b -14,b -0 (b )w) and we have 6d(b 4 ) = b d 6d () and 6(1) -0 (b)w) = b 1 ( b )6(,j). Iterating the renormalization is equivalent to letting b

216

Fully developed turbulence

tend to infinity in the renormalized flow, Le. if the frequency and the wave vector of the renormalized modes are fixed, letting q and w tend to 0 in the initial flow. The existence of a fixed point of 7Zb is reflected straightforwardly in terms of scaling laws on the structure factors (since then E'b(.) = C(.)) and consequently, on the spectral 0, co 0 for every flow density. These scaling laws are satisfied in the limit q converging to the fixed point under the iterated action of 14. Moreover, the whole of the general linear (and even non-linear) analysis of a renormalization operator (§ 3.1) can be applied to Rb in a neighborhood of this fixed point. To make these scaling laws explicit, we thus need to determine the fixed point to which the renormalized flow converges as b tends to infinity, and examine the behavior b oc of the parameters Re(b), K o (b) and A o (b). These parameters occur only in the quantity z = A o .V7 0 Re3 / 2 , which appears as the "true" perturbative parameter when we seek the solution V as an expansion in powers of A o . The typical result is the following: there exists a critical dimension dc , depending on generic properties and on the large-scale statistics of §, such that:

lim -1(b) = 0

b—Poo

if d > d,.

If d > dc , renormalization transforms the flow into an apparent flow for which the perturbative expansion 2 71 powers of z is valid (as is the case for a laminary regime): the hydrodynamic description is correct for the renormalized flow, so also for the initial flow in the limit (q —* 0,w —> 0). The equidistribution of the energy between the spatial modes (q< A) is thus verified for the renormalized flow in the limit b oo, which can be written limq_03 E(q) qd-1 when turning back to the initial flow. If d < d„ we typically have lim6„, z ( b) = f > 0: renormalization brings to a non-trivial fixed point which cannot be obtained by direct perturbative methods with respect to the parameter z. The hydrodynamic description turns out to be insufficient to describe the statistical properties of the large-scale dynamics. The renormalization procedure can be achieved in this case by setting c = d, — d and taking a perturbative expansion in c, for which the tools of graph analysis work quite well (§4A.3). Thus the fixed point and the associated exponents can be explicitly expressed at the leading orders in c < 1. Renormalization thus enables us to determine a wide class of models for which the spectral density of the flow behaves like E(q) ,--, q d-1 when q 0, in accordance with the predictions of the deterministic hydrodynamic description. These models z(b) =- 0 of their parameter z. A are characterized by the behavior as lirn more complete statement of the result requires a detailed description of the models associated to the various universality classes, which we considered as a complement above and for which we refer to Forster, Nelson and Stephen [1977]. The method generalizes to other partial differential equations describing the evolution of continuous media (such as Burgers' equation (Burgers [1974]) or the diffusion equations 132) and perturbed by a stochastic source. 132 The diffusion of a quantity of density a(, t) in a fluid in motion, with given field of velocities V(fc,t), is described by the equation Ot a (V.V)a = D6a.a where D is a constant coefficient of diffusion. Burgers' equation is given by 0-1-(V.V)9 vA1-7 4- g supplemented by 17 A 17 = O. The vanishing of this curl implies that 2C7.17 )V = ‘7(11V1I 2 ) and we can consider Burgers equation in any dimension d, not necessarily 3.

5D.3 Re normalization for large-scale dynamics

217

Limits of the spectral analysis of turbulence

Renormalization methods based on the spectral and statistical analysis of the turbulent flow give only partial results, because they require that the turbulence be homogeneous and stationary, which is a necessary condition for the Fburier modes to have vanishing covariances, without which the analytic computations cannot be completed. This statistical hypothesis means that transient regimes cannot be considered; moreover the role of boundary conditions must be ignored, and possible inhomogeneities of the flow and their macroscopic consequences cannot be dealt with. Now, observation shows the importance of the boundaries and of the obstacles in a turbulent regime; moreover, it indicates that certain spatio—temporal structures, although localized in time and space (we speak of spatio—temporal intermittency), interact each with each other and thus play an essential role in the development of turbulence. The fractal (and so strongly inhomogeneous) nature of turbulent structures also appears to be fundamental. The diversity and the complexity of the phenomena involved in fully developed turbulence also explain why, to date, there exists no entirely satisfactory scenario describing its appearance.

REMARKS AND BIBLIOGRAPHICAL NOTES

Some general introductory books on fluid mechanics are Landau and Lifschitz [1959] or the exhaustive presentation of Monin and Yaglom [1975]. Navier—Stokes equations are studied in detail in Temam [1979]. For more specific expositions of hydrodynamic turbulence, see Batchelor [1953], the very accessible course by Tennekes and Lumley [1972], Swinney and Gollub [1981) which concentrates hydrodynamic instabilities, and Lesieur [1987] or Lesieur and Metais [1989] which is directed towards the study of turbulent structures. The most recent advances can be found in the book of Frisch [1995]. Historical references which marked the study of fully developed turbulence are the articles by Richardson [1922], introducing the notion of a cascade of energy from the scale of the injection down to the scale of the viscous dissipation, Kolmogorov [1941] formalizing the preceding idea, Landau [1944] criticizing Kolmogorov's model, and Kolmogorov [1962] answering Landau's objections. Further advances can be found in the articles of Mandelbrot [1976], emphasizing the fractal nature of turbulence, and of Frisch, Sulem and Nelkin [1978], dealing with spatio—temporal intermittence observed in fully developed turbulence. More recent studies of the hierarchical structure of fully developed turbulence are given in the basic articles by Benzi et al. [1984] and by Frisch and Parisi [1985] or in Meneveau a,nd Sreenivasan [1987]; they introduce the concept of multifractality (7.2) to describe the fractal structure of turbulence and the spatial repartition of the singularities more accurately. The use of wavelet transformations (§7.3) to detect these multifractal characteristics is detailed in Argoul et al. [1989] and Arneodo et al. [1993]. The self-similarity of turbulence is discussed in Nelkin [1989] [1994]. Recent orientations are presented in Sirovich [1991] and in Zakharov, L'voy and Falkovich [1992]. The renormalization procedure presented au §5D.3, giving the large-scale properties of a turbulent fluid is based on work of Forster, Nelson and Stephen [1977]; see also the conference proceedings edited by Forster [1978]. An alternative method can be found in Yakhot and Orszag [1986]. Similar approaches in the framework of dynamic critical phenomena are

218

Fully developed turbulence

proposed in Hohenberg and Halperin [1977], and more recently in Barabasi and Stanley [1995) for the analysis of the so-called KPZ equation describing surface growth (Kardar, Parisi and Zhang [1986]). It can be generalized to the study of the asymptotic dynamics of solutions of partial differential equations (Medina et al. [1989], Avellaneda and Majda [1991], Bricmont and Kupiainen [1992], Goldenfeld [1992], Chen et al. [1994] ).

Stochastic diffusion Space—time

renormalization methods can be used in the study of diffusion

processes. Starting with the example of Brownian motion, whose diffusion law is called "normal", we show how the use of a renormalization operator, acting on the characteristic function of the process or on the family of its probabilities of transition, reveals the asymptotic scale invariance and the universal properties of the diffusion. This approach unifies the discrete and continuous descriptions of random motions. The fixed points of the renormalization operator appear as self-similar stochastic processes. Amongst them we distinguish stable laws, with independent increments but infinite variance, and fractional Brownian motions, globally Gaussian but co/related in time, and we discuss their relations with critical phenomena. We examine in detail the example of diffusion in a weakly disordered medium, where renOrmaliZati011 arguments enable us to specify the conditions under which the disorder does not destroy the normal diffusive behavior. Non-Markovian self-avoiding random walks are studied in supplement GA in the context of polymer physics. In appendix I, we give a brief introduction to the notions of probability theory used in this chapter.

6.1 Spatio-temporal evolution 6.1.1

The example of Brownian motion

Brownian motion is the name given to the random motion of a particle when it is statistically homogeneous, isotropic, stationary and without memory. This name comes from observation, made by the biologist Brown, of the erratic motion of a grain of pollen in water trapped inside a volcanic rock, too old for the motion to be attributed to the fact that the grain was alive; Brown was thus led to seek for a purely physical origin for the motion, and explained it by the successive shocks given to the grain by the molecules of water in thermal motion. The random aspect of this motion is a consequence of the difference between the resolution (i.e. the minimum scale) a at which the grain is observed, large enough for the grain to be considered as pointlike, i.e. without internal structure, and the molecular scale A < a on which the motions of the molecules of water thus also the motion of the.grain are deterministic. On the scale a, only a statistical description makes sense, since it is impossible to know and to take into account all the microscopic mechanisms: the shocks of the molecules of

Stochastic diffusion

220

water on the grain (whose size is much greater than the size of the molecules) appear as stochastic influences; the statistical characteristics of the molecular motion, which is controlled by temperature and is thus known as thermal motion are thus reflected onto the motion of the grain. Indeed, we can check that the statistical characteristics of the trajectory of the grain depend on the temperature of the fluid medium. The statistical properties of Brownian motion can be reproduced by discrete models of Brownian random walks or by the continuous model of the Wiener process

(Wiener [1976]).

The discrete model: Brownian random walks The discretized (in space and time) model split the trajectory of the particle into successive steps (dn ), > i, each of them linking nearest neighbor sites on the hypercubic lattice (aZ)el and having a fixed duration 7; these steps are a realization of a sequence of random variables [A ,]>1, assumed independent and identically distributed according to:

Prob (42 = aft)

1/2d

Prob(A= an )

=0

if

an 0 aft

where the unit vector u describes the 2d spanning directions of the lattice. By construction, this random walk is statistically homogeneous, isotropic and stationary. We write p(2,4i'n} rt-1-1, 7) for the the probability density: p(2, ,) ( n ,

r) = a —d Prob(iL=

it is called the dementary probability of Normalized by ad EgE(aZ)d transition and determines the other probabilities of transition by convolution (since 0+i , jr) = the successive steps are independent):

= ad(j+i) E

Pa(c), r) kqn

r) pa(°,r) G9j--1

r't -IL 11 7 )

11

pncyi,

Yi+

1 7)

1
1)-tuples of (aZ)d. The distribution of the end-towhere the sum is over the (j — i n over j consecutive steps, of total duration jr, does end distances iC1 = not depend on the point of departure or on the initial instant nr. The isotropy ensures that < fCi > = O. The independence of the successive steps implies that =i = ja 2 = Djr where D = a2 Ir. DETAILS AND COMPLEMENTS: PHYSICAL JUSTIFICATION OF THIS MODEL The motion of the grain, described on a mesoscopic scale a on which the grain is pointlike, is a consequence of the local and instantaneous inhomogeneities of the configuration of the velocities of the molecules of water: the impulses they transmit when they hit the grain, located at at the instant t, have a non-zero net result Ai:, t), which is perceived on the scale a as the reatization of a random variable. Since the evolution of the water towards thermal equilibrium takes place over microscopic time-intervals, we can assume that the con fi guration of the velocities of water molecules follows the centered Gaussian distribution obtained at thermal equilibrium (Maxwell distribution). This distribution of molecular velocities transmits

6..1 Spatio-temporal evolution

221

its stationary, homogeneous and isotropic character to the stochastic process 23(i, t), and consequently to the motion of the grain. Moreover, the fast relaxation of this distribution ensures that the water keeps no track of the collisions of its molecules with the grain OR the mesoscopic time scale 7: this absence of memory of the fluid medium and its homogeneity justify the hypothesis of statistical independence of the successive steps 133 of the grain. lf the velocities of the molecules hitting the grain at the same instant t are sufficiently decorrelated, the distribution of the resulting impulse f)(Z: , t) remains Gaussian and transmits this statistical characteristic to the motion of the grain. Changing the value of the resolution a modifies the observed trajectory since the microscopic motions appear smoother for larger a. A remarkable property observed by Perrin [1916] is the self similarity of the trajectories, indicating that the properties of Brownian motion are more consequences of the way in. which the successive collisions cooperate than of the physical details of these collisions, related to the nature of the molecules and to the structure of the grain. This remark gives a clue for the existence of universal scaling laws stemming from the organization of the elementary mechanisms from the molecular scale up through the observation scale. This conjecture is supported by the fact that the non-universal characteristics related to the nature of the fluid (here the water), to its size L, and to the boundary effects, disappear in the thermodynamical limit Li a > oc); those related to the initial conditions are smoothed out after a short transient regime, negligible over a long observation time, which justifies studying only the permanent -

—

diffusive regime.

The continuous model: the Wiener process The second point of view, which is less intuitive but more universal, considers 0, T. 0. This the trajectory [t t] of the grain in the continuous limit a random function is one of the realizations of a temporal stochastic process [W(t)] E R with values in Rd . In order to reproduce the statistical properties of the Brownian trajectories, this process must be: — statistically stationary and spatially homogeneous; isotropic, so centered:

< W(t) > = 0

for every real number t;

independent and with Gaussian increments; — of finite variance so that all the moments of order -

such that

2

exist;

< 1W(t = 1 ) — W(t = 0 )11 2 >=

D; de° — continuous (in the quadratic mean, so also in probability). One can check that these conditions completely determine a unique stochastic process, parametrized by the real (positive) number tr and called a Wiener process; it satisfies:

< Ewl (t )

< [1415(t)

(s)l[wk — wk (s)j > Wi (0 )][Wk(s) Wk(0)1 > =

cl (t — s) = 5j k t7 2 1t — k cr2 1s1) ,

sl

Thanks to the statistical stationarity, the increments of W are described by a single density of probability (normalized to 1); setting D = da2 , we obtain Here a "step" is the uniform straight trajectory of the grain between two changes of direction visible at the chosen resolution(i.e. minimum scale) a: it depends on a. 133

222

Stochastic diffusion

d dl2 —dr212Dt =f) =-- PD (,t)ddF where PD(i" ,t) = (27rDt e

d P rob (Wt+,

Using the fact that the density of probability of independent random variables is the convolution 134 of their individual densities, the independence of the increments leads to the following group-theoretic structure:

Vt,s ER,

PD(•,t)*PD(.,$)= PD(-,t+s)

on Rd .

The normal diffusion law In the continuous as well as in the discrete description of the random motion X(t) of the particle, the temporal statistical independence of the successive steps implies the following scaling law, which is exact over any length of time t:

Vt
D(t)

< 11X (t)

—

2 (0)11 2

> = Dt

The diffusion coefficient is a non-universal characteristic of the motion, related for example to the mass of the grain, or to the nature of the fluid or to its temperature; its value is D = der' for a Wiener process X = Wc, and D = 0 2 /7- for the Brownian walk of probability of transition pa This time dependence of Brownian diffusion, without bias or correlation, is called the normal dgffusion law . This name extends to any random motion described only asymptotically (as t —> co) by a diffusion law D(t) Di (with 0 ( D < co).

<>

DETAILS AND COMPLEMENTS: ASYMPTOTIC BROWNIAN BEHAVIOR

The elementary steps (of fixed duration r) of a Brownian random walk are statistically independent, identically distributed and of finite variance DT. Therefore, the law of large numbers and the central limit theorem can be applied to their sum .g(t)— X(0) (where t E TN). They state that t -1 [X(t)—X(0)] converges to 0 almost surely as t tends to infinity, and that t -1 / 2 [X(t)—g(0)] converges in law (cf. App. I) to the centered isotropic Gaussian law of variance Dld in each direction, i.e. to PD(.,1 = 1), This result is obvious for the Wiener process ITV of diffusion coefficient D since t 1 [1,17 (t) — LV (0)] has law t d Pp(ti t) which weakly converges to SW (constant law 2 0) when t tends to infinity. Similarly, the random variable t -1 f 2 [W(t) — W(0)) has law 2 t d f 2 Pip(i\rt, t) = PD (2 , t = 1). This identity of the asymptotic behavior of the Brownian random walk and of the Wiener process, i.e. of their consequences observable on a macroscopic scale, means that in order to reproduce Brownian motion on the mesoscopic scale, one can use either the continuous description (Wiener model) or a discrete-time model (Brownian random walk).

The convolution * is defined for two integrable functions f and g on Rd by (f * g)(t) (g * f)(t) = f f f (y)gM6 d (t — g 2)d dO di f f(g)g(t — orig.

I"

6.1 Spatio-temporal evolution 6.1.2

223

Random motion and diffusion laws

Extending the particular case of Brownian motion, the motion of a particle in a medium of dimension d can be described by deterministic equations on the microscopic scale A, describing the dynamics of all the molecules of the medium and their interactions with the particle under consideration, whose internal structures may be explicitly taken into account; —

— a stochastic motion on the mesoscopic scale a» A; — a restricted number of deterministic but phenomenological partial differential a, deduced by appropriate averaging from equations, on the macroscopic scale L the descriptions on smaller scales. The stochastic nature of a diffusion is thus more related to the scale of the observation and to information about the medium available on this scale than to the intrinsic physical mechanisms of the motion of the particle.

Mean-square-displacement and diffusion law The mean-square displacement 7)(t) of a particle is defined to be the average of II(t) "i(0)1 2 over all the realizations [t(s)]5 >0 of its motion between the instants 0 and t, where the starting point .i(0) can a priori vary. This quantity can easily be measured by observing a large number of statistically independent trajectories of the particle. Its asymptotic time dependence is called the diffusion law. Brownian motion serves as a reference point: a diffusion law D(t)"4 (t oo) is said to be normal and the factor D limt ,,, D(t) E JO, oo[, if it exists, is called its diffusion coefficient. We speak of anomalous diffusion's if its asymptotic law is different from the normal law. Usually, it is still of the form 136 D(t) . For a stationary process, we have D(t ± t o ) < 2D(t) 2D(t 0 ) for all real t and to, i.e. It + t o r < 2itr 21t01 7 , which implies that 0 < 7 < 2. Experimental investigations show that the departures (7 1) from the normal diffusion law can be due to the presence of random inhomoyenetties in the medium, or to a bias in the motion of the particle, or to memory properties of the particle or of the medium; the theoretical study summarizes these observations by stating that the value of -y depends on the temporal correlations between the points of the trajectory. Weak (or absent) correlations give the class 7=.1 of diffusive motions, including Brownian motion. Strong positive correlations lead to 7> 1 by constructively organizing the successive steps (iii)i> 1 , which thus tend to be aligned in the same direction: such a motion is called persistent or superdiffusive, and the extreme case 7 2 occurs in the deterministic motion Ai = Ai_i of a free particle. Strong negative correlations organize the steps in a destructive way, by directing each of them in the 'Generally speaking, a property described by scaling laws is called anomalous when the exponents occurring in these laws do not have the values obtained in the simplest typical models. This difference reveals the existence of other influences or of particular phenomena, so that we can also use the adjective "abnormal", meaning something less specific. 136 -y can always be defined as the threshold value separating the exponents fi'>.-y such that lim t _,, t —OD(t),- 0 from the exponents f? < 7 such that lim t ,,, t — PT,(t).00. More strictly, we write V(t) t if and only if lirrit—,, log D(t)/log t exists and has value 7.

Stochastic diffusion

224

opposite direction from the previous step, so that 7 < 1; we call this antipersistent or subdiffusive motion. The extreme case 7=0 occurs when

DETAILS AND COMPLEMENTS: AN EXAMPLE OF NORMAL DIFFUSION The diffusion law of a stationary process [f(tit>o is normal temporal correlations between its elementary steps[}' = X j-r are surnmable.

We decompose

.g.t — X0 =

—fir J.1

k.S •

whenever

By the stationarity

the of

[..kt]t ).0, the variables [Vi]i>./ are identically distributed, centered and with correlations >= C7 (j) E R. Assuming that E +r, ICT (j)I < oc, the computation of the mean-square displacement D(t) =<11Sft —,411 2 > gives: -

D(Nr) =

(N

C(j)

lim t -1 D(t) = r -1 t-f ÙO

-N<j
E cr(j) = D < co jEZ

A generalized central limit theorem shows the convergence in law of the stochastic process {(Xt — .X0)/Vik>0 to the centered Gaussian law of variance D as t tends to infinity. The departures from Gaussian statistics (for finite t) can be estimated by means of large deviations results137. If we take _a (minimal) temporal step qr instead of r, the elementary variables ko +ii with correlations Cqr (i) -= < >. The su.mmability of become = El., i>i, the real sequence [C(i)]>1 implies that of the real sequence [Ceir (i)].(>1: thus the diffusion coefficient D does not depend on the (arbitrary) choice of the temporal step T. If the correlations are not summable, we either observe a normal law D(t)/t = D E 10, oci if the correlations adequately compensate each other, or we may observe corrections insufficient to modify the exponent which remains 7 = 1imt ,logD(t)I logt = I. but with D= O or D = oc, or else we may observe an anomalous diffusion law with exponent

-y 1.

The influence of the minimum scale on the observed trajectory The track in Rd left by the particle depends on the spatial scale a at which its motion is observed. Indeed, on the mesoscopic scale a, we perceive only average consequences of a large number of microscopic steps: decreasing a transforms the trajectory into a broken line where before we perceived only a rectilinear motion. DtT, if it is valid for small t, ensures that the mean time The diffusion law D(t) needed to travel a distance a is given by r(a) D -1 /7 a2 f7 ; the velocity measured on the scale a is thus v(a). D 117 al -2 /7 . Note that the temporal resolution r (i.e. the minimum time interval which can be observed) must be less than r(a), otherwise it is the value r rather than a which determines the observed trajectory, and the smaller the value of r, the more complicated the trajectory appears. The natural choice is 7" = r(a). The number of steps (of length a) taken over a time T depends on a via N(a) , T1r(a) ,-, TD 117 a-217 ; the length of the trajectory measured -Y al-2 17 . It follows that L on the scale a is thus given by L(a)= aN (a)=Tv(a) T 137 See Ellis

[1985].

6.1 Spatio- temporal evolution

225

and v diverge as a tends to O except in the deterministic case 7 = 2: the velocity and the curvilinear length of a random walk are not defined 138 ; they depend in an essential way on the resolution a, and increase as a decreases. This is the reason why we study the mean-square displacement D(t); it is independent of the spatial resolution a and of the temporal resolution T.

