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On Hilfer’s objection to the fractional time diffusion equation E. Barkaia,, I.M. Sokolovb a
Department of Physics, Bar Ilan University, Ramat-Gan 52900 Israel Institut fu¨r Physik, Humboldt-Universita¨t zu Berlin, Newtonstr. 15, 12489 Berlin Germany
b
Received 6 October 2005; received in revised form 26 February 2006
Abstract Hilfer [Physica A 329 (2003) 35] claims to give an example of a continuous time random walk (CTRW) model with longtailed waiting time probability density that approaches a Gaussian behavior in the continuum limit. Rigorous limit theorems, derived previously, show however that in the limit of long-time such a CTRW converges to a non-Gaussian behavior. We discuss two types of continuum limits for the CTRW model: the fractional continuum limit and the one introduced by Hilfer. We show that the fractional limit yields the correct long-time behavior of the CTRW, while Hilfer’s continuum limit does not. We discuss a general approach to find a continuum limit of the CTRW process. r 2006 Elsevier B.V. All rights reserved. Keywords: Continuous time random walks; Fractional diffusion equation; Continuum limit
1. Introduction The fractional time diffusion equation [1] is of interest in the context of stochastic modeling of anomalous diffusion processes [2–4]. The equation reads qa Pfr ðr; 0Þta , Pfr ðr; tÞ ¼ K a r2 Pfr ðr; tÞ þ a qt Gð1 aÞ
(1)
where qa =qta is the Riemann-Liouville time derivative. For free boundary conditions and for initial conditions centered at the origin, the propagator in Fourier–Laplace space is Pfr ðk; uÞ ¼
ua1 . ua þ K a k2
(2)
The inversion of Eq. (2) in d dimension, to the r, t domain, was considered in Refs. [1,5]. As well known, many random walks can be modeled, under well-investigated conditions, using the ordinary diffusion equation. In a similar way, several authors [2–12] claimed that continuous time random walks (CTRWs) with power law waiting times distribution, could be modeled based on the fractional diffusion equation. Recently, Hilfer [13] challenged this claim and considered a continuum limit of the CTRW. Corresponding author.
E-mail addresses:
[email protected] (E. Barkai),
[email protected] (I.M. Sokolov). 0378-4371/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2006.05.022
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He claims that for a certain type of long-tailed waiting time probability density function (PDF) cðtÞ, the continuum limit of the CTRW is Gaussian, and not the propagator of the fractional diffusion equation (see details in the next section). This means that the generality of the fractional diffusion framework was questioned. Further, if Hilfer’s objection to the general validity of the fractional diffusion equation is accepted, claims made in the literature are wrong. Note that certain limitations (and rate of convergence) of the fractional framework are investigated in Refs. [9,14]; however, roughly speaking, these limitations are not as strong if compared with the claim made in Ref. [13]. In what follows we proceed to show that Hilfer’s claim is connected not with the properties of the CTRW or the fractional diffusion equation but with his rather arbitrary definition of the continuous limit. An existence of continuum limit is intimately related to the long-time behavior of the CTRW [15,16] and to limit theorems for probability distributions of the CTRW process [17,18]. Any suggestion for a continuum limit should be consistent with known long-time asymptotic behavior of the Green function of the underlying random walk. As we will show in Section 3 the correct approach is based not on the manipulation of parameters in the waiting time probability density function cðtÞ and of the jump length distribution as considered by Hilfer. Rather the continuum limit is found by rescaling of time and space, or identically by rescaling of the random process itself, in a way which is compatible with the existing limit theorems and long-time asymptotic behavior. As usual, the continuum limit does not work well for short times when full details of the process must be taken into account. 2. Continuum limit of the CTRW The integral equation for the probability Pðr; tÞ of finding the CTRW particle at r is Z t Z t X Pðr; tÞ ¼ dr0 1 cðt0 Þ dt0 þ cðt t0 Þ lðr r0 ÞPðr0 ; t0 Þ dt0 . 0