<>

DETAILS AND COMPLEMENTS:

FRACTAL

DIMENSION OF THE TRAJECTORIES

The highly entangled and self-similar aspect of the trajectories of a stationary diffusion process [X(t)]t>0 with values in Rd motivates the search for possible fractal characteristics 139 . However, because of the random character of the extent and of the patterns of these trajectories, their individual geometric analysis does not really make sense: an "average fractal dimension" must be defined from statistical characteristics of the motion. The time t is a curvilinear parametrization of the trajectories, independent of the scale of observation; it can be interpreted as their "mass" M. The length ID(t) gives their V can be rewritten M( r) r2 1 , from which we average extent r. The diffusion law deduce that in the quadratic mean, the trajectories have dimension140 : dr = 2/7. Together with the expression for L(a) given above, this result shows that the average curvilinear length of a trajectory depends on the segment a used to cover it, and obeys the scaling law L(a) al - df, just as for any fractal curve of dimension dp. Finally, we note that dF does not vary with the dimension d of the medium (the (Effusion law being fixed). We have dF > d if 7 < d/2: in this case a typical trajectory fins in the whole space and even crosses itself infinitely many times.

6.1.3

Formalization: stochastic processes and random walks

In this chapter, we consider only the mesoscopic, hence stochastic description of the diffusing particle. There are two possible points of view: one can consider the set of observable trajectories of the particle; each of them appears as a realization [t * c t ] of a temporal stochastic process [gt ] >0 whose global distribution determines all the statistical properties of the random motion. A priori, [X]t>0 is a process continuous in time and it has values in Rd , but we can reproduce the existence of a finite spatial resolution a and temporal resolution T by extracting a discrete sequence [X„]r2.>0 and requiring it to take values in (aZ)d. This last discretization comes down to identifying all the points of Rd located in the cell of volume a d surrounding it with a single point of the lattice. The increment An = X( n _o r , describes the n-th random step of the particle. The sequence [Aij i,>0 is called a random walk; its elements are equally distributed whenever the —

—

138 This is the physical expression of the non-differentiability of the process [Xt]t>o : the law P(t) , D0 shows that [iC, + ,9 — .X t]/O does not converge in the quadratic mean if —00. 139 See §7.1, Gouyet [1992] or Falconer [1990]. "° dF is more than a geometric characteristic of the sets of points swept out by the trajectories, since it describes a spatio-temporal property, which depends on the visiting order of these points (which could, besides, be visited more than once). Rather, it is a dynamical exponent, and dF > d can be interpreted in terms of the covering of the trajectory by itself.

226

Stochastic diffusion

stochastic process is (statistically) stationary. Knowing it is equivalent to knowing the process, since )?„ =

— The notion of random trajectory [t it ] can be set aside and replaced with the canonical notion of probability of transition p(, s). If the space is discretized, p(g, tit, s) is defined as the conditional probability that the particle is at g at the instant t, knowing that it is at at the instant s; it becomes a density with respect to the variable 9 if the space is continuous. The probabilities of transition p(g, t , s) are reduced to p(t, g, t s) if the diffusion is statistically stationary, and to p(s, — if it is statistically homogeneous; in the latter case, p(s,.,t) is the distribution of the step . t — X. If the motion is Markovian, it suffices to give the explicit probabilities (called elementary) associated to the temporal step At = T in the discrete case or to the times t E [0, .7 ] (for arbitrary r >0) in the continuous case. We restrict ourselves to the permanent (i.e. statistically stationary) regime of the diffusion. To return to the viewpoint of stochastic processes, it suffices to specify the distribution go of the initial position X0 of the particle: the density describing the probability of presence in 9 at the instant t, giving the law of the random variable .kt , is given by:

qt(g) = f P(t,g,t)q0(i)dd

satisfying

gt (g)dd = 1

It coincides with p(t o , g , t) in the case where the starting point is determined: XD almost surely since then go(i) = 6d (i — t o ).

The model of the ideal walk A simple model of stochastic diffusion of a particle in a medium of dimension d is the random discrete walk, of temporal step r, lying on the hypercubic lattice (aZ) d . The discretization is fixed by the minimum temporal and spatial scales r and a of the description. The choice of 7 determines a, which is taken to be equal to the root-meansquare distance covered by the particle over r. If the variance of the random distance covered over T is less than a2 /4, it is legitimate to suppose that the elementary steps link only nearest-neighbor sites. The hypothesis of statistical station arity is valid since we are describing only the permanent regime of the diffusion. If the medium is large enough, its equilibrium state is not disturbed by the particle, which is thus called a test particle. If r is much larger than the relaxation time of the medium, then the medium keeps no track of the passage of the particle; if the particle also keeps no memory of its past, then its diffusion is Marlcovian 141 on the scale r and its motion is thoroughly

It is only under the additional hypothesis of homogeneity that Markov's property becomes equivalent to the independence of the successive steps. 141

6.2

Space-time renorrnalization

227

given by the elementary probabilities of transition 142 :

t

ti ) normalized by

introduced in §6.1.1 e( and independent of ti by stationarity. The Brownian walk is moreover isotropic and homogeneous (§6.1.1). The probabilities of transition over larger intervals of time are expressed by pa,r(i,9,n7) = [pa,,(., 7-]()n, 0 where g E (aZ)d) by the convolution * a is given (still with

[P

a

q]( , 9)

E

ad p(i,) q(±,

)

(di an elementary cell)

ze(aZ)d Expression of the mean-square displacement of the particle The definition of the mean-square displacement of the particle D(t) is given by D(t) = < 11-k(t) — X(0)112 > where the average is taken over the global statistical distribution of the process [X(t)] t > 0 . The relation p(i. o , ,t)qc) ( 0 ) = Prob [X(0) = ±o , X(i)=- 1'1 shows that D(t) is obtained by averaging D( ±- 0 , t) over the initial position I'd) , of given distribution qo(4):

D(t) = fD(5,;0,t)q0(2o)d di'o

where

D(io, t) =

411 2p(4,.i, Odd.i

6.2 Space—time renormalization 6.2.1 Renormalization of Brownian motion The asymptotic properties as t —4 oo of Brownian motion are well-known ( 6.1.1), and we will use it as a toy-model to illustrate the renormalization methods which are expected to be relevant in the context of stochastic processes. Here the renormalization transformation must be spatio—temporal, designed for the associated invariance to reflect the scaling properties of the Wiener processes, which will eventually appear as its fixed points. The analysis of the action of renormalization in a neighborhood of these fixed points will then enable us to trace the emergence of Brownian motion, if it is present, over a macroscopic time interval of temporal processes, and to relate them to a Wiener process. Among the scale changes which simultaneously contract lengths and times, the only g/k, one which leads to a non-trivial limit when applied to Brownian motion is P? nr) are in 142 The label a, frequently omitted, recalls that the "probabilities' pa,,(t, fact densitaes defined on the lattice (aZ) d , so that spatial integration involves the term this normalization, in which the volume a' of an elementary cell appears,

shows up the obvious link between discrete and continuous situations E g E ( aZ ) d a dPO f9ertd f ()ddg. The label 7 is the time step: the temporal variable is restricted to the integral multiples t= nr (n E N); the probability pa,r(., .,nr) is elementary if YE = 1. -

Stochastic diffusion

228

t t/k 2 ]. This is the only one which preserves its (exact) normal diffusion law D(t) = Di. Henormalization can act either on the characteristic function (p(f,t,t) of its increments:

= ço(ti k , k 2t)

(71(p)(f.

where

c,o(f.t, t) = <exp(ift.[X t - X 0] ) >

or on its probabilities of transition p(f, t):

(R k p)(F ,t)

D(R k p,t) = k 273,(p,k2t)

k d p(kf , k 2 t)

-

The relation between the two operators Tk and Rk is straightforward since if 9c, is the characteristic function associated to p, then Tk90 must be the one associated to Rkp. If N o. is defined on (aZ) 4 , of temporal step r, then Rk pa,, is defined on (aZ 1 k) d and has = Tprobability temporal step r/k 2 its limit as k tends to infinity, if it exists, is a continuous of transition on Rd x R. Note that [Rdk > 0 and [71,dk > 0 have a group-theoretic structure for all (the "renormalization group") since: Rk, Rk 2 = Rk 1 k 2 and Tk i clTk2 strictly positive real numbers k 1 and k2 . These are actually one-dimensional Lie groups (isomorphic to (JO, oo[, x)), with infinitesimal generators A and B ( 3.4.1) , respectively given by: :

A= (d Rk dk) (k = 1)

(Ap)(f , t) = [d p + 1 f.Vfp 2tatp](17., t)

B = (dTk I dk) (k = 1)

(18 (p)(5 t) = [-1U.7 .5 cp + 2t (p](fa , t)

- -

Results of the renormalization analysis Each Wiener process, with probability of transition PD(f,t), is the fixed point of all the operators (R) > o; one can check that its characteristic function (pD(ft, t)= exp( Du 2 1I2d) is a common fixed point of the operators (71 )k > 0. The stable manifold of Pp under the action of Rk is the set of the probabilities of transition p whose successive images (RP) >o converge to Pp, in the sense that the stochastic process generated by lir0 converges in law as n tends to infinity to the Wiener process with distribution Pp; this convergence commutes with taking the quadratic mean, and so with the computation of D. It thus appears as the universality class of Brownian motion with diffusion coefficient D since all its elements asymptotically satisfy the diffusion law D(p, t) Dt (at least for t E {k 2 nt o , n < 0}): -

Ern (k2n to) -i.D(p, k 2n to ) ri —■

Ern icT i D ( Rkpri

- D ( pD, to ) = D = 1

fl—

We would have obtained the same result by reasoning on the stable manifold of (pD under the action of Tk. In particular, the convergence of the sequence [Rnk p(2, ,) ] 0 >0 to PD as n tends to infinity reveals the identity of the physical phenomena underlying the discrete model of the Brownian walk and of the continuous Wiener process of law Pp: indeed, renormalization modifies the perception and thus the description of the phenomenon, but not its physical nature. Here, the random walk generated by p(2), corresponds to the description on the mesoscopic scales (a, r), whereas Wiener processes, having no finite characteristic length since they are fixed points of

6.2

Space-time re normalization

229

renormalization, may be used at any scale; we just checked the satisfactory property that they have the same behavior over a macroscopic time. Via the study of the action of flk or of Tk we not only determine the discrete or continuous stochastic processes whose diffusion is asymptotically normal, but also explicitly give the value D of their diffusion coeffi cient and relate them by a scaling transformation, iterated an infinite number of times, to the Wiener process of law PD Among the perturbations of this Wiener process, those which are brought towards PD by renormalization are asymptotically related to it. Conversely, a perturbation of PD amplified by renormalization destroys Me normal diffusion law.

6.2.2

Asymptotic behavior and renormalization of the process

The above operators [Tialk>o, which are adapted to Brownian motion, can be generalized to perform renorm - alization analyses of the large-scale (in space and time) leading scaling behavior of a stationary process [t]€R with values in Rd. The renormalization must include the small-scale details of the motion in a transformation of the process, so as to make its statistical characteristics become observable on macroscopic scales. It is supplemented by a joint contraction of the times (by a factor of K) and the lengths (by a factor of k) in order to preserve the apparent minimum scales of observation. This very standard procedure need not be made explicit on each realization of the process, which would require the use of difficult results on almost sure convergence, but will be directly performed on the characteristic function WI, t) of its increments:

t)

< exp(itl.[X t+e

—

X6])

>

D,E Rd, t E R,

which is independent of 0 by stationarity. Indeed, 9c, is a statistical characteristic of the process depending only on its global distribution, which is computable directly from the observations; further analysis of the effect of renormalization is based only on the properties of convergence in law (the weakest of all stochastic convergences) of the processes, and on the convergence of their moments of order 2, which is equivalent 143 to the uniform convergence on every compact subset of Rd (for fixed t) of the characteristic functions and of their second derivatives (with respect to f./). The function 40, runs over the set 144 :

0 =

40(11 , t ) : FLd x R

C, (p(0,t) -- 1, cp0,0)

1 19,0,01 <1

(p(., t) uniformly continuous on Rd and of positive type

f

on which we define the renormalization operators [7k,K]k>o,K>0 by:

[Tkxçoj(a, t)

(/k, Kt)

K > 0, k > 0, 90 E (I)

See App. 1.2 and Lukacs [1975]. 144 A function ip is said to be of positive type if it satisfies EL I -u >0 Z3 4,(p( for every fixed real number t, every integer n, every n-tuple of elements ft,,, of ad and every n tuple of complex numbers zi ...z, (here * denotes the complex conjugacy). It follows that cp(—ti, t) 143

,

-

Stochastic diffusion

230

O. DETAILS AND COMPLEMENTS: THE RENORMALIZATION GROUP rrk,,K1k>o,K>0

k

The operators [Tk,rdk.>.0,K>o, para.metrized by the a priori independent scaling factors > 0 and K > 0, satisfy the group-theoretic relation: Tk i ,K i o Tic a ,K2 = TIc 1 k3,K1K2

in particular

Tlc,K = 71,K ° 71,1

They form a Lie group of dimension 2, isomorphic to the group (10, +oo[, x) 2 , and whose Lie algebra is generated by the infinitesimal generators:

B1

ad

B1 = (d71,1 I dk)(k = 1)

[Biço](ft, t) =

B2 = (d71,K I dK)(K =- 1)

[132 1(1-1, t) = k8ts00, t)

t)

B2 commute and the subgroup [7jk>0 is a Lie group of dimension 1, isomorphic

to 0 0, -1-00[, x) and with generator Ca =

saB2.

This renormahlation reduces the study of the asymptotic behavior (observed over a time which tends to infinity) to a study in a finite (temporal) domain of observation, by substituting:

Ern 71"K so(., t) for fixed t

n —*

for

lirn [X(t) — (OA co

oo taken individually on each stochastic process P-Ct — 2-Y(0)j tE R is The limit t thus transferred onto the iteration n —,00 of the operator Tk i k in O. The analysis of the trajectories (Tknic so),„, >0 in 40 should determine the values (k, K) leading for an adequate starting point ço -E I to a non-trivial limit P 00 , E 0. Setting a = log K/ log k, this limit is a fixed point of Tk,k., which via a change of variables can be sought in the form:

—0=0k(fit l i",

logt) if t >0

and 9,„ a (t,t=0)-al

y) defined on Rd >< R and periodic The solutions are given by the functions of period a log k with respect to y, uniformly continuous in ZE on Rd (for fixed y), of positive type, such that Ok,,,(0, y) = 1 and Ok(i, y)I < 1 so as to have ça,„„ E The fixed points of all the operators [Tk ,k«lk>o are thus of the restricted form = 04sg(t).filt1 1 /1 where sg(t) is the sign of t (the function (1),„ does not depend on k or on y). These fixed points define the notion of a stochastic process [(t)], E R which is self-similar in law, such that the characteristic function of its increments, containing all information about their statistics, satisfies the scale invariance:

t) = 900 (kt-i, t)

Vk > 0, Vt > 0, VEt, E Rd ,

which intuitively suggests that the amplitude of the increment from 0 to t depends on M I R'. If the moments of order 2 of g(t) — X(0) exist, ço,„ a is twice differentiable

6.2

Space—time renorrra anzation

231

with respect to ft; the Laplacian (with respect to i, for fixed t) of the scale-invariance relation at ft = 0 gives: —

= — k2 <11t Xo = 12 >


i.e. /WO = k 2 D(t) for all k > O. Setting D F._ D(1), the diffusion law of a process self-similar in law for an exponent a is exactly /NJ) = Dit1 2/a • The renormalization procedure is completed by the analysis of the operators [Tk,ko] k >0 in the neighborhood of one of their common fixed points p o,,, a . The elements of the "basin of attraction" of soc,,,„, which converge to under the iterated action of the renormalization operator, are associated to stochastic processes which have the same asymptotic behavior: when the time of observation t tends to infinity, the random variables t -1 / [Xt — X0] converge in law to a single random variable 1-7 , with characteristic function voc,,,(.,t =1). If the convergence of the sequences {Tk,kavh>o to commutes with differentiation with respect to û the stochastic processes associated to the different functions so of the basin of attraction of „0 have the same asymptotic diffusion law D(so,t) D1ti 2/a (as t > oo), where D 1); the rate of convergence can be estimated and the error terms bounded. The technical aspects of this statement are detailed below. ,

—

(>

DETAILS AND COMPLEMENTS: ASYMPTOTIC BEHAVIOR

Let us explain how the study of the action of the operators (Tkx)k>o,K>o on the characteristic function sox0, t) of the increment X t +9 — X 8 of a stationary stochastic process [..keL E R with values in Rd enables us to determine their asymptotic behavior as t oo, and the associated diffusion law. Set:

2,(t) = t — '/[i t— fc o ]

(t> 0)

The renormalization is expressed on the characteristic function 145 11,bz by 1,bz.,(fi, t)=

of

2,„, related to

sox

sox (i't, t), by introducing an operator Tk,k.:

kco,1'2.]0,t)

[7ic ,k.vx](11,t -1 ia , t)

filk,

5 2 1 (ft,t)= I2 (û, et)

This relation a posteriori justifies the choice of the auxiliary process Zr,, constructed precisely so that the action of the renortnalization affects only the time dependence of its characteristic function tP t). If the pointwise limit rk,k 0, 40 X0/t)=00,0 (û , t) exists, then the limit v,,,,,, is a fixed point of all the operators ,,...k,k.„,k (7. ) ).0, so it has the form Woo,o(0 , t) = 00, [80) 5 1tI lia]- If Oct (fi) is continuous at it =.-- 0, so as to be the characteristic function of a random variable Ya , the expression of [ilk,k.02. ](t7t, t) shows that the above limit expresses the convergence in law of Z(t) to 17- ",- as i tends to infinity:

Vt > 0

[ lim rk kŒ(pati, t) = 0 a (fit 1 /a) k ---• co

< ?

lim 11) g (ta , t) =

The value of a for which [rk,k.V,V(ui 1)]k >0 converges in 4) to a non-trivial fixed point as k tends to infinity thus determines the "scaling law" of the process [X t .go] tE R as the time —

145

z

Zc (i) and (Xt +0

go)t l f a have the same distribution of probability, so they have the

same characteristic function.

Stochastic diffusion

232 interval t tends to infinity:

fCt+9 — X0

t i t' cx

(t

00, in law)

This scaling law describes the cooperative behavior of the successive increments: these are organized in such a way that the cumulative effect of N consecutive identical steps of duration r and amplitude 1 is that of an effective step of the same apparent duration time r (after rescaling) but with amplitude NV'. The study of the convergence of the iterates [TPK (px],„).0 thus relates the asymptotic behavior as t 00 of Lt- t+8 — X9l e>0 to that of the self-similar process [t l in"c 1t > 0 of characteristic function 900 ,„, by specifying the adequate exponent et. If the process P-C-t] te rt has finite variance, the function 90x D(t); similarly, is twice differentiable with respect to the variable fi and —6,92(0, t) cc with — A(Tk,kce9x)(0, t)= k -2 7)(k at). If it is legitimate, the exchange of the limit k differentiation with respect to the variable fi gives the diffusion law of the process P-Ct ltE R, namely:

him k -2 13) x(k't) =

k —a. co

6.2.3

= 1)

D g(t)—ItI 2larY,.(1) (t

00 )

Self-similar processes

Fractional Brownian motion We obtain a particular family of fixed points of the group [Tk,k.]k>o by requiring that the stationary process (Xt )1>0 with values in Rd have Gaussian isotropic increments. Their characteristic function is thus of the form 9 ( 5, t)=exp(—u 2 a 2 (t)/2). The self-similarity in law Tic i kc r = p of these processes requires that 904,0 = 0(uiti l1a); the general form of these solutions, which are called fractional Brownian processes, is given by:

40,0 = exp[— au 2 iti 211]

where

H = 11a

> .72 (0 = It1 211

The general relation 9(-5, t)= 90,0* ensures that 9 is real; a is thus a real, positive parameter, so 19(ft, t)I < 1. The real number H is called the Hurst exponent of the process (Hurst [1951]). The mean-square displacement is D(t)= —0u2 u9(° , 0= 2 ° It P H . According to the general result (§6.1.2), stationarity implies that H < 1. The expression of the correlations

l 1 2x < [VV(t) — W(0)] . [W(s) — W(0)] > = a [ t12=11s

it s 2.fi ]

shows that the increments are positively correlated if H >1/2 (persistent motion) and negatively correlated if H <1/2 (antipersistent motion); they are independent only if H =112, which corresponds to Brownian motion. The exponent H thus describes the nature and the importance of the temporal correlations of the stochastic process. The discussion above of the general diffusion law D(t) ,-, t7 applies here (with -y =2H); H=1 applies to the deterministic rectilinear motion of a free particle and H =0 to a "trapped" motion with a variance c 2 (t) = 2a independent of the time t.

6.2

Space-time renormalization

233

DETAILS AND COMPLEMENTS: FRACTAL DIMENSIONS The definition of dr as a dimension of mass is based on the scale invariance D(t) ,--, r, rewritten r 2 M(r) 7 where r = \//7(t7 is the mean (linear) size of the trajectory; it gives the value dr = 2/7 ( 6.1,2). For the fractional Brownian processes of exponent H introduced above, the diffusion law D(t) t 211 gives dr = 1/H > 1, We find dr = 1 for the deterministic motion of rectifiable trajectories (H = 1), dF = 2 for Brownian motion (H = 1/2); the situation dr > d, where a typical trajectory intersects itself an infinite number of times, is obtained for very antipersistent motions such as if < 1/d. If {gt jt>0 is self-similar in law, the invariance 90/k, PO= 9(4, t) of the characteristic function 9(ft ) t) of the increments and the spatio-ternporal scaling law it reflects, enable us to introduce a dimension of similarity in law dp, = a. This second definition is the only possible one for a process with infinite variance. This scale-invariance in law expresses the fact that [Xt+9 fi-6]t -1 / has the same distribution of probability as a random variable Yle„ independent of t and of 0: thus we recover the relation M(r) ra j but in the sense of a convergence in law (weak convergence of the distributions of probability). When they both exist, the dimensions dpi and d'F. coincide. Finally, we can analyze the distribution of probability q(, t) of a motion over a time t, i.e. the distribution of the random variable .gt — fCO 3 like any measure on Rd (§7.2). This approach describes only instantaneous spatial scaling properties and not a spatio-temporal self-similarity; it is useful only in order to quantitatively describe the transient properties.

Stable laws We obtain another particular family of fixed points of the group [7ie,k.]k > 0 by requiring that the associated stationary processes [X] >o are continuous in law and have independent increments. Let us restrict ourselves to a real-valued stochastic process. The characteristic function ço(ti, t) of its increments, which is continuous s) yo(?4, s)o(u,t) by the independence of with respect to t and satisfies 9(u, t the increments, should be sought in the form 9(u, t) = Ro(u, 1)r. The self-similarity in law implies that 9(u, t) = refers 146 to the sign of u. The (fluicr) where the sign solutions, which are called stable laws 147 , are given by

9(ti, = exp( - a ± tlur)

t > 0, u E R

Since the process [Xt]t>0 is real, 9(-u,t) = [oeta, Or implies that a+ = (a - ) * (where * denotes complex conjugacy); the bound 19(u, < 1 shows that Ra± > O. The value yo(0, t) 1 implies a > 0 so that u yo(u, t) is continuous at u = O. To be a characteristic function, this function can must also be of positive type, which gives the upper bound <2. Brownian motion corresponds to a = 2. In all the other cases (a < 2), 9 is not twice differentiable at u= 0, and the associated process thus has infinite variance: the "'For processes [Xtl ee R labeled by R, we would have (p(u,t) = 0 1(t1 lula ) where the sign ± refers to the sign of tu since 4,o(u,t) (p( a, —t). ' 47 The repartition function F of the random variable Z associated to a "stable law" is indeed stable under convolution in the sense that for all real numbers al > 0, a2 > 0, b1 and 62, there exist A > 0 and B such that; b < z). Fa2 ,b2 = F A,B where Fa,b(z) E Prob[aZ —

Stochastic diffusion

234

distributions vt (x) of the increments, defined by vt (x)dx = dProb(Xt+ , - X, = are spread out around their average x 0, and decrease too slowly at infinity for x2 to be integrable; this is called a long-tail distribution. This divergence of the variance can be interpreted as the absence of a characteristic size for the fluctuations: even large deviations from the mean-values can be observed with a sizable probability. DETAILS AND COMPLEMENTS: CRITICAL ASPECTS OF A STOCHASTIC PROCESS

Let U(, t) be a spatio temporal stochastic process describing a mesoscopic consequence of microscopic phenomena. In the absence of macroscopic constraints which are inhomogeneous or time-dependent, an adequate choice of the mesoscopic scales (a, r) makes it homogeneous and stationary (so of uniform constant average < U ( ,t) > U0). The elementary increments t) = U(2-3,t + 7) — U(, t) describe the random nature of the evolution of U at the point x and the fluctuations o5U(i, t) = U(2, t) Uo describe the random nature of the local and instantaneous state of U; thus W(±,t) = 0 for all t if and only if U(Y;) is fixed and SU ( ,t) = 0 if and only if U(, t) is a deterministic quantity. The stochastic processes W(i, t) and 8U ( , t) are homogeneous, stationary and with zero average. Assuming for the sake of simplicity that U is real, we construct the temporal correlations for fixed (depending only on s): -

(s)i < r(o) < (472>

F(s) = < W(, t s) W(i,t) > and the spatial statistical correlations for fixed

C(9) = < 45U(i + 9,t) 611 ( ,t) >

t (depending only on 9):

IC() < C(0 ) = < (W)2 >

observe the evolution SN = 1,v(, Jr) = U (ã ,1Vr) U(.i, 0) (with zero average). If the correlations r(s) have a short-range rstat < oo and are thus summable, we showed in §3.2.1 that the (strong) law of large numbers applies and gives limN, o, SNP/ = 0 almost surely; by a generalized central limit theorem, using F(s)

•

For fixed

i, we

r(0)e-121/7.t.t if 1sI

—

co, we estimate: where DT=

ISNI < (9(<Sk > 1 / 2 )

E

r(s)

<142 >

Tstat

—00<s<400

The situation is then non-critical. In fact, it is critical whenever the correlations r(s) are no longer ummabie, which happens

— if the variance < W2 > diverges, which is the case if W(i- , t) obeys a stable law; in this situation, even the law of large numbers is not a priori satisfied; if the correlation time diverges. This is the case if W ( :, t) is (for fixed i!) the increment over a time t of a fractional Brownian motion of exponent H > 1/2. Indeed, for —

fixed r, we compute: 2r(s) = < W 2 > r-2H [(s r) 2H + s - rF H - 2s 2H] = 2H(2H - 1) < w2 > (rlisi) 2(1-H) + ( 7 /1 .5 1)3-2H

(if

Isl > 7-)

00) of the non-integrable function r(s) « S -2(1-11) corresponds to r,, t .t =oo. This behavior leads to <Sk r(s)(N—Is1)—N2H: we The slow decrease at infinity

(s

6.2

Space—time renormalization

235

always have limN—, 00 SN/N = 0 almost surely (since H < 1) but the standard deviation of SN no longer varies as 'VNT but as (Arr) 11 with the time /VT of the evolution.

• For fixed t, we observe the random variable Uv = fy U(,t)ddtIV (with average U0). The critical characteristics show up on the fluctuation Uv — U0 when the correlations CM are no longer summable: Uv remains random instead of almost surely converging to Uo as V oc, as happens in the absence of correlations. — If 5U(, t) is a stable law, its infinite variance reflects the high probability of observing a large deviation of liv from its average U0, which alone can be more important than the standard deviation °(VC(C)ad/V) which would be given by the central limit theorem if it could be applied to tiv.

The stable laws thus describe critical situations where the local instantaneous fluctuations (in a volume ad) are statistically independent and non-organized but of amplitude large enough to change the macroscopic observation.

t) is the elementary spatial increment (AT, = a) of a fractional — If for fixed t Brownian process of exponent H > 1/2, these fluctuations 6U (. , t) are spatially correlated on every scale. If d = 1, we compute: C(y) = H (2H —1)C(0)

01102(1—H) + O (a/103-21.1

The leading behavior C (i ) e's C(0)(a/11911)2(1—ff) remains valid in any dimension whenever > V 2(11-1) / d ; the macroscopic quantity /iv is thus 119H a. It follows that much more dispersed around its average U0 than in the absence of spatial correlations (when

»

<4

H = 112). The fractional Brownian motions of exponent H > 1/2 are related to critical phenomena because the cooperative behavior they describe is self-similar and generates scale-invariant macroscopic properties.

6.2.4 Renormalization of the probabilities of transition The renormalization procedure for stochastic processes discussed in §6.2.2 in the framework of characteristic functions can also be used on their probabilities of transition. We will consider only discrete random walks on (aZ)d, which is justified by the mesoscopic scale a of the stochastic description of the motion. In order to use renormalization for determining the exponent of a diffusion law D(t),t 2 ier reflecting the self-similarity (—>ki, kat) of the random walk, the main idea is to increase the spatial resolution by a factor of k and to find by what factor k' the average time T of an elementary step (of length a) must be multiplied in order to obtain a statistically identical random walk. We encounter the usual stages of real-space renormalization: — a spatial and temporal coarse-graining, obtained by considering "macro-sites" associated to the cell of side ka and "macro-steps" made up of K = kt/ elementary steps linking these macro-sites; — the computation of the probabilities of transition of this decimated walk;

236

Stochastic diffusion

— a spatial scale change, with ratio k to preserve the spatial resolution a, and a temporal scale change of ratio K =k' , to be determined, so that the iteration of this

renormalization leads to a non-trivial random motion (with an average size which is neither zero nor infinite in finite time). The mathematical implementation of these steps consists of the construction of an operator acting on the elementary probabilities of transition:

[14 ,K(p)1(i

k > 0, R> O.

k d p(lc , 4, Kt)

Here p is defined on the lattice (aZ) d , so 74 x(p) is defined on the finer lattice (a/k Z) d . Similarly, the minimal temporal scale of Rkjc(p) is r/K if that of p is r. The fixed points of Rkx are thus functions of continuous variables (i, 9, t) c Rd x Rd x R.

Consequences of renormalization on the diffusion law

The advantage of this renormalization appears in the transformation of the meansquare displacement (over a time t):

k 2 7:1 [7?4,1(p),t)=D(p, Kt)

k 2 n DN'k', K (p),t]=D(p,Knt)

which iterates as

The asymptotic behavior as t co of the function D(t) is thus transferred to the limit as n oo of the sequence [R.S: K (p)],>0 of probabilities of transition generated by the iteration of 74,K , If 9, t) p*( 9-, t) is a fixed point of k , K , the following exact diffusion law is satisfied:

Vt < co

D (p* , t) = D (p* , = 1) t 2

where

a*

= log K I log k

Suppose the stochastic process generated by p* is of order 2, so that 1)(p*,t o ) < oc. If the images [R ik' jc(P)1n>o of p under the renormalization action converge to p*, or equivalently if 74,k(p) converges to p* as k tends to infinity (the convergence must be uniform so as to be able to exchange the limit n co with the spatial integration), we have:

lim t_2RD(p,i)

=

lim t 2

t = t o knoel

n oo -

(t o arbitrary)

nlirn D [741 ,kOE(P)) tr]

= tcT 2/Cr* D[ liM R rk4a,(p), n —Jo°

to)]

t (Via* D(p*, t o ) = D(p*,t = 1)

D E10,00[

If p* is a fixed point of 14, k . such that D(p* ,t = 1) D E]0,00[, then every probability p belonging to the basin of attraction of p* leads to an asymptotic diffusion law D(p, t) D t 2I . The study of the trajectory n TIZ h.(p) in the space of the probabilities of transition gives the value of a* since it converges to a non-trivial walk only if a = a*. Indeed, the mean-square displacement D(R.Z k,(p), t = 1] converges to 0 as n tends to infinity, to 0 if a < a*, and diverges if a '> a*; it converges to a finite value D E- (p*, 1) only if a = a*. This result shows that all the probabilities

6.2

Space-time renorrnalization

237

of transition p belonging to the basin of attraction of p* have the same diffusion coefficient D = D(p* , 1) Ell), oo[. We naturally recover the result obtained in § 6.2.2 with the operators [7k,K]k>o,K>o, since the characteristic function of the stochastic process generated by 7:1k ,K(p) is Tk jc(c,o) if cp is the characteristic function of the stochastic process generated by p.

<> DETAILS AND COMPLEMENTS: THE RENORMALIZATION GROUP [TZk,K]k>0,K>o The operators P2k,Idk>0,K>0, Parametrized by the a priori independent scale factors k and K, satisfy the group-theoretic relation:

7241,1C1 ° Rk2,K2 =

141k21, K1K2

in particular

12k ,K R,i . K 074 , 1

They form a Lie group G of dimension 2, isomorphic to the group (, JO, -Foo[, X ) 2 and whose Lie algebra is generated by the infinitesimal generators

A 1 = (d7Z)

dk)(k = 1)

A2 = (dRixidK)(K = 1)

[A 2p](t,

) = [(d +

+ Y•Vg)P](t) Y)

[A2P](t3 a3 9) =t(atP)(1 , 2--3,0

We are seeking to evidence a joint spatio—temporal behavior. We want to find what relation ,K(k)(p) as of the type K (k) = W 148 leads to a non-trivial behavior of the transforms ans_orms t k K(k)t] of the diffusion. 00, revealing the asymptotic scale invariance This invariance and the corresponding diffusion law D(p, Dt 2/a are associated to a one-parameter subgroup viz L—k,kcv jk>0 of G, of generator G = A1 + CIA2 since A1 and A2 commute,

6.2.5

The example of diffusion in a disordered medium

We conclude our presentation of the renormalization of random walks with an application to the diffusion of a test particle in a (weakly) disordered medium, aiming at giving a description of the asymptotic behavior observed there. The state of the medium is quenched, ensuring the statistical stationarity of the diffusion, but the random spatial repartition of the diffusing sites destroys the homogeneous, isotropic and deterministic character of the probabilities of transition of Brownian motion. The question is whether diffusion is still "normal" under certain conditions on the dimension d of the medium and on the statistical properties of the disorder, i.e. whether some diffusive motions in disordered media belong to the universality class of the Brownian motion observed in the absence of disorder. The family [Rk of renormalization operators was constructed precisely in order to solve this problem (§6.2.1). Assuming that the disorder is weak enough to be treated as a small perturbation, the local linear analysis of the action of these operators around Brownian diffusion (of order 0) gives the answer: if the perturbation is amplified by Rk , the disorder destroys the normal diffusive behavior. Conversely, all the diffusions which 148 The form K= ka of the dependence K , K(k) is a. consequence of the group theoretic constraint If (10), K(k)"

Stochastic diffusion

238

are brought by Rk to a Brownian fired point are asymptotically normal; the disorder comes in only in the coefficient of diffusion. We emphasize the dual origin of the stochastic nature of diff usion in a disordered medium; the particle moves randomly from one diffusing site to another, as in a Brownian walk, but these sites also have a random aspect (state or position) which is reflected in the random character of the probabilities of transition describing the motion of the particle.

<> DETAILS AND COMPLEMENTS: THE ROLE OF THE DIMENSION d The work of Sinai [1982] in dimension 1 is one of the first advances in the subject of diffusion in a disordered medium. Usually, in physical situations, the one-dimensional systems are simpler than those in higher dimensions; often they are even analytically solvable. However, in diffusion phenomena these systems play a very particular role: their analysis does not clarify higher-dimensional diffusion because their behavior is atypical and actually more complicated. The explanation of this lies in the very severe constraints imposed on diffusion by the restriction to a single dimension: a single obstacle suffices to block the diffusion, and removing a single diffusing site separates the space into two disconnected parts. Sinai (log t) 4 which obtained (with no conditions on the degree of disorder) a diffusion law D(t) is very different from a normal law. The example we will explain here follows from work of Bricmont and Kupiainen [1990], and shows via renormalization that the diffusion law is normal in dimension d> 2 if the disorder is weak. The debate remains open on the behavior in dimension 2 or in the case of strong disorder.

Formalization of diffusion in a disordered medium In order to quantify the random disorder of the medium supporting diffusion by a parameter c > 0, we use a phenomenological model, not specifying the random characteristics of the diffusing sites but directly describing their consequences on the probabilities of transition; we should recover Brownian diffusion if c = 0 (order 0). This model uses the discrete formalism presented in § 6.1.3: the Markov property is preserved since the disorder is quenched, so diffusion is still entirely determined by the elementary probabilities of transition par(, r) where the indices (a, r) recall that the elementary step is of time 7" and length a. But these probabilities are now:

-

inhomogeneous; they depend on

9

-

•i and

t: the distribution of the step

g depends on the starting point re. Consequently, the random walk does not have independent increments; -

— random; the disorder is characterized by the distribution of probability of the random function pa,,, not by one of its realizations 7-3a,,, We introduce the parameter

c into pa,, (writing p), and require that the variance of the random variables 9, T)] zwE (a Z) be uniformly bounded by c2 a -2 d, In this case the medium is said to be c-disordered. It is important to distinguish between the average -‹ >relative to the distribution of probability of the random function p(:, ), and the statistical average relative to the random walk in (aZ)d obtained for a realization p-(aE, T) of I» ) a,r•

6.2

renorrnalization

Space-time

239

In particular, we will compute a mean-square displacement for each realization

its

value is a realization of a random variable D(pV),, t), which must be averaged with respect to the distribution of p(a.',?r to obtain a global statistical characteristic

t)

-- of the diffusion. A possible model is presented below.

DETAILS AND COMPLEMENTS: A TYPICAL MODEL

Since the disorder is assumed to be weak, we shall use a perturbative approach whose order 0 is the random walk with independent steps in an isotropic homogeneous medium specified by the elementary probabilities of transition:

, , r)

E (aZ 4 ),

[2dac] -1

if Ilk - Yll = a otherwise

=O

At order 0, the diffusion is thus exactly normal since at every finite instant t = nr, we have: < M2> (nr) = no,' = Do nr where the diffusion coefficient is equal to Do = a2 /r. If c > 0, the elementary probabilities of transition can be written:

G (aZ ci ))

g, 7)

ki27.(x,

a -d

[(2d) -1

, 9, 7)]

=0

§1 = a if otherwise

This perturbed model is supplemented by the following conditions: 9, r) are random variables by means of (i) for all x, 9 E (aZ d ), the quantities which the statistical properties of the disorder are expressed;

(ii) we take q,,,,, (ï; , T) O if P-9110 a, which restricts the elementary transitions to nearest neighbor sites in (aZ 4 ), as for the ideal walk of order 0; -

(iii) the normalization of p(ot, )? is ensured by requiring that almost always; (iv) the positivity of p. is ensured by requiring that (2d

—

57

—YaZ)d qa,r(21, 7-) = E(

1) > 2d c

T) > 1 ;

r) and g m (x , r) are assumed to be statistically independent if ± V; the this constraint is justified if statistical correlation length of the disorder is less than a; it expresses the act that the disorder does not induce any correlations between the sites, nor, consequently, between the steps coming out of them. In the opposite case, an organization in the structure of the medium may appear, which by having repercussions on the diffusion of the particle would radically modify it; (vi) qa, r (k,g, r) is taken to be isotropic around the joint distribution of the 2d non-zero random variables [qa, r (k , , r)] 9 is invariant with respect to the rotations of g - •i preserving the lattice. For fixed these 2d random variables are thus identical and independent of 9 , so that by (iii), ---= 0 where >- is the average 149 over the statistical distribution of the random function qa,r ; this condition expresses the fact that the disorder is statistically isotropic around every point and that it does not induce any favored direction for the diffusion;

(y) q e,,,(.i ,

-

149 We shall also write >- for the average over the statistical distribution of 14,, induced by the distribution of qa,7.

Stochastic diffusion

240

(vii) the condition -<exp[eq„,,, 9,7)]>- < exp [92 ] for every real number 0 implies that -< [qa ,r(i, , 7)] 2 >- < 1 and ensures that the hypothesis of weak disorder is actually

measured by e <eo < 1, since the variance of the random variable p (aOf, 7 kx,-y, r) is uniformly bounded by ( 2 a -21 ; the relevant perturbative parameter is thus E. The random nature of pa(',), does not mean that pa(c),, fluctuates: once the realization T) is chosen, the configuration obtained is fixed throughout the duration of the diffusion of the particle. This implies that we can obtain only statistical information on it; one of the motivations for studying the diffusion in a disordered medium is actually to find out how to relate the observed characteristics of the trajectory of the particle to the unknown statistical characteristics of the disorder present in the medium. The realizations 0,E,?,. are elementary probabilities of transition (t, 9) 1--, 7.i,9,7) in the usual sense, belonging to the function space .11 4,-r) defined by the constraints (ii), (iii) and (iv). The probability distribution of p(a€, ),. is that of the realizations in .TI ci . If (1) is a real functional defined on (e) \ -x(p)›.- is the statistical average of the real random variable Okp-) Fa,r, a,?) the quantity --<(Ys with respect to this distribution. A realization 7-46,), E ..F4,6;- is in general inhomogeneous, so that a trajectory of the particle coming out of = 0 and its translation coming out of = io - are not equiprobable. The meansquare displacement in a given reali2ation Ito.) thus depends on the starting point. However, if the constitution of the material is a priori homogeneous 15° and if the boundary effects can be neglected, the statistics of the disorder can be assumed to be homogeneous. In particular, the elements [(., g- ) 1—, ptc, ), (i , 9, TA and [(a., 9) ---. j4. ),-(.i -I- ao, 9 + 1:1, r)] of .F,V, ,) are equally distributed realizations of p(ac, ),, so that the statistical average --C » recovers the homogeneity of the motion: nr, 0) >-

-e -‹ ad Eue(az)d y 2p(:, )T (0,9,n7)>-

= -=

< /31 (p(at, ), ,

-

nr) .io)>-

/3(p(a% nr)

The "doubly averaged" mean-square displacement (over the steps with probability i, r) and then over the realizations pt. of the disorder) is thus defined without ambiguity since it is no longer random and no longer depends on the starting point.

(>

The analysis via renorrnalization Determining via renormalization which diffusions in an e-disordered medium have normal asymptotic behavior amounts to specifying a set P(f, D0) of random functions, containing the discrete models /-42E, ), whose values a and r are relaied by a2 = Dor , and Markov processes obtained in the continuous limit T --> 0, such that:

FM E P(e, Do)

klirn Rkp ( c) = PD,

almost sure L2 convergence

'In the sense that the inhomogeneities creating the disorder are introduced without bias, and so are homogeneously distributed.

6.2

241

Space time renorrnalization -

oo[ such that for almost every realization i.e. if p(c) E P(c, D0 ), there exists D, fi(f) of p( 0 , the sequence of Markov processes [54(t)1 k >t generated by the families [RkiPlk> i of elementary probabilities of transition, converges in the quadratic mean to the Wiener process with distribution PD,. Taking the limit k oo in the identity D(RkgE), t)=k -2D ( p),k 2 t) gives

lim t -I 73(p(e) ,t)= D(PDe , t = 1) = D E JO, oo[ for almost every realization fP) of 1)( 0 , since the convergence takes place in the quadratic mean. The strength of the renormalization arguments is that they show that the diffusion coeffi cient D,, already identical for almost all the realizations /5(') of p(€), does not even depend on the random function p() belonging to P(E, Do ), but depends only on the degree of the disorder measured by the parameter c and on the (unperturbed) Brownian diffusion coefficient Do. The set 2(e, Do) thus appears as the universality class of the Wiener process PD,. The proof takes up the arguments already used in the construction of the universality class of PD, in the set of deterministic and homogeneous probabilities of transition. We emphasize here the new difficulties related to the a priori inhomogeneous and random character of the elements of P(E, Do ); the main one is the possible presence of "traps", which must be evaluated and whose influence must be restricted by additional hypotheses on 'P(c, Do).

Traps In the case of a diffusion on a lattice (aZ)d, a trap is defined to be a domain A of the lattice which the particle has a finite probability 0(1) of entering during a given time step 7, and a very low probability, bounded by I] < 1, of ever corning out. The existence of traps follows from the asymmetry of the probabilities of transition r) Pa,r (17, r), so indirectly from the inhomogeneity of the medium. Figure 6.1 below shows a trap consisting of two nearest-neighbor sites. The intensity of the trap A can be measured by computing the average number of steps T(A) during which the particle remains trapped in A, knowing that it is there at the initial instant. For the trap of figure 6.1, one shows that:

71 (A) = (1 -

ri if ri < 1

whereas

T(À) = 0(1) if rl = 0(1)

One checks that this order of magnitude T(A) = 0(1 / 77) remains valid for every trap A having no sites in its interior and such that the probabilities of transition associated to the outgoing steps lie between two values on the order of ri < 1. The order of magnitude becomes T(A)= 0( -1 ) if A has interior sites which the particle can only leave by taking at least J steps each having a probability of realization on the order of q<1. To control the slowing induced by the traps and to avoid its certainly destroying the diffusive behavior conjectured in P(c, Do), we must restrict the probability of the trapping realizations of p. by adding the following condition (H):

> 1 large enough I51, p(ae, )i E P(e, Do) -

kin> 0, Prob[p,(, 9, r) < ] <

A

The detailed renormalization computations, based on a perturbative procedure in e, prove a posteriori to be valid when taking A 1/c (Bricmont and Kupiainen [1991]). 151

Stochastic diffusion

242

for every pair 9) of nearest neighbor sites in (aZ) 41 . This condition ensures that the 9) are rare, and the more intense is the trap realizations of p- presenting a trap (i.e. the longer it holds the particle), the rarer they become, so that altogether, the net influence of the traps does not modify the diffusion law.

Trap (asymmetric diffusion) is a The nearest-neighbor pair of points (•) A = , trap if the probabilities of transition satisfy for example (in dimension d): • , y) = M,X) =1– rj , with n < 1 Figure 6.1

Li rt.

7f--7 •

•

p(.i ,

)

-

= i1/(2d— 1), p(g,) = ri/(2d— 1),

• , = pi =

V) = = 0(4 the average number of steps

Starting from t = 0 in A, taken without leaving A is T (A) = (1 – n )/ 1

1/77.

DETAILS AND COMPLEMENTS: INFLUENCE OF THE TRAPS ON THE DIFFUSION

T(A)r during which the diffusing particle remains in a domain A knowing that it is there at t = 0 does not depend on its previous history since its evolution (of elementary time step r) is Markovian. The quantity T(A) is thus a relevant characteristic for measuring the slowing of the diffusion due to the trapping of the particle in A. In order to estimate the influence of traps, we must relate T (A) to the bounds on the probabilities associated to the oriented bonds coming out of the trap A. Let us introduce the conditional probability Qk(A) that the particle has not yet left A after k steps, knowing that it is in A at the initial instant; the probability of leaving the trap in one step at the step k k 1 is Qk(A) – Qk+1(A), so that: 71(.4) = Ek >i k [Q k(A) – Qk+L(A)] = Ek>i Q (A). — The exact computation of Qk (A) is easy for the trap in figure 6.1: since the particle is one of the two sites of A, it has a probability 1 — n of staying there for one more step, so we have Qk(A) = (1 — 77) k and consequently: T (A) -= (1 – .77)/71 vn 1. — If none of the sites of A is internal, i.e. at each of them the particle has a (weak but non-zero) probability bounded below by 1)> 0 and above by n i < 1 of leaving the trap, we The average time

have:

(A) <(1 – 71)k and so 1/171 (1 — n ')/f < T(A) < (1 — TI )/71 1/77. Thus it is not the size of A but the bounds on the barriers of probability delimiting A which determine the average time T(A)r spent in the trap .4. — If A has internal sites, a particle on any one of them has a probability 1 of still being (1 — 7f) k

Qk

A at the next

step, which of course increases the time it passes there. To explicitly estimate this effect of the internal sites and give an upper bound for T(A), the idea is to separate A into concentric layers A = LJ=0A , such that the particle has: a. probability bounded below by 0
—

6.2

243

Space—time renorrnalization

less internal layer .A1-1, —zero probability of passing in one step from A.7 to a layer Ai_k if k> 2, —a. probability greater than no >0 of leaving the external layer Ao, —probabilities 0(1) of taking the inverse path (towards the interior of A). Averaging if necessary the different probabilities of transition over all the sites of their starting layer to obtain uniform values in each layer, we get a linear recursive formula which can be written with matrices as: [C2k+i] -= M[Qd

with

[Qk] =

where Q (kn) is the probability of being in An at the instant k without having left A, (tot) oo, we have Q = knowing that the particle is in A at i = O. Asymptotically as k is the maximal eigenvalue of the matrix M, given by — )? where (1 — n=0 Q k det(M rd + aid)=0 and A<1. If we only want an estimate of A<1, it suffices to solve the relation linearized in A=0. The explicit computation gives: det(M

Id)

= (-0jorion, _11, 1 and

[d(det(M — Id+ Ard))IdA](A=0) =0(1)

which proves the general result T(A) = °([non]. ...m,„_11 -1 ) if A has jo layers. We recover the fact that the average time spent in the trap does not depend on the size of the trap but only on the various lower bounds on the probabilities of sliding along the outgoing paths. Do). Let us This estimate of T(A) justi fi es the hypothesis (H) imposed on p( e) E consider a trajectory generated by a random function p(oe?,- E P(6- , Do) satisfying (H). For every value of 77, each of its N>1. steps has, according to (H), a probability bounded above by 71A of entering into a trap of "intensity" > O. The realizations of the probabilities of transition of different starting sites are independent by (v), so the probability that the particle re-enters a trap of intensity n > 0 is bounded above by Nn A (neglecting the possible correction due to the correlations between the probabilities of the steps coming out of a site visited several times). If the trap has J < Jr. levels of internal sites, it stays there for a number of steps 0(n —J ) during which its displacement is negligible, whereas this duration corresponds in the case of an ideal walk to a mean-square displacement D(rn—J )--, T.Dn —J . This "catching up" must be subtracted from the mean-square displacement of the real walk after being weighted by the probability of entering into a trap: D reai(Nr) DN7 - (1

—

77A—Jmax )

We deduce from this that trapping of the particle does not affect its normal diffusive behavior when A > Jrnaz (here A is the fixed constant in the condition (11)).

Linear analysis of the renorrnalization In the procedure presented above, the fixed point PDE of Rk, selected by the iterated renormalization of the realizations of p(e ) EP(€,D0), depends on an unknown parameter DE , which it is actually one of the goals of this analysis to compute, once the normality of the diffusion is proved. To avoid this unknown quantity appearing in the equations, it is more convenient, in order to obtain conclusions about the

Stochastic diffusion

244

renormalization flow, to linearize Rk around the ideal random walk p(2),. rather than around the continuous fixed point P,51:

= R k p(2),

where p(:),, =

DRk(PV).qa,T ± • •

This linearization coincides with the perturbative analysis with respect to c and is justified by the hypothesis of weak disorder c < co ‹< 1. Since the random walk is Markovian, R k p is entirely specified by its action on the elementary probabilities of transition: Rie p(c,, 7)

k dp(ki,k9,k 2 7)

= kd

E

E (aZ/k) d , 9 E (aZ/k) d

where

—

adue2+1)6( 4 _ki)8(ik 2

II i<j
where ad EÏE(aZ)d

f(i)6(i — 4) = f(o). The linearized operator can be written:

[DRk(p (a°, r) ).q](i, y1 7 ) = = kda2d E

E

, k, (k 2 — n Or)

nr)

q(i, if, 7) pki,

0
a

n steps

ci.\./k 2 k2

—

n

n —

—

1

1 steps

• k9-

The summation is reduced to pairs of nearest neighbors (i, ii) E (aZ) d x (aZ) d , The asymptotic behavior of the linearized flow generated by the renormalization is obtained in the limit as k ---+ 00 (since R7, = Ric .). Since the random function Rkp is defined on (aZ/k) d x (aZ/k)d, the limit k oc if it exists, must be a continuous "doubly" stochastic distribution P: each of its realizations P is the global distribution of probability (i, 9, i) , g, t) of a continuous and stationary spatio—temporal process with values in Rd. ,

6.2

Space-time renorrnalization

245

Convergence results The quadratic mean convergence (as k tends to oo) of the deterministic term Rkp a(c,,) of order 0 to the distribution PD,, of the Wiener process with diffusion coe ffi cient Do = a2 /7 ensures that: hm D(R k p(23.,t k ) = Do = a2 /7

k —*op

where 1 G t k =

< 1 + rk -2

Since the mean-square displacement D is linear with respect to the probability distribution, we can write for each realization it,), of p:

D(Rk115V-r )tk) -= D(Rkie),,tk)+D([Rkft?,

—

RkP (2, ,) ],tk)

Taking the average -< Y over the realizations p-lie,)r and using tk = 1 (since E [1, 1 rildp, it suffices to prove that converges tk to a D([Rkp () Rkp (°) ], t k) ›finite real number 6, to obtain the convergence: urn 131 (1)(at, )T

n co —

nr

nr)

Ern [5(RkPVT

= DE

DID + 6'

We then show that if d> 2 and if c G c o < 1, it does converge and leads to a diffusion coefficient D E = Do+O(c2 ) which is identical for all the random functions p(E) of the set P (c , D0). Under the same conditions, one can show that the sequence RkiP) converges weakly to PD, for almost all realizations j3(E) of any random function p(E) E P(c, D 0 ). In this sense, the limit PD, and the associated asymptotic behavior are universal, of

universality class "P(E, Do). REMARKS AND BIBLIOGRAPHICAL NOTES Historical works on Brownian motion are due to Einstein [1905] [1926] and Perrin [1916] for its physical aspects, and to Levy [1948] and Wiener [1976] for its mathematical formalization . Further important articles on this subject are collected in Wax [1954 ] . For more recent surveys, see Hida [1980] and Knight [1981], Fractional Brownian motion was introduced by Mandelbrot and Van Ness [1968] and by Mandelbrot and Wallis [1968], [1969]. About selfsimilar processes, and more specifically about stable laws and the associated asymptotic properties and limit theorems, we refer to the monograph of Gnedenko and Kolmogorov [1954], to the article by Sinai [1976b] and to the review by Bouchaud and Georges [1990]. Fractal properties of stochastic processes are approached from a mathematical view point in Falconer [1990], On the problems of stochastic diffusion, we cite for example the very complete survey of Havlin and Ben Avraham (1987], or the more accessible one by Bouchaud and Georges [1990 ] ; Gouyet [1992] give a more physical presentation, detailing the fractal characteristics observed. On the more specific question of diffusion in a disordered medium, important results were obtained successively by Sinai [1982], Marinari et al. [1983], Fisher [1984], Dnrett [1986], Bramson and Durett [1988] and Bricmont (1991], The renormalization procedure presented in P.2.5 is drawn from the work of Bricmont and Kupiainen [1990], [1991]; alternative approaches can be found in Shlesinger and Hugues [1981], Luck [1983) or Derrida and Luck [1983].

Supplement 6A Polymer physics The physical study of a polymer charz is closely related to the theory of stochastic processes (chapter 6) if it is described as the trajectory of a random walk, and to statistical mechanics (chapter 4) if it is considered as a set of interacting monomers. The first approach is based on the notions of ideal chains and of self-avoiding (random) walks (§ 6A.2); these latter walks reproduce the critical properties of real chains and renormalization methods can be designed to describe the statistical properties of their configurations (§ 6A.3). In the second approach, the renormalization methods elaborated in statistical mechanics are applied to a phenomenological Hamiltonian of the polymer

(§ 6A.4).

BA..1

Polymer physics

A polymer is a complex molecule made up of identical molecular "patterns" called monomers assembled together. If a monomer establishes chemical bonds only with its two nearest neighbors, the polymer is modeled from a physical point of view by a linear chain whose links are the monomers; moreover, choosing the minimal scale of the description to be greater than the length ao of the monomers, these appear as one-dimensional constituents whose internal structure is imperceptible. Regrouping, if necessary, 'several basic patterns in each elementary link (we still call these elementary links monomers), we can assume that ao is greater than the persistence length of the polymer, Le. the minimal length which two neighboring sections must have in order for there to be no restrictions on their relative orientations, so that any angle can occur between two successive monomers. The number N of monomers considered is always very large, even in this general sense of the Word "monomer" (of the order of 105 for polystyrene). Physicists are first interested in the statistical properties of a chain after it has reached its statistical equilibrium in which the distribution of its configurations no longer depends on time. We shall restrict ourselves here to the situation observed at thermal equilibrium in a solution of polymers which is sufficiently dilute for the mutual influences of chains on each other to be negligible. The search for statistical laws describing the entire chain in the limit N oo means that microscopic details without macroscopic consequences can be ignored, the main emphasis being laid on the universal properties. We can represent flexible polymer chains as trajectories of random walks, and this relates their study to that of diffusion processes; conversely, this modelization gives a physical interpretation to the results obtained for "abstract" random walks. We can also describe these chains as families of interacting subsystems, in thermal equilibrium

Polymer physics

248

at a temperature imposed from outside, so that the statistical formalism of the canonical ensemble is well-suited. This double aspect of polymers means that they lie at a crossing point of various methods, in particular of various renormalization methods, namely those designed to study stochastic processes (§6.2) and those introduced in the framework of statistical mechanics and field theory (§4.3).

8A.2

Polymer chains and random walks

To study of polymers within the formalism of random walks presented in §6.1.3, we label the N monomers i = 1...N in order of succession on the chain. The chain is described' as a broken line whose successive angular points [M o 0 with values in Rd ; the rectilinear realization of a discrete stochastic process segments kij =ii Ti —']i' form the trajectory of the random walk with the successive steps [Ai = Xi — Xi >1 corresponding to the elementary increments of the process [nix). We define two distances between the monomers i and j: their Euclidean distance 1±j in Rd and their chemical distance lj 4, measured along the chain. By hypothesis on the solvent, the distribution of the configurations is homogeneous and isotropic, reflecting the statistical invariance of the polymer under global translations and rotations in Rd . Thus we can fix 5C0 0 almost surely. Since the monomers are identical, it is legitimate in the limit as N co to ignore the particular status of the endpoints and to assume that the distribution of the configurations ]). E Z (where [] €z) is invariant under shifting of the indices j—+ j+ jo , i.e. by translation along the chain. The steps [A]i > 1 are then identically distributed centered random variables, and the correlation depends only on the chemical distance lj — i. Note that this limit N — > oo appears as a thermodynamic limit (§1.3.3). A study via renormalization of the finite-size corrections, depending on N and due to the presence of these endpoints, is proposed at the end of § 6A.3. We simplify the model by inscribing the chain in a hypercubic lattice (a0Z)d, which restricts the orientations of the steps to the 2d directions of the lattice (figure 6A.1).

The ideal chain model In the above framework, this model assumes that the random variables [AA >1 are independent, i.e. it assimilates the configurations of the polymer to the realizations of a Brownian random walk with values in Rd (see §6.1.1). It is exactly solvable and thus gives a reference analytic description, which we use, for example, as the zero order of perturbative approaches (§6A.4). This model is moreover exact if d > 4 or if the solution of polymers is very concentrated (a "melt" of polymers). We write ao for the root-mean-square length of each of the N steps and we take :?0 a 0 almost surely. The hypothesis of independence of the successive steps is reflected in a remarkable way on the statistical characteristics of the polymer: is2T7 p u to a small ambiguity, since we can associate a monomer to a segment (so a chain

consists of N monomers) or to an angular point (in which case the chain consists of N 1 monomers). The latter point of view is actually better suited for describing the interaction of two monomers via a potential h(ilij (§ 6A.4). The first point of view is better suited to the description of their correlations .

6A.2

249

Polymer chains and random walks

— the mean elongation

<XN> is zero: <XN > = 0;

— the end-to-end distance R(N) =< IPCNi12 >1/2 is given by the relation R(N) = acriff, which is exact for all N;

— the central limit theorem applies to the sequence (111)j >1 and shows that converges almost surely to 0 as N tends to infinity, and its fluctuations around /N gN 0 are Gaussian, of order 0(1/V7V). It follows that the density of probability Piv(f) of the elongation X N of the ideal chain is asymptotically Gaussian, with variance o(2) N; remark that PN(f) is exactly Gaussian (for every N) if the elementary steps already obey Gaussian statistics; since the configurations of the ideal chain are equiprobable, an "inverse microcanonical hypothesis" leads us to take a constant reduced (dimensionless) energy N Le, independent of the configuration; —

— generally speaking, the density of probability PN (f) can be written NN ( ) /MN where Hiv(f)d 4 F- is the number of configurations of endpoints 0 and f' (up to Al among the ArN configurations of N steps. The statistical entropy is defined by SN(f) = log ATN (0 and is related to the thermodynamic entropy by SN(f) = kB SN (F) • The explicit formula SN() = :57 N (0) ± log[PN()1 PN (0 )] , applied to the ideal chain, gives the asymptotic value of its entropy:

5---,N(r) _

SN k (f.

)

d

N (0)

d r2

"N(n\

2R2 (N)

24N

(N oo)

the reduced free energy FN = UN - SN can be deduced from it as follows:

FN (r)

PAT

(0)

dr2 2R2(N)

fIN(0)-F

dr2 2a2N 0

(N --+ oo)

f (f < k E,Tia o ) applied to the endpoints of a chain satisfies the relation: f = [VFN](f = <X N >), which leads in the ideal case to an average elongation given by: < ( N > = fR 2 (N)IkTd, which is linear in f and in N; A weak tension

— the (Euclidean) correlation function gr(i) is defined in the general case to be the probability that there is a monomer at knowing that there is one at = 0; it depends on the modulus r only, by isotropy. Its computation in the ideal case gives gN(r) r2-4 if r < a 0 /V. Its Fourier transform N(q), called the structure factor, then behaves like ip(q) q -2 if qao/i » 1. Note that - 7(4) is proportional to the scattering cross section (for light or neutrons) where _-_- kf - k describes the deviation of the incident ray with wave vector k, , if emerging with wave vector kJ . -

<>

DETAILS AND COMPLEMENTS: THE VALIDITY OF THE IDEAL CHAIN MODEL

The ideal chain is a rather rudimentary model of a single polymer since it neglects the impenetrability of monomers. Overlapping trajectories are present in the model but not in reality, and they bias the results unless their statistical weight is weak enough, in which

Polymer physics

250

case the global statistical properties are not sensitively modified by taking into account the constraint that a monomer should not overlap another. This is the case in the limit oc if the dimension of the space is large enough (d > d, = 4). It is also the case for as N a solution which is very concentrated or for a polymer melt (a "solution" consisting purely of polymers), since then the intertwining of the different polymers causes a screening of the repulsive interactions of a polymer with itself, which thus play only a. negligible role.

••••••• 1011•11••• IMMUNE

111111....11 ER COMM ao

Figure 6A.1 - Self-avoiding random walk

ao is the resolution (i.e. the minimum scale of the description), chosen to be greater than the persistence length of the polymer. (1) the polymer is considered as a broken line made up of N identical segments, identified with the steps of a random walk. (2) it can also be described as an ordered sequence of N+1 points [j]o112 is the root-mean-square distance between the two R =< MN of the endpoints chain; it obeys a scaling law R(N) ao Nu in the limit co. Flory's theory gives N = 3/(d + 2) if d < 4 and recovers the value ii = 1/2 of the ideal chain (Brownian random walk) if d > 4.

The model of the self-avoiding random walk

The hypothesis of independence of the monomers of the ideal chain is often too strong. Taking into account the couplings between the monomers without entering into their detailed modeling, the model of the self-avoiding random walk corresponds to a Brownian walk in which the trajectory is forbidden to pass through a site already

6A.2 Polymer chains and random walks

251

visited. This model is supported by the very short range of the couplings, whose only consequence is to prevent two monomers from being in the same place.

The excluded-volume parameter The preceding model must be supplemented with a quantitative estimation of the impenetrability of the monomers. The classical description, due to Flory and corresponding to a mean-field approach, introduces an internal parameter 14, called the excluded-volume parameter it measures the importance of the repulsive interaction assumed to be identically distributed between all the pairs of monomers sufficiently close to each other in the space Rd . The parameter u has the dimension of a volume and is a statistical characteristic of the polymer; it is given explicitly by u = 2 Eint [cIv Rd(N)] -1 where Éin t is the (dimensionless) total repulsive energy, cN is the quadratic average of the local density of monomers in the ball of radius R(N) and d is the dimension of the space. In a "good solvent", i.e. one which is neutral with respect to the monomers, the monomers feel only their mutual repulsion, and u is maximal (for a given polymer). If a repulsive interaction appears between the monomers and the molecules of the solvent, u decreases until it actually becomes negative in a "bad solvent", where this repulsion is stronger than the repulsion between the monomers. This influence of the solvent can be modeled by an effective interaction, attractive between the monomers whose distance is r E [ro, rib zero if r > r1 and (infinitely) repulsive if r < ro. The passage of u through the value 0, observed for example when thé temperature varies, is called the 6 point of the polymer in solution; starting from this transition point, u is negative and the polymer can fold over onto itself entirely (De Gennes [1975], De Queiroz [1989]). Consequently, we consider only the case where u is strictly positive.

<> DETAILS AND COMPLEMENTS: FLORY'S SCALING RELATION R(N) N' In Flory's theory, the reduced free energy 1.- 4N of a chain of N monomers is obtained by adding to the free energy of the ideal chain the total reduced repulsive energy Zti n t taking into account the average effect of the couplings. This quantity is given by ki n t = WagC 2 Rd 12 as a function of the reduced (dimensionless) excluded volume parameter w au and of the still unknown end-to-end distance R. The main approximation (of mean-field type) is the identi fication of c2N with the square of the mean density of monomers in the sphere of radius R, i.e. -

cls 30c (F) >

laN

A

.;

oc()R —d dd 17,

<

> 2 = R -2d N 2

r
r
J r

The free energy of the ideal chain is approximated by Fr dR2 /2N4, up to an additive constant. Minimizing the sum FN = R2 iNc1/42) wagN 2 R—d with respect to R (for fixed N and Iv) leads for d < 4 to the result:

R N (

)

ao tv

Il(d+ 2 )Nv —

where

v

3 d+2

(d < 4)

If d > 4, this value R(N) maximizes FN, so it is not what we want. We expect the minimum to occur for R(N) > aol5 since the couplings added to the ideal chain (for taking into

Polymer physics

252

account the non-overlapping constraint) are repulsive; this bound on R(N) shows that if d > 4, the repulsive couplings only cause a weak perturbation of the total free energy acV/T1 (provided of the ideal chain and do not modify the ideal scaling law R(N) of this approach is correct). The major drawback that the approximations used above are that it ignores correlations between the elementary couplings and a for on their possible collective behavior; when these correlations play an essential role, other methods, for example renormalization, become necessary.

Critical aspects of real polymer chains Real polymer chains are generally very different from the ideal situation; their convoluted shape can make points of arbitrarily distant labels (i, j) actually be very close and thus strongly correlated. The statistical correlation < A14 > no longer decreases to 0 as the chemical distance lj —il tends to infinity. This critical characteristic is well reproduced by a self-avoiding random walk (where the chemical distance is interpreted as the walking time). Indeed, it is time-correlated and the temporal range of the correlations diverges; it represents the motion of a particle having an infinite memory, i.e. remembering all of its former positions. Studying just a piece does not suffice to describe and understand the structure of the whole chain: only a global vision makes sense here. This likely critical characteristic motivates us to seek asymptotic scaling laws (in the limit as N co) for the different statistical quantities of the chain, i.e. the end-to-end distance R(N), the probability density PN(f), the correlation function g N (F) and the structure factor iN(4). For example, Flory's theory gives the leading asymptotic behavior R(N) aoNv with y = 3/(d + 2) for d < 4 and I/ = 1/2 if d > 4; this latter value v = 1/2 supports the validity of the model of the ideal chain in dimension d > 4. Flory's law also gives < 11.gN/N11 2 > the quantity j-C N /N still converges to 0 in the quadratic mean whenever d > 1, but its fluctuations around 0, of order N— ( 1- - "), are more dispersed than in the case of the ideal chain if d < 4.

0, DETAILS AND COMPLEMENTS: CONNECTEDNESS

CONSTANT AND EXPONENT

7

To better describe the critical aspects of a random walk, we construct for z > 1 the generating function G(z) EZ =0 A1N z —N , formally analogous to the partition functions used for phase transitions: the role of the energy levels is played by the number N of monomers and the role of the inverse temperature /3 by log z. Assuming that all the admissible con fi gurations of N steps are equiprobable, their number )1/-N is the degeneracy of each "level' N. The fact that G(z) is decreasing and the boundary values G(z =1) = +cc and G(z = oc) = 0 ensure that there exists a unique threshold value 1 < z c < oo such that G(z) < ao if z > zc and below which G(z) diverges. The value z, is called the connectedness constant of the chain; it describes the asymptotic behavior of Ariv via the limit lim JVN/N1 = z c >

N—. co

Thus, z, > 1 can be interpreted as the asymptotic mean number of admissible orientations possible at each step. One then shows that the divergence of G at z = ZG obeys the scaling taw

G(z)

(z — ze ) Y

6A.2 —

ArN =

Polymer chains and random walks

253

For an ideal chain on a lattice where each site has m nearest neighbors, we have m N and zc = rra; the exponent takes the "normal" value y = 1.

— A self-avoiding chain does not overlap itself, which removes at least one of the m possible directions on the lattice; we have Ariv < M(M — O N-1 , SO z1 < rn — 1. One shows z eN which agrees with G(z) (z — z,) -1 if z is near ze ; that asymptotically, JVN "-0 the exponent y depends only on the dimension d. The proof of the scaling law satisfied by G uses a geometric renormalization which also applies in the study of percolation (§7A.3). Its principle is to interpolate the admissible configurations of the random walk by broken lines of Nlk macrosteps each made of k successive elementary steps (for fixed k > 2) (figure 6A.2). The essential stage is the description of the admissible configurations of a macrostep and of the constraints imposed on their assembling; for example they must be self-avoiding. They should make the set of the possible configurations of the chain of macrosteps as similar as possible (up to a change in the size by a factor of k) to the set of the admissible configurations of the initiai chain. The generating function G(z) = E [2) z N P)) of the initial chain, where the

N(v)

sum is over all the admissible configurations [i..] with steps, is suitable for taking a partial trace, after which the summation is over the configurations of the renormalized chain (made of macrosteps); G is thus related to the generating function RkG of this chain. The determination of the fixed points and eigenvalues of this operation Rk enables one to prove the scaling law G(z) (z, — 2) - 1 and yields the values of z, and of 7.

The various characteristic lengths of a polymer In conclusion, let us note that it is important to distinguish carefully between:

— The range ro of the physical interactions between the monomers: measured in terms of Euclidean distance, this length is very short, on the same order as the length a o of a monomer, whether the chain is almost ideal or on the contrary critical. The model of the self-avoiding random walk reproduces via a purely geometrical constraint the effect of a "hard-sphere" potential describing the infinitely repulsive contact interaction of two impenetrable and undeformable spheres of radius r o /2: V(r) = +oo if r < ro

V(r) = 0 if r> ro

The statistical correlation length j along the chain: this curvilinear distance, measured by following the labeling, is a statistical quantity, defined to be the characteristic length of the statistical correlation of the steps i and

ci

>

j. It gives the order of magnitude of the number of monomers (counted along the chain) whose position and orientation depend on those of a given initial monomer. If we consider the chain as the realization of a random walk, j is the temporal correlation range between the increments. The divergence j co expresses the critical character of the chain.

— The Euclidean statistical correlation length defined to be the characteristic length of the pair-correlation function gN (F.), it gives the order of magnitude of the Euclidean size of a segment consisting of j monomers, so it diverges with J.

Polymer physics

254 6A.3

Geometric renormalization methods

The critical nature of a polymer is not a consequence of the form or of the (always very short) range of the interactions between the monomers, but of the organization of these interactions, revealed by the divergence of the Euclidean and curvilinear range of the statistical correlations. This organization is ignored in Flory's theory, which takes into account the binary couplings but not the correlations between them, and one thinks naturally of using renormalization tools to analyze its consequences on the overall scale of the chain. It turns out that renormalization can be applied within a global statistical description, without its being necessary to perform a detailed analysis 153 of the contributions of the couplings in the different configurations of the chain. Its principle is to account for the effect of the correlations existing between larger and larger groups of monomers into a modification of the statistical parameters, in successive stages. We present two typical renormalization models, which via iterations bring the extreme cases of quasi-ideal chains and self-avoiding chains back to solvable models.

Renomalization of a quasi-ideal chain For a quasi ideal chain, for which the statistical correlations between its N monomers remain in the short curvilinear range J < N, renormalization reduces the apparent range of these correlations and brings us back to the situation of an ideal chain. This is done by replacing the initial chain by a chain of N I k macromers of curvilinear extent k » J, sufficiently large for them to be considered as statistically independent; then we express their average size b (or their variance b2 ) as a function of the length an of the monomers and of the correlations eliminated from their explicit description. The gain in this procedure is the independence of the obtained "macromers", which allows us, for example, to make use of the relation b \tic. The renormalization amounts to a change in the minimal scale of the R(N) description: at the resolution b instead of a o , the polymer appears as an ideal chain. -

<> DETAILS AND COMPLEMENTS: SOME TECHNICAL DETAILS The end-to-end distance of a chain (1-1.0 1
EE<

Aie i=1

where C1 -E-<

N-1

>=

i=1 :7=1

E

(N—i/DC,

1=1—N

> depends only on the chemical distance I. One shows that: liM

N

R2 (N) N

+00

+ 00

Ci

b(2)

whenever

E1Cd< oo

—00

This result makes the term "quasi ideal more precise: it can be applied to any chain whose statistical properties are invariant under index shifting and whose correlations are summable. borN is then asymptotically satisfied as N co. The value bo The scaling law R(N) -

' 53 Although this is the approach considered in § 6A.4.

6A.3

Geometric renormalization methods

255

appears as the effective length of a monomer, once the effect of the statistical correlations is o since the repulsive taken into account; it is greater than the real average length ao = N/Gr the characteristic scale of interactions between monomers tend to "unfold' the chain. If j is the correlation function 1 l''' Ch the sections made of k >> j monomers can be considered to be statistically independent. Indeed, their correlation is zero if they are not consecutive; if they are, then it is bounded independently of k since it involves only the segments of length j situated at their endpoints: its relative value is negligible if k is large enough. These sections, of average size b = bo jc, form the elementary segments of the renormalized ideal chain replacing the initial quasi-ideal chain.

o

10

Figure 6A-2 - Renormalization for polymers The basic principle, comparable to the method of spin blocks (§ 4.3.1), is the construction of macromers of k monomers (here k = 2) forming an effective polymer which is simpler to analyze. The diagram represents two successive decimations. They are supplemented by a transformation of the parameters of the chain. The renormalization is iterated until a fixed point is reached, or at least a situation whose statistical properties are more suitable for computations, for example perturbative ones. It reveals the self-similarity of the chain and the scaling laws satisfied by its statistical characteristics. If the chain is quasi-ideal, renormalization brings us back to the situation in which the macromers are statistically independent and form an ideal chain. If the chain is self-avoiding, the renormalization can be made explicit as a transformation of the average size a of a monomer and the excluded-volume parameter u; it leads to a non-trivial fixed point if d < 4 OT to the ideal-chain model if d > 4.

Numerical approach to renormahzation The numerical implementation of renormalization consists in randomly constructing a large number of admissible configurations of the real chain, of No monomers of average size al) , then performing the geometric renormalization illustrated in figure 6A.2. The statistical analysis of the decimated configurations gives straightforwardly

256

Polymer physics

the average size a l of the N1 = No/k macromers. The advantage of this method is that one constructs the transformation ao a l without having to explicitly perform the partial summation of the correlations, which avoids the error caused by the fact that these sums can be expressed as functions of the effective parameters only in an approximate manner. The fact that renormalization preserves the end-to-end distance R enables us to deduce u from the knowledge of ,j,,„x successive iterations =No k -i, ai)j<j,.n .,, for je and n such that n j o <jm , the exponent I) is given by: log(ajo /ao)

Or

jolOgk

u=

log(an+io/aio) n log k.

Testing the independence of I) with respect to jo and n in the second expression proves the validity of the scaling law and ensures that the asymptotic regime is reached after less than jû iterations.

Renormahzation of a critical chain The typical configurations of a critical chain present numerous /oops. Here we consider renormalization in the framework of the global statistical description of the chain, where the interactions between the N monomers of average size a are described only by the excluded volume parameter u. Renormalization transforms the parameters a and w va -d , by integrating the effect of the statistical correlations into them. Once we have chosen a decimation factor k, the renormalization Rk transforms the triple (N, a, w) into (N1 , a i , w i ) or N1 = N/k, where a l is the average size of a macromer and v i = w i g. is the excluded-volume parameter of the renormalized chain. The value of a l takes into account the correlations between the monomers inside a given macromer, whereas that of w I depends on the correlations between distinct macromers. R.k can be written: -

= NI k = aVrc [1+Ak(W)] wk 2— ( d 1 2 ) — Wk(W)] a? = coj is the size of a macromer in an ideal chain. If w > 0, the repulsive interaction (depending on w) between its k monomers tends to "unfold" it, which is reflected in the factor [1 + Ak (w)] correcting a? in al , where Ak > 0 if w >0; if w =0, we recover the ideal case, so Ak(w----.0) = 0; v i is an effective parameter measuring the overall energy of interaction when it is equally distributed among all the pairs of nearby ( in real space) macromers. It thus implicitly takes into account the organization of the couplings between the monomers of two interacting macromers. In Flory's theory, the interaction of two macromers involves k 2 pairs of monomers assumed to be equivalent and independent, which gives =k 2 u. However, the macromers are deformed by their mutual repulsion: the number of pairs effectively interacting decreases, which is reproduced by the factor [1- Wk (w)] correcting ta? in u l , with Wk > 0 if w >0; moreover Wk (w= 0) = O.

6A.3

Geometric renormalization methods

257

The molecular study gives access to the real microscopic couplings between the monomers, which allows one to compute (at least numerically using molecular dynamics methods) the functions Ak and Wk if k is chosen small enough. Iterating nk generates a sequence (Nj , ai, wi ) i >0 . A qualitative argument suggests the existence of a non-trivial fixed point for certain very convoluted polymers, called the "blob model" (the typical example is given by polyelectrolytes, i.e. polymers whose monomers are ions): the molecular structure and the functions Ak and Wk deduced from it ensure in this case that the macromers of sufficiently large order j > jo are voluminous enough in Rd to behave like hard spheres: t varies as al, so wi =taia.T d tends to a finite limit w*, which is a solution of

ws k2-(d12) [1

—

Wk(W * A =

<=>.

ID *

= 0 i.e. Wk (W * ) = 1 - 0 -4) / 2

Since Wk> 0 as w >0 and k> 2, there exists no non-trivial fixed point (w* > 0) unless d < 4. Identifying w2 with w* is legitimate for j > J large enough, in which case the renormalization equation for a simplifies to: ai +i = ai \rk: [1 + Ak,(w")] = rkai

where rk = \ifc11.

Ak (1.0 * )] >

The sequence (a)>i is thus asymptotically geometric; this result expresses the selfsimilarity of the chain at large scales I> aj >a0 0 2 . To explicitly determine the scaling law satisfied by the end-to-end distance R(N,a,w) as N tends to infinity, we use the fact that it is preserved by Rk and can be written in the form R(N, a, w) a 0(N, w). Asymptotically, ai +1 — rkai and wi +i wi w*, so that Ri = R)+1 can be written rk0*(Ni lk)=0*(Ni ) where 0* (N)E 0(N, w*), from which we deduce that:

0*(1V) N'

where

is =

log rk, log k

R(N,a,w) , aj (N1C -J )" ac(k, w)N v

where c(k, w) is independent of a and N. The renormalization analysis, here supplemented by a molecular study, thus proves the scaling law R aN' which is valid for every a and w if N is large enough, and it also gives the value of the exponent is. If d> 4, the only fixed point is w* = 0, corresponding to an ideal chain: we recover the result stating that all chains are ideal in dimension d > 4 (r = Vi does imply that is = 1/2). The quasi-ideal chains are the chains with Wk 0: renormalization makes their parameter w tend to w* = O. In that case, the parameter w appears as an inessential quantity, which a posteriori justifies the renormalization of quasi-ideal chains presented at the beginning of this paragraph and acting only on the average size a of the monomers. In this procedure, w* =0 implies that rk =Vii, which confirms that the quasi-ideal chains belong to the universality class of the ideal chain; this result is quite satisfying physically since we showed earlier that the two models can describe the same polymer at different scales. The existence of a critical dimension d, = 4 above which the solution is known enables one to consider using perturbative methods in order to solve the situations d < d,, the (small) adequate parameter then being e = d, d = 4 — d. —

258

Polymer physics

DETAILS AND COMPLEMENTS: FINITE-SIZE EFFECTS IN THE FREE ENERGY

The existence of -a fixed point w* of R. allows one to describe the dependence of the reduced free energy F(No, wo) on the number No of monomers of the chain, in particular the distorsion stemming from the particular environment of the endpoint monomers. Writing for the middle of the j-th segment, .-FI(No, wo) can be written:

F(No ,wo ) = — log f

.1 e -1."'0. - 0 (21— zwo ) ddii

, , 2 N0 ) is the reduced effective Hamiltonian of the chain in a where 7-1No , w ,3 configuration ...4/0 ), depending on the parameters No and W. When No is a multiple of k, the decimation is reali7ed by constructing N1 = No/k macromers made up of k successive segments and centered at the centers of mass itivi ) of these k-tuples. The transformation of the associated excluded-volume parameter is the component wo w1 of P. The transformation of F is obtained via a partiai integration eliminating the variables iivo ) describing the No monomers and replacing them with the variables (i ll ) the N, describing macromers. To this effect, we insert: —

H

E

_ k-1

j=1

ik j+g iddifi z

0
into the integrand of .frI(No, wo) and exchange the integrations, keeping those over (VI ...40 and integrating the ones over „ Without needing to explicitly perform the computation, which avoids knowing the Hamiltonian 71 explicitly, we obtain a relation of the form:

F'(Aro,wo) — P(Ari,w,)

No

(k 1) f(wo) - g(w o ).

g(wo) is a contribution related to the particular status of the endpoints, so independent of N. The other term follows from integrating over (k 1) degrees of freedom in each of the NI. = No/k macromers; f(wo) would be constant in the ideal case. Like the functions Ak and Wk above, f and g can be deduced from a microscopic analysis or from numerical simulations based on molecular dynamics. Starting from a value No = 10) °, we can iterate the renormalization Te,k jo times. If No is large enough, we can assume that the fixed point w* is reached after J < jo renormalizations and neglect the transient steps j < J; using k -2 te— i° = (k the approximation 1) -1 this gives a relation showing that the effects of the endpoints increase with N as log N: -

-

(No , w o)

P(N

,

= 1, w*) N f ( w* )

lologgN: g(wt)

6i1.4 6A.4

Polymers in statistical mechanics

259

Polymers in statistical mechanics

Taking the repulsive interactions between the N monomers of a linear polymer into account via the excluded-volume parameter u and the average size a of a monomer appears as a "mean-field" approach, since u is defined from overall statistical properties of the chain. The major drawback of this model is the global and deternunzstie nature of the parameters and a: they do not vary along the chain or with the configuration of the chain, and this is noticeably different from the situation in which we have a real self-avoiding constraint. The analyses given in the two preceding paragraphs, even those improved by renormalization, do not give any role to the loops and to the possible inhomogeneities of the spatial distribution of the monomers, To obtain better statistical predictions, it is necessary to describe the couplings between the monomers by modeling their interaction potential. This second approach formally resembles the situations encountered in statistical mechanics in the study of critical phase transitions. The study of the statistical properties of polymers, or more generally of random walks, is now based on the computation of the probability density PN (i) of one of its endpoints N = i knowing that the other is at .1 0 O. The initial step is to express PN(i) as a path integral whose integrand involves a phenomenological Hamiltonian locally reproducing the self-avoiding constraint; the integration "variable'" is the path ±(t) describing the configuration in the real space R d after a continuous extension of the discrete description (i;i)0
DETAILS AND COMPLEMENTS: EXPLICIT COMPUTATIONS As in the case of spin lattices, the discrete configurations (ii)0
In fact it is a function ±(0); for more details on path integrals, see Schulinan [1981] or Kleinert [1990 155 See 4.3.2 or Ma [1976 . 156 See for example de Gennes [1974 154

].

]

260

Polymer physics gives the probability density Po(i, 7):

P0(,

(11)0(Ali T) =

=

Pl

dI 2

d

exp

271-Dr

z2d -- ) 2DT

In order to reproduce the self-avoiding constraint and also any possible influence of the solvent, we introduce an effective (dimensionless) Hamiltonian H([il, 7), appearing in a "Boltzmann factor' exp[— H([], 7)] weighting the elementary probability dPo( 7) of the Brownian random walk in order to give that of the self-avoiding walk, denoted by dP( 7 ):

7) = e -11([47) dP0( [i]

dP([i] The conditional probability

P(, r) is thus expressed as a 11 "'7) dP0(ki

P(2, 7) =

Li, 7)

path integral: (with i(7) =

7)

[t ]

The statistical hypotheses on the polymer lead us to choose

([4

r

7) = fo di

f

t h(1*(t)— x(s)II)ds —

1 — 2

H([], T) of the

form:

fo fo /1(11s(t) — s(s) I ndi-ds T

The phenomenological homogeneous and isotropic potential h(r) is chosen with very shortrange To, so that h(11± — gH) is zero unless i and 9 are very close, separated by a distance smaller than ro; we speak of a (point of) contact of the chain with itself. Expanding the Boltzmann factor as a series transforms P(i, 7) into a sum of contributions [in (i, T)] n >o, where h(i , r) = P0 (2 , 7) and the (In)n>1 are given by:

ln(i, 7)/07. ..fordt i dsi •-• dP0( [i] h is the term of order 0 since it would coincide with P(2, T) if no constraint were imposed on ;421 +1

the chain. The term In describes the correction with respect to the statistics Po of a Brownian walk, due to configurations having exactly n points of contact. We change the labeling of Sn) into an increasing sequence 0 < 01 < ...< 02 n < r and set each (2n)-tuple (t1, s 1 , . . = ±(9j), with the convention that o=O and 92,2 +1= i. The (2n +2)-tuple is fixed, so that we can exchange the tinte integrations and the path integration on the subset consisting of the continuous trajectories [t] which interpolate the 2n + 2 positions )i=0...2n+2 in this order. This stage is essential since taking the path integral on this subset first involves a discrete Brownian ra,ndom walk on each interval 10j, 9, +1], where it gives the OA. It remains only to perform the time integrations and contribution Po(i +1 — 27L : one spatial (no longer functional) integration over the 2n variables

(-1r f 'I

r

n• j Jo * • *

2n

2n

f(9 ... 9 2n)

=

.

where f is given by:

f01 • - • Oan }del • - • d92, 1

ddYj 1=1

_

91 + 1 - 91)

H 41=1

0( 2k ) fl,

▪

6A.4

Polymers in statistical mechanics

261

writing rfo for the permutation transforming Pi function f(O i ...02n ) satisfies: for ... for

fo i

=1...2n into (t 1 , sl, . ,

t

,

Every

02rode1 . de2n =

do„ Jo

d92 J.:, do,

1

where the sum is over the set 82 n of (2n)! permutations of {1, 2, ..., 2n}. The time integration can thus be restricted to the ordered (2n)-tuples. The integrand f here has a particular form, since it must be invariant under the permutations exchanging 2j and 2j -I- 1 and also under those exchanging the pairs (2j, 2j + 1) and (2k, 2k + 1); the set of these permutations and of their products forms a. class SI, of order (2n0. The integration then involves as many distinct terms as there are permutations in the quotient S2,/SL, i.e. distinct terms in the factor (E, Es„, ...). Each term appears with the multiplicity (2n n!). A diagrammatic analysis helps to compute P(i, r) by graphically representing the various contributions, thus making it easier to count and estimate them. For all j E {1,2 ...2n}, and 'jr_+,1 are joined by a line corresponding to the factor Po(pi + i — yi 3 Oi+i — 95); each factor h(lIgi — is represented graphically by a "bridge" between gi and The number n of contributions h determines the order n of the diagram and indicates in which term In it appears: (-1)I is the sum of the contributions of the distinct diagrams of order n, each counted exactly once. An example is shown in figure 6A.3.

•

•

•

_• • • N_•••• • " •• • • • •

• 1

•

•

92

Figure 6 A.3

•

I

93

■■■

•••••••■■

•

94

- Diagrammatic analysis

The integrand in the contribution represented by the graph (with go = 0, .1 and 9i E Rd ) is given by

Po(Yi+i —

93 )

0<j<4 It contributes to

P(, r), in 12, 04

d04

Jo

after integrating (0 < 91 < ...< 04 < 7)

02 d03 1 163 c 92if del f

ll d'i ldd92dd 9.3dd 94

Polymer physics

262

In the case of a self-avoiding random walk, we observe that the graphs produced by this analysis are identical to those coming from an n-vector Landau—Ginzburg model with n = O. This correspondence enables us to straightforwardly transpose the results and methods associated with this model to polymers, and in particular the renormalization techniques used in conjugate space to achieve the explicit computation if the expansion above does not converge fast enough. Let us give some of the results obtained in this way; their statement b) of P(i, t), given by: involves the Fourier—Laplace transform

cm,

e — bi 13 (, t)dt

b) = Iff , dd el"

Using renormalization arguments analogous to those explained in chapter 4, that there exists a critical value b such that:

Q( q= 0 , b) lb b,) q1-2

(if lb— (if q

one can show

6,1 0)

which we regroup into a universal scaling law:

Q(4, h)

Q[(g(b)) -1 ]

where

e(b) ,-,

bc i —v

0 and The function Q(z) is analytic with respect to its real argument z; it is non-zero at z must behave like Q(z) .z 1-2 at infinity, which leads to the conjecture that 7 = (71 2). —

In the case of a Brownian random walk, the explicit computation is possible and gives:

(2,0,0 = [b + q2 D/2dr 1

SO

4- (b) 2 D /2bd

The critical value of the parameter is b, = 0, and the exponents have the values 7 = 1, = 0, v = 1/2, satisfying the conjecture y = v(t) — 2). In the general case (non-zero Hamiltonian), returning to real space—time enables one to deduce the behavior of P in the limit z oc, t oo from the behavior of Q as q —} 0 , lb — bcl Ch

P(Zt —v )

P(.ï, t) =

V', where in Thus the characteristic length of the random walk under consideration is e can also show general y differs from the value v = 1/2 obtained for a Brownia.n walk. One that the probability P(= 0, t) that the trajectory passes again through its starting point, satisfies a scaling law:

=

t a-2

(t >

Just as in statistical mechanics, there exists a critical dimension (here d,= 4) above which the mean-field approach is valid and leads to "normal" values for the exponents y, r?, ti and i;e; which coincide with those obtained for a Brownian random walk. For d < 4, these exponents can be given explicitly as a perturbative series in powers of c = 4 — d. Knowing P (i , t) gives the reduced entropy:

t) =

=

t)

log[P(i, t)IP(i

0,0]

— With no self-avoiding constraint, all the configurations have the same reduced energy U0; the presence of this constraint means that it is necessary to take into account the additional

6A.4

Polymers in statistical mechanics

263

H ([i5], t) coining from the repulsive interaction between the monomers for each configuration [4, so that the internal energy can be written: reduced energy

Ci (i, t ) We then deduce

=— f [r ]

d P( [

] I i , T ) [ rfo + 11 ( [ ±], t ) I

the reduced free energy: P(i, t) = 0 - (i, t)

—

(i, t).

o

REMARKS AND BIBLIOGRAPHICAL NOTES

The basics of polymer physics can be found in Flory [1969], [19711. For the description of the scaling properties of polymers, the fundamental reference is De Gennes [1984]; he explains certain "geometric" renormalization techniques introduced in §6A.3. The "Hamiltonian" renorrnalization presented in §6A.4 by analogy with the renormalization of spin systems is detailed in Ma [1976]; the use of diagrammatic methods in this context is treated in Kleinert [1990]. Specific references about the Flory exponent /./ are Cotton [1980] for its experimental evidence and Le Guillou and Zinn-Justin for its computation using fieldtheoretic renormalization methods. On the simulation of polymers via the Monte Carlo method, we refer to Brender et al. [1983] or Sokal [1989 ] for basics and statistical foundations, Kremer and Binder [1988 ] for lattice simulations and Leontidis et al. [1995] for simulations in the continuum. Des Cloizea-ax [1975], De Gennes [1975 ] and De Queiroz [1989] consider the so-called "0 point" transition. Standard real-space renormalization for polymers is presented in Maritan et al. [1989]. Some extensions of renormalization are suggested in Family [1984 ] , Oono [1985], Jug [1987] or, for the study of branched polymers or gels, in Stanley et al. [1982] and Duplantier [1986].

Fractal structures A book about fractals would deal with most of the topics where we have already encountered renormalization, merely because the physical principles underlying the presence of fractal structures are the same ones which make renormalization methods work. The essential notions of scaling laws and of scale invariance, of self-similarity and of universality appear in both situations (§ 7.1). A fractal structure is a visual rendering of the characteristics ensuring that renormalization methods are relevant for the analysis of the system in which it appears; the expression of its self-similarity guides the choice of the formalism and the construction of the renormalization operator. This chapter continues with the study of fractal measures; their local singularities can be hierarchically described by their dimension spectrum, determined by multifractal analysis, or revealed by renormalization analysis (§ 7.2). The chapter ends with an introduction to the wavelet transform: this local spectral decomposition is based on the scaling properties of the objects analyzed, so that it is natural to consider its relations with renormalization. The notions of measure theory used in this chapter are recalled in appendix 1.

7.1 7.1.1

Fractal geometry Critical aspects of fractal structures

Fractal geometry and renormalization share a common physical framework: the class of critical phenomena. Indeed, these phenomena are revealed by the appearance of spatial or temporal fractal structures, whose presence gives a. sure and simple clue to the adequacy of a renormalization analysis. Indeed, by definition, fractals have a structure which is self-similar on every scale in the sense that the characteristics on the scale i can be deduced from those on the scale 12 by a dilation. Fractals give a concrete form to the scale invariance on which the construction of a renormalization operator adapted to a given system is based. They provide visual examples of collective critical behavior, relating very different scales within a self-similar hierarchical organization: they are critical objects in the sense that they have no characteristic scale. A new kind of geometry is necessary to study fractals, because the usual tools of description and measurement are inadequate: for example the length of a fractal curve depends on thc resolution (minimal scale of the steps) with which it is observed (figures 7.1 and 7,2). Fractal geometry provides a guantitatzve description of the self similar structure of the phenomenon, by introducing various scaling laws whose exponents are related to -

Fractal structures

266

the fractal dimensions, generalizing the Euclidean notion of dimension to non-integral values (which become integers when the fractal is trivial). Renormalization methods provide a natural way to determine these dimensions and to describe their universality.

DETAILS AND COMPLEMENTS: LOCAL AND GLOBAL STRUCTURES In a real scale-invariant system S, typically a critical system, we must distinguish two "nested" structures with very different properties: local structure (or pattern); it does not present any critical characteristic; it is specific to the system and depends heavily on the details of the interactions and of the physical mechanisms involved; —

the small-scale

— the global self-similar structure, which reflects the existence of a collective, coherent and hierarchical organization of the elementary constituents of S; this is the backbone of the critical phenomenon. It is related to the nature of the statistical correlations (3.2.1) and does not depend on the specific details of the short-range physical couplings, hence it is not very sensitive to perturbations. It typically presents fractal characteristics, related to the critical properties of S. To describe the system completely (globally and on every scale), it suffices to know these two aspects, i.e. the small-scale patterns and the hierarchical structure according to which they are assembled. We will be most concerned with the global structure, because of its universality, its robustness and its physical role in the overall behavior; indeed it is the global structure which ensures the transfer of information:

— from small scales to larger ones: a perturbation of the basic pattern is sent by repercussion up through all the higher scales "along the global structure"; — from large scales to smaller ones: we observe an immediate adaptability to even sudden changes of boundary conditions or sharp variations of some global parameters; stabilization occurs thanks to the appearance of additional levels in the hierarchical structure. The robustness of such a mechanism explains the abundance of fractal structures in natural phenomena, in particular those which obey strong constraints simultaneously on small scales (internal constraints related to microscopic mechanisms) and on macroscopic scales (constraints imposed by the exterior environment). The global hierarchical structure of a fractal object is revealed by the action of scale transformations, appearing as adapted renormalization operators: consequently we will be able to deduce quantitative results on the nature of the collective organization, independently of the details of the local patterns.

Mathematical fractals The notion of fractal is only well-defined for mathematical fractals c Rd,which are sets of points generated by a specific algorithm or by an explicit definition. The simplest example is the dyadic Cantor set illustrated below (figure 7.1). Many other fractals can be constructed via, recursive procedures; they are lacunary if we dig holes in the basic pattern (as in the Cantor set) or on the contrary convoluted and intertwined if we complicate it (curve or Koch flake, figure 7.2, Von Koch [1904]). Other examples are Julia sets (Julia [1918], Douady [1986]), defined as the boundaries of basins of attraction of the fixed points of discrete evolutions (§ 5.1.1), attractors of chaotic

7.1

Fractal geometry

267

dynamical systems (§ 5.2,2) or the Mandelbrot set 157 related to asymptotic properties z2 c (in the complex plane C). of z

Step 0 Step 1 Step 2 Step 3

=I

MI

i■

i■

■

■

MN

MI

Step 4

• •

• •

• •

• 1

• Ir

••

••

• •

Step 5

1111

1111

1111

II II

II II

II II

IP I!

PI !I

Figure 7.1 - A lacunary fractal: the Cantor set

We start with Co = [0, 1]. At the rt-th step, we split each of the N„ = segments of length a, = 3' which make up C, into three equal parts; we eliminate the N, central segments to obtain C,+1. The iteration to infinity gives the Cantor set,: Coo

= ni-« iim ct, co

n >0

Cn

C,„, perceived with the resolution a, coincides with C. The length of Cn is L n Arn ct,= (2/3) n , equal to its Lebesgue measure mgC,), which proves that rrtL(Coo ) = O. The similarity dimension Ds is defined by: N(ka) = k -D IV (a n ) hence

D5 =

(if k = 3 -i)

log /V, log 2 = <1 — log a n log 3

Ds
( 0 /a9 1-1s L(d) since L(a)

L, = a

D s. We obtain a dimension of mass equal to Ds according to the scale invariance: mass ( n Co, 2 mass ( [0, 3 - " 1 1 n c,0 ). ).

157 The Mandelbrot set is the set of parameters cEC having the following property: the set of points whose trajectories under the action of z'-+ z2 c do not go to infinity is connected (Mandelbrot [1080], Douady 1 1986]).

Fractal structures

268

We say that Y is self-similar if, for certain values k > 1, the structure k..F obtained via an isotropic dilation of F of factor k consists of k D s disjoint parts which can be deduced from Y. by a similarity (i.e., a composition of translations, rotations and symmetries). This property requires a structure on every scale in F which must thus extend or become subdivided at infinity. The exponent Ds > 0 is called the similarity dimension of f; it coincides with d for a Euclidean structure of dimension d (for example a hypercube) which is trivially self-similar, This type of "ideal" fractal structure can be interpreted as a fixed point of a geometric renormalization whose operators [Rdk are exactly the dilations relating Y. to the kDs components of k.,F. When .1 appears in the model of a real system, the renormalization should be supplemented by a transformation of the physical parameters and of the mechanisms generating this structure, ,

0 (1)A

Generator:

Figure 7.2 - A convoluted fractal: the Koch flake The Koch flake 100 is constructed starting with an equilateral triangle ..T0 of edge ao, The generator of the recursion transforms a segment of length a into a broken line made of 4 segments of length a/3. At the n-th step, the flake F,, made of Arr, 3 x 4" segments of length a, = 3 -n a 0 , has length L r, =(4/ 3)n Lo ; this length diverges for n —too, so that is not a rectifiable curve. Locally, F, is made similar to by a dilation of factor 3 and possibly also a rotation and a translation; Y is thus exactly self-. similar and its similarity dimension Ds = (log N„)/(— loga n ) is equal to Ds .= log 4/ log 3, which is also equal to its dimension of mass. Ds > el =1 reflects the fact that this curve is convoluted, obtained by complexifying a basic Euclidean pattern of dimension d= 1; a consequence of this is the growth of the length L(a) of F as the minimum scale a of the observation decreases: L(k a) = ki-Ds L( a) (for k _ 3-j).

7.1 Fractal geometry

7.1.2

269

Real inhomogeneous fractals

Unlike the mathematical fractal defined analytically as a set of points of Rd, the "real" fractals obtained as results of experiments or of numerical simulations are defined with a minimal scale a (for example the resolution of the picture, the step of the numerical simulation or the sensitivity of the measuring apparatus). They thus appear as a union Ya of disjoint cells of volume ad. Such a discrete structure F can be said to be fractal if it is scale invariant in one of two following senses 158 (for fixed a): 1) in a glollal sense if the number N (a, r) of cells of a tiling of side r necessary to cover Ta scales as N (a , r) s r - D1( 4 ). In general, N(a, r) must be averaged over the various possible tilings, translated by A± where 11,6■ 11 r). The quantity N(a, r) decreases as r increases, so the real number Di (a) is positive; it is called the covering dimension of Ta or its capacity. It is a global fractal dimension of Fa , smaller than d since N(a,r/k)< kd N(a,r). It describes the dependence with respect to the linear scale r> a at which the d-volume V(a, r) of Y„ is measured; this apparent volume is then equal to V(a, r) = rdN(a, r d-131 (a). It increases with r except when D I (a) = d, in which case ,F„ is Euclidean. Its lower bound is V (a, r = a), and is reached when the covering coincides with Fa. in a local sense if the number n(a, r, 13 ) of disjoint elementary cells of _Ta contained in the ball of radius r and center ±13 scales as n(a, r, o) (ofvlumead) 1),(a,e ; in general we need to smooth out this quantity n(a, r, .io) by a local average over Zo in a ball of radius ro:z.dd a. The real number D2 (a, i.o) lies between 0 and d since < kdn(a, r, 0) ; it is called the local fractal dimension at o of .F,, n(a, kr, 2)

In these two qualitative definitions, the symbol • means the existence of a linear part' in the graph of — log N or log n as a function of log r between the values r,„ > a and rMr > rro bounding the scales at which the structure Fa is perceived as fractal; the slopes are D 1 (a) and D2(a, (:)) respectively. For k varying between suitable bounds at fixed a and r, we have N (a, kr) D '(a) iV (a , r) and n(a, kr, i;) ) kp 2 (a i'°) n(a, r, 0). The two dimensions D i (a) and D2(a, io) are, by their very definition, experimentally accessible. Note that these graphs must be smooth at scales Ar < a to erase the discontinuities due to the discrete character of Ta ; at scales r < a, at which the structure of la is by definition that of the balls of Rd , their slope is d. In the zone r > rm, their slope again has an integral value do (independent of o E .7, if this fractal is homogeneous): we speak of a fractal curve if do = 1, of a fractal surface if do = 2, and so on. The fractal is called lacunary if Di (a) < d o , and convoluted if D 1 (a)> do ; its fractal character is imperceptible if D 1 (a) = do . The fractal .Ta is homogeneous if and only if D2 (a, i o ) is independent of E Ta . In this case, the product N (a, r)n(a, r, 1=p) is approximately equal (for all o E to the number Ara = N (a , r = a) of cells a d making up .Fa ; thus it is independent of r, which we use the distance "sup" in Rd given by d(x,y) = supi
Fractal structures

270

implies that the two dimensions D1 (a) = D2 (a) are equal, If .Ta becomes homogeneous on a scale ro < rm , the number of cells of Fa is the same in each cell rg of a covering of Fa , and so is equal to the spatial average < n(a, r, 0) >spat over the centers of the cells a d making up T. We can then write .Afa = N(a,r) < n(a, r, i o ) >; the concavity of the logarithm ensures that < n(a, r, io) > > r , and we deduce from this that DI (a) > < D2 (a) >. In conclusion, we point out that the fractal properties of a natural structure are defined only approximately, locally and in a domain of scales which is bounded above and below; moreover, they are generally only statistical properties, which become observable and well-defined only by averaging over different subdivisions, for the global quantities such as N(a, r), or over different centers for the local quantities such as

n(a, r, ,t 0 ).

Self similarity of a real fractal structure -

By varying the scale a chosen for defining the fractal object, we obtain a family [Fa]. This family is said to be:

1) globally self similar if N(ka,kr) r) for all k such that r and kr are in the adequate domain of scales. The exponent a can be interpreted as a large-scale similarity dzmension of the fractal: .F„, dilated by a factor of k and covered by N(a,r) cells of radius kr, is identical to the union of k' parts similar to .Tk a , each covered by N(ka, kr)= N (a, r) cells of radius kr . -

2) locally self similar if n(ka,kr, o )R J n(a,r,o) when k varies between suitable bounds (for a, r and o fixed). -

-

These notions of self-similarity are easily adapted to experimental check. If the dimensions are well-defined, we have the equivalences:

1) [Fa] 0 globally self-similar .4=> Di(a) = a independent of a. Indeed, plugging the definition of D1 into the relation of global self-similarity, we obtain N(ka, kr) k' N(a, r) k' r-D i(a) from which we deduce that: Di (ka) 2) [Ta ] , locally self-similar D2(a, io) independent of a. Indeed: n(ka, kr, o) kp,(ka,to n ia, r/k, "aj o ) by definition of D2 (ka, o ) and then kp3 (ka''On(ka, r, by self-similarity. We also have n(ka, kr, i;o ) n(a, r, k,D2 ( a>o)n(a, r/k., la ) by first making use of the self-similarity. Comparison of the two expressions gives D2(ka, io) = D2 (a,i 0 ). The self-similarity is important as it makes Di(a) (or D2(a, i. 0)) appear as a quantity independent of the scale a at which the fractal is constructed, therefore intrinsic to the physical system in which it occurs. In this case, one can express the dependence of N (a, r) and n(a, r, (:)) not only with respect to the resolution r of the analysis but also with respect to the resolution a of the definition:

7.2

Fractal measures

271

1) The relation N(ka,kr) ,--, k'N(a,r) , k -D IN(ka, r), deduced from the global self-similarity and from the definition of D I , shows that N (a , r) no longer depends on a since D 1 = a. (r/a)D2(0) where ra(ka, r, i 0 ) 2) We write ra(a, r ) i; 0 ) n(a ) r/k, ()) by self-similarity). (since n(ka, r, .i0)

k -D2 ('Û)n(a, r,

A mathematical fractal .F0 appears as an ideal structure, defined at the scale a = O. The family [yd a where F is a covering of .F0 by cells of volume a d tends to the limit Y0 as a tends to O. The exact self-similarity of Y0 implies the exact self-similarity of the family [Yak both locally and globally. The covering of Fa by cells of side r coincides with Fr so that N(a,r)-= N(0, r); consequently, the covering dimension D i of a > 0 and coincides with the similarity dimension D s of Y0 . isndept

More complex fractal structures Among the extensions of the notion of fractal, let us draw attention to: nested fractals, for which the covering dimension depends on the scale r of log N(F, r), we thus observe breakpoints of the the analysis. In the graph of log r slope at r / . rm , so we obtain a different dimension in each domain [ri, rj + i] of scales. Thus, a global vision of F is filtered by the choice of the scale r of the analysis, and the resulting situation can be said to be plurifractal. —

— inhomogeneous fractals, where the local fractal dimension D(x) depends on the point x but varies continuously with x; such a structure requires local analyses (such as renormalization or wavelet transforms, — superimposed fractals, where the local fractal dimension D(x) depends on the point in a very irregular way: for each value D, Ix , D(x) = D} is a very lacunary fractal set. Multifractal analysis was designed in order to describe for each value of D its intertwined fractal distribution in the x-space; it gives a global but filtered vision, since it describes only the points x with a given singularity D(x) = D (§ 7.2).

7.2 Fractal measures 7.2.1

Local dimension and dimension spectrum

In this paragraph we consider the fractal analysis of Borel measures on Rd when they are more complex than the measures dm() = pMdd defined by a density pW > 0 which is regular on Rd . The first extension is to the situation where a density p can be defined but may not be regular; an example is p(x)-= ixi - (0 < a < 1) on [ 1, 1]. In a more general case, the support of rn can be lacunary (fractal), and even the notion of density disappears. We define the local dimension of Tn, at by: -

.

I1M r—p 0

1o g(rn(B[:C,r)]) log r

D(m,

- m

Fractal structures

272

r1) (rn'') for r small enough. rritB( , r)] is increasing with respect to r in such a way that D(rn, ±) > O. If drnW = p(i)d' (where 0 < p(±) < co) then D(m, i) d.

Qualitatively, we write m[B(i• , r)]

m has a singularity at ±o if D(rra, 4) < d: visually, this corresponds to the presence of a localized mass at 4. If p(,) — xoII - where 0 < a < d, then = d — < d but D(rn, .i) = d if ± D(m, ±0. A limiting example is an atomic , where D(rn, 0 ) E 0 and D(rn, = oo (if ± measure drra(i) = (5(i — 3 0). a d> d; generally, (a > 0) then D(m, c)) — Conversely, if p(±) ^d D(rri, > d reveals a lacunary measure at 40 . The local dimension D(m, ia) is thus an exponent which quantifies the singularity of m at 4; the singularity becomes stronger as D(m, c)) tends to O. D(m,.i0 ) = d if In has a density p (0 < p < oo) in the neighborhood of 4 whereas D(m, 4) =-- oo if does not belong to the support of m. DETAILS AND COMPLEMENTS: EXAMPLES OF FRACTAL MEASURES — On a real fractal .Fa defined as a union of cells a d , the measures rn a are entirely defined (on the discrete a-algebra generated by the cells a d ) by the weight of each cell. A uniform measure will describe only the geometric aspects of Ta ; an inhomogeneous measure, giving different weights to the different cells, can also describe dynamic aspects (for instance the visiting frequency of the cell or the reactivity of a site). The local dimension is only

log ma [B(± 0 , r)]; empirically defined, as the slope of the linear part of the graph log r the local dimension D(rna , 4) coincides with the local dimension D 2 (a , ±0) of ,Ta if m a The family [(1-a , m a )] >.0 is said to begives the same weight to all the cells ad of consistent if for a' > a, Ya , (a' > a) is the covering of 1a by cells (d) d and if m a , is the restriction of m a to the coarser cr-algebra generated by the cells (al d ; in this case, the limit mo of the sequence [rn a] a> 0 when a tends to 0 is a fractal measure, of support Fa equal to the intersection of the sets [Zs]a>0• — A class of fractal measures is obtained by modeling on the construction of the Cantor C]O, 1! to the set and either keeping a symmetric dyadic iteration but giving a weight left-hand part (coded by c = 0) and a. weight 1— to the Tight-hand part (coded by E = 1). At the n-th step, the interval coded ci , E n will have a weight:

Tnn([6 1 • • En]) =

[6,(1 — o)+ (1— c.7 )t31

a ([0,1 ] ) = 1

1<j
or by keeping a uniform weighting giving a measure 2 — n to each interval of the Cantor set of order n, but making an uneven subdivision of the intervals tx,s --= [x, s + ay] U -fy] (a 1/3, b 1/3). Among the 2' intervals of Cri , ay, X + (1 — b)yi U [a; + (1 — b)y, 614 have length ai — Physical examples are the local (volume) density of a porous material or the local surface density of a deposit. — In the strictest sense, the adjective "fractal" is used to designate a measure m which is invariant a.nd ergodic with respect to an evolution when its local dimension (still called the information dimension, and mpalmost everywhere constant by ergodicity), is strictly less than the covering dimension of its support (or capacity). This situation reveals the lacunary nature of the attractor of the evolution and is typically encountered in chaotic dynamical systems. (.>

7.2 Fractal measures

273

The example dm() = reveals the absence of regularity of the local dimension D(m,i;) with respect to For a more general measure m, and for each real number D > 0, we can define the set ED = {±, D(711., Z) D} of points where the singularity of the measure rn has exponent D. These sets are in general irregular, lacunary and very intertwined with each other, so that their measuretheoretic description gives only trivial information. This is, for example, the case if m is an invariant and ergodic measure with respect to an evolution, with local dimension constant rn-almost everywhere, equal to D,: we have rn(ED,n ) = 1 and m(ED) = 0 if D Dm , which is not helpful. Thus, it is in fact the fractal properties of the sets ,FE:D .ID >0 which must be studied as functions of D to obtain a description of the singularities of ni and of their spatial distribution. The tool for realizing this global and hierarchical description of m is the dimension spectrum which associates to each possible local exponent D the fractal covering dimension f(D) of ED; note that it does not directly describe the fractal properties of rn or of its support but rather those of the sets in which rn has a given local dimension. Concretely, we subdivide the support of m into cells of side a and write ii(a, D)dD for the number of cells such that the local fractal dimension of the measure at their centers lie between D and D dl); then

f(D)

log v(a, D) a—>ID —

log a

or more qualitatively

v(a, D)

f (19)

For each singularity of rri, the dimension spectrum gives a quantitative characteristic describing the lacunary nature of the set of points where such a singularity is observed; the smaller the value of f(D), the more lacunary the set of points. Let us remark that the dimension spectrum plays the same role with respect to the measure ni as the critical exponents which are used to classify critical phenomena.

7.2.2 Multifractal analysis The goal of multifracial analysis of a measure m of support contained in a compact subset X of Rd of diameter L is to determine its dimension spectrum'' a 1. f(a), describing the spatial distribution of its local dimensions a > 0, by relating them to computable quantities, for example:

Z (IV , q) =

inf

partitions AN

E 7,,,o5”

and

1<j
F (q) = rim N

log Z(N,q) log N

where AN =
fo.

150we

adopt here the standard notation a

4,1(a)

instead of DP—, f(D) as above.

274

Fractal structures

DETAILS AND COMPLEMENTS: PRINCIPLES OF THE PROOF

To show how f(a) can be computed via the knowledge of F(q), we express f(c) via the partitions AN; the number of elements A 3 E AN of center xi such that a(a 3 ) = a is given by: Ni(a) a} card { j, 1 < j < N d , a(ij ) Plugging the definition of the local dimensions [a(±3 )]1
00

1

F(q) = lirn

log

N—roo

Z (N , q)

N

log

N - q a N 1 (a) da

We complete the computation using a method of "steepest descent"; this method is valid a q , and is based on the in the limit N co if f(a) — qa has a unique maximum at a expansion:

is.rf @l q )—q `x ' exP[—(a — a q ) 2 1r(a q )1(1ogN)/2] [1+ r(a)(log N)(a

ce q ) 3]

Termwise integration gives the contributions of successive orders (to an arbitrary order if we co): carry on the expansion of the remainder r(a)) to the quantity (N 00

Nf (*)da = N 1 (a)_ gŒci V2r[lf" (a q )I(log N)] 1 / 2 [1 + 0(logN) -1 1

Plugging this result into F(q) shows that transform with (q, a) as conjugate variables:

df

F"(qa)

and

T st,(ce 9)

t(ag)

dF dq

F(q)

(qa) = a and

F(q) and —f(a) are related by a Legendre

_a

d(—f)1 4

= qaq f(Ceq) 01=0(

q (q)ig=g.., f(a) = [F(q) q ddF

F(qc,) aqû

[a

da

)

The rigorous mathematical proof, showing the existence of the limits, the regularity of the functions and the uniqueness of the maximum a q , is more arduous Note that F (q) can be introduced as a threshold value via the following relation:

inf

partitions A N

E rn(k )§, r;F(q)

1

1<j
where 7'1 is the linear size of the element Ai of the partition AN (made up of Nd elements). A more constructive criterion (Hentschel and Procaccia [1983)) is given by:

inf

partitions .A N

E

nt(A.)q 7.7 — -

i<j
00

if y > if y <

—

—

F(q) F(q)

Note that Z(N, q) is analogous to the partition function of the canonical ensemble encountered in statistical mechanics: log N can be interpreted as the volume V of the system and q replaces the inverse temperature ,3 imposed by the thermal bath. F(q) is obtained in the thermodynamic limit; it is the dimensionless free energy (the free energy .F divided by k B T) per unit of volume, i.e. a pressure. f corresponds to the (dimensionless) reduced volumic entropy and a can be interpreted as the internal volurnic energy:

q=-

F=

V

f=

k

a= v F=U-TS

F aq - f

275

Fractal measures

7.2

Figure 7.3 - Multifractal analysis and generalized dimensions

Generalized dimensions We introduce generalized dimensions D(q) (or Renyi dimensions, [1970]) via the relation F (q) (q — 1.)D(q), They are related to the fractal properties of the measure and are all equal to d for the Lebesgue measure. D o = f(a o ) is the Hausdorff dimension of the support of the measure rri, also called the capacity of the measure, it corresponds to a (fractal) covering dimension (Hausdorff [1919]). D I is the information dimension of the measure (often called the dimension of the measure without any qualifying adjective). Figure 7.3 illustrates the following relations: at any q: ag = F'(q), f' (a9 ) ag is decreasing;

—

q

q and F(q) = gag — f(a g ). The function

F'(0), f(o) = Do, f(a o ) = 0; - at q = 1: F(1) = 0, a Eai r(i) = f(ai) = Di, Pal) = 1; — one checks that ai
at q

0: F(0) = —Do,

a 7, ao =

-

local dimension of the measure is equal to a*, identical at every point of the support of rn, whose covering dimension is f(a*) = Do = —F(0). The measure m is called multifractal if aq is not constant, in which case D I < Do . Multifractal analysis relates f and F via a Legendre transform, and thus gives a constructive tool for studying the scale properties of the measure m,

Fractal

276

structures

7.2.3 Renormalization of a measure Let m be a measure on a set X C Rd. To evidence a scaling law of the form r)] r D (m , '°) as r tends to 0, one constructs a family of transformations [.Rk,E,„A]20EX,k>0,A€R, appearing as renormalization operators acting on the measure rn. Rjc ,g o ,A is defined via the relation:

E X,

d[Rk, t,,,A(m)]W = k A dm[k -1- (i., - i; o )

increases the "apparent resolution" by a factor of k in the neighborhood of i o since the renormalized measure of a ball B(±o, r) is related to rn[B(o, r/k)].11. rn is discrete, defined on cells of size a, Rk A ,,A(rn) is also discrete, defined on cells of size ka. The dilation of the lengths by a factor of k is accompanied by an increase in the "mass" by a factor of kA, so as to reveal the scaling law defining the local dimension D(rn, 4). This transformation is invertible: = Riik,x„,A. The family [Rk,g,,,A]k >0 has thus a group-theoretic structure isomorphic to 00, cc], x); it is equivalent to iterate Rk , g, , A or to replace k by its powers: R O A = Rk3,.A. The relation: Rk, o ,A

Vn. E N,Vr > 0, [Rka rto , A (rn)][B(io,

k

nA in[B(i o ,rk - n)]

shows the equivalence:

Rk,20,A r+/ * = m

„ k nA re[13(4, r)] Vn E Z, Vr >0, i2*[B(0, re)]. log in* [B(i;o , r)] hm log ro

For the measures m* which are fixed points of Rk, g0,A, the local dimension D(rn, thus exists and is equal to A. A typical example is dm() = particular, the Lebesgue measure is a fixed point of all the operators [14, - 0 ,A-d]k>o• All the measures belonging to the stable manifold of Rk . ,„,A at rir, i.e. such that t. 0 A (m) = in*, have local dimension A; D(nt, 0 ) = A at It is also sufficient that this weak convergence takes place for measures restricted to the A (m) is either zero if neighborhood of ;t o . On the contrary, the limit A< Dfrra, to), or not defined (infinite) if A> D(m,i3o). The Vahle A= D(m, i o ) is the only one giving a non-trivial asymptotic behavior under the action of renormalization.

((> DETAILS AND COMPLEMENTS: GLOBAL PROCEDURE The drawback of the renormalization operator RIc, o ,A is that it is local, i.e. adapted to revealing scaling properties of measures only in the neighborhood of the single point Another, global and more constructive renormalization is possible; let us describe the principle of this approach which is still being developed 161 • Let rn be a measure of support contained in a bounded open set X C Rd (the study can be generalized to any measurable space). For a fixed real function A("±) on X and a scaling factor k> 0, we construct the renormalization of m by induction. At the n-th step, we subdivide X into disjoint cells of volume 161

Research on this subject has been done by Mandelbrot and Evertz [1994 [1992].

277

The wavelet transform

7.3

a -d , labeled by their centers

1/2n)ra X (n ) C (Ziri) d and coinciding with the balls We define the measure — if we take the distance sup supj.i...d — Y.11 on Rd . kAn()),k m on the a-algebra TO) generated by the balls [B(Z, 1/2n) r1 X] tex (n) by: ),k rnJ[B(i , 1/2n)

[11,

n xi

kAw rrt[B(,1/2k7t)

n

E X (n )

C (Z1n) d

A km is thus related to the restriction of m to the a-algebra '7-(k " ) , which is finer than R T(') if k > 1. The limit of the family (T ( n ) ), >1, i.e. the smallest a-algebra containing all the a-algebras T(n) is the Borel a-algebra E of X. The renormalization nit(,),krri is then the measure defined on T as the limit as

RA(.),km

n

= nlinl R (:().),krn

oc

of the sequence

(weak convergence of measures)

One must of course check that this limit exists and does not depend on the sequence of subdivisions ( 1 /ra, X('))„> 1 used to construct it, which restricts the domain of definition of RA( . ) , k to a set of measures M Ao j k. Since the function A(.) is fixed, (PZA(,),kik>o, 0) is a one-parameter group 162 isomorphic to J0,04, x). The advantage of this construction lies in the following results:

— [R.A(.),k][B(o, r)] involves the restriction of m to B(±o, rik); the contraction of the lengths by a factor of k is compensated by a local weighting k A( ') ; = A = eonst, then [TIA(,),km][B(±,r)] = — — the Lebesgue measure mL belongs to MA( . ) ,k whatever the function A(.), and the associated renormafized measure P—A(.),krni, is the measure of density p(i) = kA (') -d ; — If

drr() = p(I)dd

then RA(.),km has density

[TR AG),k d(t) =

R (An()) ,m is non-trivial if and only if A(i.) coincides with the local dimension — D(m, i) of the measure m at every — If R.A( . ),km* = Tri*, then m* is locally self-similar of local dimension A(.). — If k> 1, R. A 0,k reduces the possible fractal character of the measure by a magnifying is glass effect: schematically, a singularity of m concentrated in a volume a d around distributed after renormalization in a volume (ka) d after having been multiplied by k A W . — If k < 1, then on the contrary .A(,),k enables us to describe the long-distance scaling law perceived at each point. The renormalization brings to the immediate neighborhood of the mass distributed in a ball of radius 1/k times as large: R.A( . ) , km describes the massic distribution as it is perceived on the average at

7.3

The wavelet transform

The wavelet transform is a rnultiscale , localized spectral analysis, designed to describe the local scaling properties of given natural structures. Like renormalization, 162 This

group is a Lie group only if we consider only measures which are absolutely m[B(i, r] is differentiable. continuous with respect to the Lebesgue measure, so that r

278

Fractal structures

it is based on the scale invariance of these structures, and several further analogies will be made precise in this paragraph.

7.3.1

Transformation formulas

The spatial inhomogeneity of the fractal characteristics of real structures often makes global methods of analysis inadequate or even inapplicable: these methods give only average indications which, even if they correctly take into account the effect of the local inhomogeneities, cannot describe their possible spatial (or temporal) organization. For example, the dimension spectrum gives a hierarchical description of the singularities of a fractal by describing the more or less lacunary structure of the sets of points of given local dimensions, but it does not give any information on their spatial pattern. Only an image of the fractal which is simultaneously both spatial and hierarchical is able to reveal this organization, which is particularly important when we seek to understand the mechanism of formation of the structure or in problems of pattern recognition. To make up for this inadequacy of global spectral analysis, researchers developed a method of local spectral analysis: the wavelet transform, It has two aspects: — an analytical procedure, whose goal is to associate a family of local spectral components to the function A(t); — a synthetic procedure, whose goal is to characterize the local scaling properties of these components and to determine the local fractal dimensions directly on them, then to reconstruct the function A(t). Schematically', wavelet analysis consists of a decomposition of A(t) according to a basis of functions [gc,], called (analyzing) wavelets having a finite support in order to give a local character to the analysis and being adapted to the detection of a particular pattern in A(t). The local spectral components are constructed as follows:

S(A,

„, b) = b—d / 2 JA()9 rt(

X

—

b

Zo

ddt

bd/ 2 A(to bi)g)dd

The wavelet transform is extended to measures by replacing AWol dt with dm(). Because the support of gc„ is finite, the scaling factor b enables us t,o choose the size of the domain of observation around t o , The usefulness of this parameter b to display the local scale invariance of A (at t 0 ) and the flexibility it brings to the analysis is intuitively obvious. In dimension d> 1, the function g o, generally contains a rotation (indexed by d — 1 angles). So one can perform not only a local but also a directional analysis of A. Visually, the wavelet transform acts like a microscope whose optics is described by the wavelet g, which can be oriented and moved around (by varying t 0 ) over the object being studied, and whose magnification can be controlled (via the choice of b). The spectral components obtained keep track of the scale b and of the spatial distribution of the local structures they describe.

We do not detail the conditions of admissibility which must be satisfied by the spanning functions [2,,,], nor the conditions of regularity and integrability on A(t); for these questions we refer to Holschneider [1988] or Meyer [1994 163

7.3

279

The wavelet transform

DETAILS AND COMPLEMENTS: COMPARISON WITH FOURIER ANALYSIS

In order to clarify the specificity of this transform, we compare it to the Fourier transform. The Fourier transform (App. IV.1) corresponds to the choice Y,.0 =0, b =1 and g a (i)= 0:6 1 ; since the functions [g a ] a are periodic, their support is all of Rd. This transform is used to show the periodicity of A:

[Vi, A(i)=A(i. i 0 )1 is equivalent to (S(A, (1)=0 unless äE2(o)={ , ã.o E 27rZ)] It gives a useful analysis of the function A(i) only if the leading behavior of the function is periodic or quasi periodic on the whole space, since then the transform 8(ã) presents very marked peaks, corresponding to wave vectors ci associated to the periodicity of A and to their harmonics. In that case, Fourier analysis is relevant as it simplifies the description of the main behavior of the function A(i) by replacing it with a sequence of spectral components which is, at worst, countable and often even finite; this sequence contains almost as much information as A (exactly as much if A is exactly periodic) and it suffices to reconstruct an approximation of A. In all the other cases, we need to know all the components to be able to reconstruct A: Fourier analysis does not simplify the study since it uses as many degrees of freedom in real space as in conjugate space. Moreover, it is not possible to perceive just from the Fourier transform of A the presence of a locally periodic behavior of A, nor to find, on a spectral component, the spatial zones which give an essential contribution, nor to determine explicitly the spatial arrangement of these domains. -

A wavelet transform can solve these problems by taking

g6 (i) = e

igo(i). The function go has bounded support, for example, of linear extent Az = 1, centered and maximal at i= 0; this ensures the local nature of the analysis by truncating the range of observation, in a way which can be regulated by the choice of b. The variable rj appearing in the spectral components allows us to center the analysis at any point of the space. We thus obtain a representation giving both the wave vectors '6 and their spatial localization to on the scale b. A familiar time-frequency analog of this representation is a musical score, which simultaneously indicates the notes (the frequencies), their duration (the temporal scale) and the moment at which they must be played. The advantage of the wavelet transform as a method for analyzing a signal is that it uncovers the underlying "musical score".

7.3.2

Local scale invariance and renormalization

One of the advantages of wavelet analysis is that it is injective and even explicitly invertible. In dimension d= 1, the inversion formula is given by: A(X)

C; 1 J R JR

SO, X0 5 g ,b) g

x — x0\

b 7 db d

xo

where Cg = 27r f +: V(k)1 2 Ik1 -1 dk , which requires that 'i(0) = O. However, we can detect remarkable properties even without using this inversion formula. Above, we detailed the use of the wavelet transform to detect localized oscillating behavior. Other extensions of the wavelet transform make it possible to detect other kinds of remarkable behavior besides periodicity. The nature of the symmetry properties which

280

Fractal structures

are accessible via simple observation of the components depends on the choice of the analyzing functions g.

0

DETAILS AND COMPLEMENTS: WAVELETS AND SYMMETRIES

The translation groups G(i0) .= {O E Rd n E Z are the symmetry groups associated to the Fourier transform in the sense that the invariance of a function AM under the action of one of them is immediately reflected on the Fourier components of A: }

[VO E

A(i+ nia )] [S(A, =-0 unless ci E Z(k-0) -.= Ce, 6.4 E 274 Nn E Z, Vi e Rd , A(2)

Aot., A] -

The group G(i0) acts linearly on the space of functions ff : R d 3. R), transforming fo O -1 . Consequently, the invariance of A can also be stated its elements via To (f) as: A is an eigenvector of eigenvalue 1 of all the linear transformations of the group g(i0 ) = {To, 0E G(i0)}, isomorphic to G(4) since To, o = To, 0 0 2 . The ability of the Fourier transform to detect periodic behavior comes from the fact that the basis [g(i) e i ' i t5E R d of the decomposition consists of eigenfunctions of all the transformations of -

0 (g) = e' .4 g a . Consequently, an eigenvector A of eigenvalue 1 has no components except on the spanning functions of eigenvalue 1, i.e. for (i E Z(4). T

Extending this point of view, we can introduce wavelet transforms to study invariance with respect to groups G of linear transformations acting in F = : Rd —4 R}: it suffices to be able to choose a complete orthogonal set [g]„ of wavelets (g a E .F) made up of of common

eigenfunctions of all the elements of Ç. In this case, we conclude that a function A E F is invariant under a transformation T of the group G if and only if it decomposes unique/y onto the wavelets of eigenvalue I under the action of T. The group g of transformations can in particular be a linear (or linearized) renormalization group.

Via the wavelet transform we can, for example, obtain the local scaling laws of the objects analyzed; this is the point which makes it a method of anaksis of fractal structures. Moreover, the spectral components of the function A under consideration keep track of the spatial distribution of the local fractal dimensions.

0

DETAILS AND COMPLEMENTS: WAVELETS AND LOCAL SCALE INVARIANCE

Let us show that we can directly read off the scale invariance of A at 0 from the dependence on b of its spectral components [S (A , ka, g, b)], where g is the analyzing wavelet, of finite support, which appears in their definition. Let us introduce:

[ht ,,,t,A](i)

b -1 / 2 A[i.0

— i70)]

A : Rd

R

This operation A 3 3- 12 10 ,bA satisfies the group law oh 0 = h x0 ,b0, 2 ; it represents the action of the affine group (consisting of the dilations and the translations of Rd ) , it is linear and preserves the norm in L2(Rd , dd ). We check that: -

E Rd ,

,

S(A,

g, b) = S(h ,bA , io

2

, g b = 1)

The wavelet transform

7.3

281

Let us introduce: 2

JA(0,

r) = < *r

til(i; + i,)] 2 dd

Set:

— A(

0)

==

A

0(X 0)

=0

and

Iz 0 ,b( 11 t0 )=h 0 ,b(A) — A(x0)

One can then check that the following three statements are equivalent:

br) = b(D() +4 ] JA, o (i'o,r)

flgo , b A 0 = b[D(1'0)+4) At. 0 S(i6 0 , 4 +

b[D ('''» I] S(A, ,

g ,b = 1)

If they are satisfied for every b and r with r < ro and br < ro, these three assertions give three equivalent formulations of the scale invariance of A at io, associated to the local fractal dimension D(±0). The exponent of the scaling law satisfied by JA (r) is given by

. The interesting point is that the existence of a (local) scaling law for the function A is equivalent to the existence of a scaling law of the same exponent for the spectral components of A. In this case, the adequate renormaliza.tion is the linear transformation:

Rt oi b(A)

P

4 0 ,041 (io,

and

b

= = b -14+D(1°) LT4 (to, br) = Rk0,b(A)( 2; ° ' r) .

being arbitrary and fixed, we have:

RT0,t1 ( A Z O

)

A

t0

( JAY° )

j

A ffla

<=> S(Ae- 0 , 13 o ± b, g, b) = b[4+D(2 ) 1 S(A xo ,

b = 1)

In the extension of the wavelet transform to measures, the local scale invariance dm(±0 bi) bl)( ° ) drn(1:0 i) is equivalent to S(m, ±0+b,g ,b) = bp( 0 )i-d12 S(m, ±a+1,g, b = 1). Here the adequate renormalization is the operator Rb ,x, p(0 0 ) introduced in § 7.3.3. Like the Fourier transform, the wavelet transforms can be defined in the sense of distributions164 , which makes it possible to extend them to the cases where the first definition gives rise to problems of existence and convergence. Thus it is possible to consider the family tg aoj a where = as a family adapted to scaling properties. Indeed, with this choice we have:

S(A2 0 , 40, a, b) P+1

4, b = 1)

(6 >

Let T be a linear transformation on a space .1" of functions of Rd in R, for example R) or Tm f 1—• f f (t)clm(t) where TA f f A(t) f (t)d d t with F = L2(R't = L1 (dm). We define the Fourier transform of T "in the sense of distributions" on the set D(T) of functions yo of Rd in R which admit a Fourier transform (,-3 el. by: r.-[.."(9)E T T)). No 164

(

restriction is imposed on T: the constraints of existence of D(.7") (Schwartz [1978], [1979]).

I" concern its domain of definition

282

Fractal structures

The following conditions are thus equivalent:

-

R o0 ,b(16 0 )

A 0 for all b > 0;

— S(A 20 , i o , a, b =1) = 0 unless a — for all b > 0, ±0 , a b)= 0 ,

unless a = D (5 0).

Note that [Tt 0 g2](x) = gcet) (i.- 0 ) is an eigenfunction of the transpose of R,t Di b; the transpose is on the linear operators of the Hilbert space L2(11.d , ddi;), whose scalar product we denote by < I >. Indeed:

b) b[cf—D(x0)1s(A 0, y 0 , a , b = 1) b — i4+D( ' °)) S(A 0 ,±0,

< Rfo ,b(A f l aTfo e„ >

-

< Ik,171±092 > < A„,104- 0 ,1,117'1 4°j >

from which we deduce that [t .F4 0 ,b][Tta gaa ] = b'n ('°% o gc,. ° . If A is scale invariant at --0, of local dimension D(), then only its spectral components on the function g° are

non-zero, i.e. on the only function whose translation Too g°OE is an eigenvector of eigenvalue 1 of 1 .1=4. In the case of a renormalization group and a wavelet transform, one finds the result obtained with translation groups for the Fourier transform. The general result can be stated as follows: if Lq c,]„ is a family such that for all a, T a g, is an eigenvector of 1 R, 0 ,b, then the only non-zero local spectral components 8(A 0 , g, b) of a function A invariant under i b are those where a is associated to an eigenvector of eigenvalue 1.

The method can be generalized to other types of symmetries, reflected in the local invariance of the function A (from R d to R) under the action of transformations of a parametrized group. Wavelet analysis then simultaneously gives the spatial and even the directional and hierarchical (as functions of the observation scale) distributions of the parameters of the local transformations preserving A, which enables us to visualize the spatial structure and the symmetry properties on different scales. For example, it allows us to detect the position and the scale of a particular given pattern. The wavelet analysis is thus an essential step in the understanding of the mechanisms of formation of fractal structures or more generally of structures presenting a local invariance under the action of a symmetry group. The close relationship between wavelet analysis and renormalization techniques can be seen in the fact that their respective domains of application have a large intersection.

0, DETAILS

AND COMPLEMENTS: PHYSICAL EXAMPLES

Let us conclude this chapter by giving some examples where the wavelet transform and renormalization methods come together and supplement each other. They prove to be particularly efficient in the quantitative analysis of scale invariance — of the accumulation of period-doublings associated to the period-doubling scenario and of the subharmonic cascade observed on the power spectrum 05.1.6,

of the fractal structure of certain strange attractors and of the associated invariant measures (§ 5.1.3, § 5.4, Meyer [1991]);

7.3

The wavelet transform

283

— of the energy cascade of developed turbulence, of the field of velocities and of the phenomena of spatial intermittency observed in this regime (§ 5D.2, Argoul et al. [1989],

Arneodo et al. [1993]); — of the fractal clusters, such as the critical (infinite) cluster of a percolation lattice (§ 7D.2); — of models of fractal growth and of aggregation.

REMARKS AND BIBLIOGRAPHICAL NOTES The fact that fractals are currently fashionable as well as really conceptually interesting, and the diversity of the domains in which they naturally arise, have given rise to a great deal of research. To begin with, we must cite the "historic" article by Mandelbrot [1967], little noticed at the moment of its appearance although it introduced the original notion of fractals. Now recognized, (Aharony and Feder [1989]), Mandelbrot participated in most of the progress in the description and the understanding of fractal structures (Mandelbrot [1977], [1982], [1986]). Barnsley [1988] and Peitgen et al. [1992] show the breadth of the domain of application of fractal geometry; the mathematical aspects are explored in Feder [1988] or Falconer [1990]. For a more physical approach to fractals, see the presentation by Pietronero [1989] of their origins and properties, the collective books edited by Runde and Hay lin [1991] [1994] or the conference proceedings edited by Pietronero and Tosatti [1986] and by Stanley and Ostrowsky [1988 ] . The subjects considered in chapters 5, 6, and 7 and their supplements are also given in Gouyet [1996], in a presentation oriented towards their fractal aspects. The article by Farmer et al. [1983], those collected in Barnsley and Demko [1986] and the book by Devaney [1990] present fractals encountered in the study of deterministic chaos. Gefen et al. [1980] consider the critical phenomena taking place on fractal structures. Stauffer and Stanley [1990] present fractals as an extension of "traditional" physics. The book by Barabasi and Stanley [1995] deals with fractal concepts encountered in surface growth phenomena. Applications of both renormalization methods and fractal concepts to the analysis of aggregation phenomena (DLA) and related models, not approached in this chapter, can be found in Gould et al. [1983], Nagatani [1987] and Nagatani et al. [1992]. Historically, the concept of multifractality was introduced by Benzi et al. [1984] and by Frisch and Parisi [1985] to describe the distribution of singularities of the velocity fi eld in a fully turbulent fluid. This concept, further developed by Mandelbrot [1986] and [1988] in relation to already known fractal geometry, turned out to be relevant for the delicate analysis of strange attractors (Halsey et al. [1986], Collet, Lebowitz and Porzio [1987]). Multifractal measures and their similarity properties are studied in Mand.elbrot [1989] and in Mandeibrot and Evertz [1991], [1992]. Other more physical examples can be found in De Arcangelis [1988], Stanley and Meakin [1988) and Stanley [1991]; Paladin and Vulpiani [1987] study the anomalous scaling laws observed in multifractal objects. A complete and recent mathematical presentation of multifractal analysis is given in Falconer [1990]; a more physical and more accessible approach can be found in the book by Peitgen et al. [1992]. The experimental aspects of the determination of a dimension spectrum are considered in the reference article by Grassberger and Procaccia [1983]; another approach, based more on numerical aspects, is proposed in Chhabra and Jensen [1989]. The waveiet transform presented in §7.3 is a stillexpanding subject; surveys of the successive advances in this area can be found in Combes [1980], Combes et al. [1988] and Meyer [1991].

Supplement 7A Percolation Percolation lattices give discrete models of disordered binary media, and they undergo a universal critical transition (§ 7A.1). This transition induces scaling laws and fractal characteristics both in the statistical properties of the percolation cluster (§ 7A.2) and in the transport phenomena on these clusters (§ 7A.4). Renormalization gives numerous and exemplary methods (§7A.3) to describe these scaling properties analytically or numerically. It is also possible to use finite-size scaling to obtain certain critical exponents.

7A.1

Percolation models: clusters and percolation threshold

The term

percolation is associated to the study of disordered binary media, in which a

local property can be realized in two ways coded 0 and 1. The small-scale structure is thus a nesting of regions 0 and regions 1, perceived as random by a macroscopic observer. There are numerous examples of percolation situations: — systems consisting of two species A and dominates or 0 if the species A dominates;

B; the coding is locally 1 if the species B

— adsorbing catalytic surfaces: the adsorbing sites are coded by 0 if they are free and 1 if they are occupied; — mixtures of a conducting material and an isolating material, where one studies the transition between global isolating or conducting behavior; — mixtures of a conducting material and a superconductor, where one studies the appearance of superconductivity on the macroscopic scale; lacunary systems, modeling porous media or rough surfaces; the empty places are coded 0 and the occupied zones 1; — polymerized gels, where the presence of a chemical liaison is coded by 1; one studies the transition of the liquid to a special phase called "gel"; — populations, where one studies the possible propagation of an epidemic; the healthy individuals are coded 0 and the sick 1. The etymological example 165 is the passage of water through coffee grounds which consist of fine particles more or less agglomerated according to the density, which can be regulated by tightening the filter of the percolator. As can easily be observed on a percolating coffeepot, and Hammersley [1957] to designate the modeling of random binary media. This type of model could be found previously in the work of Flory [1941] and Stockmayer [194 4] on polymerized gels. 165 The term "percolation" was introduced by Broadbent

286

Percolation

the time taken by the water to pass through the filter, i.e. the time it is in contact with the coffee grounds, depends on this density; moreover there exists a density known as the percolation threshold above which the water cannot pass through the filter. The natural problem arising here is to describe the agglomeration of the coffee grounds as a function of its density, then to describe the characteristics of the propagation of the water through the random inhomogeneous medium obtained.

Formalization: four models of percolation Let us consider systems of extension L in Rd, described with a resolution a
— site percolation (figure 7A.1.(a)): the situation is discretized by a tihng by identical cells of volume a d, identified with the sites of a lattice indexed by Z d . The state of a site is random, and the sites are statistically independent: each of them is occupied with the same probability p, so has a probability 1 — p of being empty;

— bond percolation

(figure 7A.1.(b)): the situation is discretized by a mesh with statistically independent and identical bonds of length a, which are present with probability pB, and therefore absent with probability 1 — pB;

— site-bond percolation

(figure 7A.1.(c)); this is a. combination of the two preceding situations; one uses a model of sites in which the bonds between two occupied sites are present only with conditional probability pBi

— directed percolation (figure 7A.]..(c1)); one uses the model of bond percolation but adds a random orientation on each bond.

<>

DETAILS AND COMPLEMENTS: PHYSICAL DISCUSSION OF THE MODELS

In model (a), a site corresponds to the minimal volume perceptible by the observer: it is occupied if its density is greater than the threshold of sensitivity of the measuring apparatus or if it can serve as a support for the diffusion of a test-particle. This model is suitable for problems of contagion in a random medium or for modeling adsorbed systems. The other models are better adapted to the study of transport phenomena; they lend themselves to the schematization of statistically anisotropic media. Models (b) and (c) were introduced in the context of polymer gels. Model (d) is used, for example, to study neural networks or invasion phenomena in porous media. A network of resistances placed randomly on a regular lattice with probability p realizes a bond percolation system; a system of directed percolation is realized by replacing the resistances by transistors.

7A.1

287

Percolation models: clusters and percolation threshold

WEISS MM. ME M O WO O =

MEE=

(b)

(a)

(c)

20111111101 Figure 7A.1 - (a) Site percolation, (b) bond percolation, (c) site—bond percolation and (d) directed percolation

111•11111 It ERNE 711131211

(a) the probability p that a site is occupied coincides with the total concentration if the lattice is large enough; (b) the probability pB that a bond is present coincides with the total concentration of bonds if the lattice is large enough;

(d)

(c) we recover (a) if all the admissible bonds are (ps =1) and (b) if all the sites are occupied (p= 1); (d) the bonds of the model (b) are oriented.

present

0. DETAILS AND COMPLEMENTS: PROBABILITY p AND AVERAGE NUMBER DENSITY The configurations of a lattice of N sites are described by [e] where ej = 1 (this happens with a probability p) if the site j E Z d is occupied and Ey 0 if it is empty. The total number of occupied sites is a random variable No(p, N, [e]); thus this is also true of the concentration c(p, N,[e ]) = No (p, N,[E])IN=L- Ej 6TIN. Applying the (strong) law of large numbers (App. cx) , this random concentration coincides almost 1.2) then shows that in the limit as N

surely with the probability p that a site is occupied:

Ern c(p, N,[f])= < f > = p

N-9.

a°

almost surely.

The central limit theorem estimates the fluctuations < [cp ,N —29 ] 2 can be transposed without difficulty to bond lattices.

>= 0(1IN).

The proof

The advantage of these models lies in their simplicity: the sites or the bonds are described by independent and identical bimodal random variables, entirely specified by their individual probability p or pB. These models are particularly well adapted to simulation methods; they were developed in parallel with numerical tools. Percolation systems thus constitute basic models for studying binary random media; they provide representatives of the universality classes observed for these media.

Percolation

288 Clusters and the percolation threshold

A cluster is a connected set of occupied sites or of occupied bonds. In case (a), the sites of a cluster must be connected by a sequence of occupied nearest neighbor sites; in cases (b) and (c), two bonds of a cluster must be connectd by a chain of occupied bonds; in case (d), two sites of a cluster must be connected by a sequence of coherently oriented bonds. Unless we explicitly say so, we consider below only model (a) of site percolation. -

In an infinite lattice, the percolation threshold is defined to be the concentration p, at which the first infinite cluster appears. Intuition would seem to indicate that p, = 1/2 by symmetry or that this threshold should be a random variable however, these two ideas are entirely wrong! In a lattice of infinite extension, this concentration has a welldetermined value, depending only on the chosen percolation model and on the geometry and the dimension d of the lattice, but not on the physical interpretation of the occupied sites or the bonds present, nor on the way in which p increases starting from p= O. The question is thus to compute p, and to describe the transition p=pc . In a finite lattice of linear extension L, we introduce the notion of a spanntng cluster, whose definition, depending on the physical context of the model and the geometry of the lattice is, for example, the existence of a cluster connecting the edges of the lattice. The concentration I3,(L) at which the first spanning cluster appears is now a random variable; its value depends, moreover, on the precise definition of a spanning cluster used and on the way in which it is constructed, for example by randomly filling the remaining free sites (which increases

tends to

will study the rate of the almost sure convergence of MI)) to p, as L infinity as well as the dependence on L of the mean and of the variance of Pc (L).

p). We

The study will deal with static aspects (§7A.2), considering geometric statistical properties of clusters and their fractal characteristics as functions of the concentration p, before approaching dynamic aspects (§7A.4), in particular the study of transport phenomena on the static structure uncovered by the first part of the study. This study is mainly motivated by the desire to understand and quantitatively describe transport phenomena, which are directly related to observable physical phenomena.

(>

DETAILS AND COMPLEMENTS: NUMERICAL DETERMINATION OF pc

To determine the percolation threshold, we will observe a transport phenomenon which occurs only if there exists a spanning cluster supporting it; the statistical study of the values Pc obtained by observing a large number of configurations (of finite extension) gives the deterministic limit p c . A first example is the contagion model used to reproduce the propagation of forest fires (Drosse! and Schwabl [1992)]. It is realized by filling each site of a square lattice N X N with probability p: for this, we independently select random numbers ei for each site i, uniformly distributed in [0,1], and we fill the site if ei < p. The fire is set at a site on the left-hand edge at the instant t = 0; in a time step to t o + 1, the sites which are lit at to light their nearest neighbors and go out. For each configuration, we obtain the time r taken by the fire to reach the right-hand edge or to put itself out. We observe that the time 7- (p, N) obtained by averaging T over a large number of independent configurations selected with the same probability p diverges for a deterministic value pc (N) dependent on N; as N tends to infinity, pc (N) tends to a value pc depending only on the geometry and on the dimension of the lattice.

7A.1

Percolation models: clusters and percolation threshold

289

A second example is that of diffusion on the preceding lacunary support. At t = 0, the test-particle leaves an occupied site d',0 on the left-hand side. Its motion, discretized in time, is + 1 , of the defined recursively: we (equiprobably) select one of the nearest neighbors, denoted particle goes the instant if it is occupied, the ti = jr; site z(t_i ) where the particle is located at there in a time step 7 and ,t(ti+i) = ki+1; if it is empty, Oi-i..1)= (ti). The quantity which best characterizes this motion is the quadratic mean displacement D (t , p) = < 1i(t)— .t011 2 >, where the average is taken over a large number of independent configurations and for each of them, over a large number of independent particles. If the lattice is large enough, we observe

t ) oo: pc — p = 0(1), then D(t, p)

the following asymptotic behavior as

—

remains bounded with time; the — if 0 < p < pc and medium is too lacunary for diffusion to occur and it traps the particle; -

pc = 0(1), then D(t,p) t: the diffusion is asymptotically — if 1 > p>p, and p normal, analogous to the Brownian motion observed when all the sites are occupied;

tcr — between these two extreme regimes, the diffusion is anomalous: we have 1)(t,p) where 0 < a < 1, and the quantity 2/a appears as the fractal dimension (in quadratic mean) of the trajectories of the particle. The greater the size of the lattice, the more the transition between the trapping regime for p < pc and the diffusive regime for p > pc is sharp and allows a threshold pc to be defined precisely. There also exist purely geometric (static) methods to numerically determine the random threshold Fc (N) in each configuration of the preceding lattice of N sites, and then to

) oo. One of the most classical methods is determine pc itself by passage to the limit N based on the Hosheu-Kopeiman algorithm (Hoshen and Kopelman [1976], Kopelman [1986]). This algorithm can be applied whatever the geometry and the dimension of the lattice, and enables us to test whether a given configuration percolates and to make a statistical study of the spanning cluster. We begin by labelling all the sites of the lattice line by line 166 . The —

first step of the algorithm is to assign a number (or label) ni to the j-th site encountered while numbering, with

-

ni = 0 if this site is empty;

-

ni = ni _k if this site is a nearest neighbor of a site j — k of label — ni = 1 supi j) with nk
166 The most general condition is that at every moment, the set of numbered sites and that of the not-yet numbered sites form two complementary connected clusters.

290

Percolation

with pi,,-1 and we take Pj,n 2 —(n+1) , with the sign — if the configuration percolates; in particular, Pj,11 Pj,ri an-d Pi > Pc if Pj,n-f-i.
large numbers gives an estimate of the average <23,(N) > J p . The results of finite-size scaling theory (described at the end of §7A.2) then allow us to estimate the distance between the value < (N)> deduced from the simulation and the exact deterministic value

lh•

7A.2

Static aspects

Consider the situation of an infinite lattice. The value pc of the percolation threshold is deterministic but depends on the specific details of the chosen percolation model and of the geometry of the associated lattice; comparison with the observed value serves as a guide to select the most relevant model. On the other hand, the transition p = pc has universal properties, which can be expressed by scaling laws whose exponents depend only on the unique physical parameter which is intrinsic (i.e. independent of the model), namely the dimension

d of the

space.

The probability of belonging to the infinite cluster The transition p= p, corresponds to the appearance of an infinite cluster: it is present for every value p>p, arid is quantitatively described by the probability P„,,(p) that it contains a given, arbitrarily chosen site, Experiments and simulations suggest that P00 (p) should satisfy:

P(p)= O

if

p < pc

Poo(P)

(P — PO P if p > pc

Poo (pc ) = 0 reveals the very Iacunary character of the infinite cluster at the percolation threshold: although it is infinite, it occupies a zero fraction of the lattice; its characteristic property is thus its connectedness on every scale (Coniglio [1982}). Experimental observation in dimension d < 3 of p — ) 13,0 (p) shows the divergence of the derivative [dP/ dp](p) at p = pc , When ,3 < 1 (so at least in dimension d < 3), the graph of Po, (p) has a vertical tangent at p,; its form, analogous to case (b) of figure 1.3, thus shows the critical nature of the transition p.= pe ; in that case, the relevant order parameter is P. The interest of this scaling law for P(p) lies in its universality: 0 turns out to depend only on the dimension d. This law is thus independent of the model and reveals a real physical property, common to ail disordered binary media. It is valid only near pe ; much higher than the threshold, the infinite cluster contains the majority of the occupied sites, so that P(p) increases as p. c(;)

DETAILS AND COMPLEMENTS: THE MEAN-FIELD APPROACH

The computation of P(p) via a mean-field method 167 is possible if we neglect the presence of closed paths in the lattice, called loops, so as to assimilate the lattice to a Beth e lattice as in figure 7A.2. Thanks to the "tree-like" geometry of this lattice, we can define the conditional probability Q(p) that a site (c) neighboring a site (o), known to be occupied, does 167"

Proposed by Flory [1941 ] in the context of polymerized gels.

7,4.2

Static aspects

291

not belong to an infinite cluster contained in the branch coming out of (c) and not containing (o). We then compute the probability Pfi nit,(p) that an arbitrarily chosen site (o) does not belong to an infinite cluster: either this site is empty (with probability 1—p) or it is occupied and none of its z neighbors belongs to an infinite cluster in the branch not containing (o) of which it is the initial vertex. This can be written:

1

—

Poo (p) = P1 11 (p) -=- (1 — p) ± pQ(p) z

The condition P(p) = 0 implies that Q(pc ) = 1. We then continue the reasoning on one of the neighbors (c) of (o); this site (c) is in the situation of probability Q(p) described above if it is empty or if it is occupied and if each of its z — 1 neighbors in the branch not containing (o) does not belong to an infinite cluster in the branch not containing (c) whose initial vertex it is. This assertion can be written:

Q(p)

= ( 1 — /3) + PQ(P)z-1

(with Q(Pe) = 1)

Differentiating this relation at p:--_- p,, we obtain the exact value p, = 1I(z — 1) where P(p) = 0 and Pc/0 1 (p, + 0) = +co. The expansion of P(p) with respect to p — pc then gives the value 168 /3 = 1. As in numerous other critical transitions, the "mean-field" computation is valid without restriction on the lattice or on the model whenever the dimension is greater than a threshold etc ; we have d, = 6 for percolation.

o

7A.2 - Bethe lattice (z = 4) Each site has z neighbors and z(z — 1) points at distance 2; this lattice is characterized by the absence of "loops': the sites (a) and (b) are connected only by a path passing through the site (o). The mean-field approach here leads to the exact result p, =11(z — 1) and )6 =1. Figure

s

One can introduce other statistical quantities which describe experimentally or numerically accessible quantities by averaging over a sufficient number of results. Like P(p), they have the advantage of obeying a scaling law with universal exponent at p = pc . 168 A Bethe lattice is conceivable only in infinite dimension: it is not so much a lattice with a particular geometry as a representative of the class d = co: the independence of )3 with respect to the geometry of the lattice does not fail and we shall write )3(cd = co) = 1.

Percolation

292

The average number of sites in a finite (non-empty) cluster We denote the average number of sites in a finite cluster by 8(p). It diverges at p,, then decreases since the infinite cluster is not taken into account; we always have S(p) > 1. where 7 is identical on both sides of p c and It satisfies the scaling law 8(p) 113 depends only on d.

Correlation functions r The correlation function C fi r,ite (p, 0 is the conditional probability that the site f.0 is occupied and belongs to the same finite cluster as the site fo, knowing that fo is occupied and does not belong to an infinite cluster; thus it is normalized by Cjirtite(P, r= O) 1. It depends only on the modulus r of F by homogeneity and statistical isotropy. FOI p > pc , we define Cinfinite(p, F) similarly for the infinite cluster (Kapiltulnik et ai. 119831). We obtain Cfi n ite (f) where the sum is over all the sites of the lattice (including = 0). S(p) = Two characteristic lengths are associated with these functions, according to:

E,

E r 2 Cm(p, F.) . [ E C[j (p, f) ] -1

where [I ] = finite Or infinite

inite(P) estimates the characteristic size of the finite clusters when the concentration is p. These correlation functions and lengths diverge at p=p, showing the critical nature of the transition p= pc . Empirical scaling hypotheses lead us to write the functions C fi, it ,(p, r) and Ci n, fi n it e (p, r) in the form:

C(r,p)

(13.[r(p) -1 ]

(r > a > 0)

zb e — a4 at infinity. Their behavior at and typically (I)(z) where 4P(z) is analytic, 0(0) for r large enough: we recover the fact that at the critical point r p=p c is thus CH p=pc, an exponential decrease rb— e — "le(P) is replaced by a power law r— e2( for r co.

Characteristic length There are several possible definitions for the

characteristic length C(p).

< p < 1) and C n fi,it e (P) We introduced the correlation lengths, i(P) (if firlte (if pc