0
(3)
r0
Here lðrÞ denotes the probability for displacement r, and cðtÞ is the waiting time PDF of times between successive jumps. As shown by Hilfer and Anton [19] the passage from this integral master equation to a fractional master equation is readily obtained if cðtÞ is a suitable Mittag–Leffler function [20,21]. Note that the first term on the left-hand side of Eq. (3) describes a particle that did not jump during the observation time t while the second term describes a random arriving at r0 at time t0 and then making an additional single jump to point r. The Montroll-Weiss equation yields the Fourier–Laplace transform of Pðr; tÞ, Pðk; uÞ ¼
1 1 cðuÞ , u 1 cðuÞlðkÞ
(4)
where cðuÞ ½lðkÞ is the Laplace [Fourier] transform of cðtÞ ½lðrÞ, respectively. We investigate the one-dimensional walk on a lattice, and then lðkÞ ¼ cosðskÞ, where s2 is the variance of jump lengths. We consider first Hilfer’s example number 1, ta1 ta c1 ðtÞ ¼ E , a;a ðt1 Þa ðt1 Þa
(5)
(6)
where 0oto1 and E a;a ðxÞ is the generalized Mittag–Leffler function E a;a ðxÞ ¼
1 X n¼0
xn . Gðan þ aÞ
(7)
Here and all along this manuscript we use 0oap1. For long times c1 ðtÞ / tð1þaÞ when ao1, hence the first moment of c1 ðtÞ, the averaged waiting time, is infinite. The Laplace transform of Eq. (6) is c1 ðuÞ ¼
1 . 1 þ ðt1 uÞa
(8)
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If a ¼ 1 then c1 ðtÞ ¼ ð1=t1 Þ expðt=t1 Þ. Before considering the work in Ref. [13], let us define the fractional continuum limit. Let cðtÞ be a longed-tailed waiting time PDF cðtÞ / tð1þaÞ for long times, and let cðuÞ1 ðtuÞa . . . when u ! 0. Then according to the fractional continuum limit lim
Pðk; uÞ ¼
2
s t!0;s!0;2t a ¼K a
ua1 . ua þ K a k 2
(9)
Note that when a ¼ 1 the diffusion constant K 1 ¼ s2 =2t is the same as Einstein’s definition of the diffusion constant of a Brownian particle, based on the random walk picture. Therefore, when ao1, K a is a fractional diffusion coefficient which is a natural generalization of the normal diffusion constant. We first check the validity of Eq. (9) using the example Eq. (6). Using the expansion cosðxÞ ¼ 1 x2 =2 þ x4 =24 we have P1 ðk; uÞ ¼
1 ðt1 uÞa ¼ u 1 þ ðt1 uÞa lðkÞ ua þ ð
ua1 s2
2
k ðt1 Þa 2
4 s 24ðt a k þ Þ 1Þ 4
.
(10)
It is now easy to see that in the fractional continuum limit lim
2 a ¼K a 1Þ
Pðk; uÞ ¼
t1 !0;s!0;2ðts
ua1 . ua þ K a k2
(11)
Thus, the fractional continuum limit yields the correct propagator for this situation Eq. (2), which is compatible with the fractional diffusion equation. Eq. (11) was in fact derived by Tunaley [15], a fact unfortunately ignored by Hilfer [13]. Following Hilfer [13] we consider a second type of continuum limit, which we call Hilfer’s continuum limit (HCL). Consider the waiting time PDF ta1 ta c2 ðtÞ ¼ E a;a (12) c2 ðt2 Þ2 c2 ðt2 Þ2 and then c2 ðuÞ ¼ 1 c2 ðt2 Þ2 ua þ .
(13)
The reader noticed that Eqs. (12) and (6) are identical when we set c2 ðt2 Þ2 ¼ ðt1 Þa . Note that c2 has dimension of timea2 . The notation in Eq. (12) is a preparation for the following limit. According to HCL: t2 ! 0, s ! 0, while the ratio s2 =ðt2 Þa remains finite. Clearly HCL is different from the fractional continuum limit. Using Eq. (13) P2 ðk; uÞ ¼
ua1 ua þ ðc
s2 k 2 2 2 ðt2 Þ 2
4
24csðt 2
2 2Þ
k4 þ Þ
.
(14)
Hence, according to HCL lim
2
Pðk; uÞ ¼ 0.
(15)
t2 !0;s!0;ðts Þa !Const 2
In taking HCL c2 remains fixed. We see that HCL is meaningless, it does not yield a PDF in the sense that when we take the limit, the PDF Pðr; tÞ vanishes since its Laplace–Fourier transform turns out to be equal zero. Now consider example number 3, following [13] ta1 ta 1 t c3 ðtÞ ¼ E exp þ . (16) a;a 2t3 t3 2c3 ðt3 Þ2 c3 ðt3 Þ2
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Hilfer [13] showed that under HCL such a waiting time PDF does not yield the propagator of the fractional diffusion equation, instead, according to HCL, lim
t3 !0;s!0;s2 =ðt3 Þa !Const
P3 ðk; uÞ ¼ 0.
(17)
Thus, we see once more that the HCL is not well defined, since it does not yield a normalized Green function. We now show that using the fractional continuum limit, the propagator of the fractional diffusion equation is obtained. We use a 1 ta1 t 1 1 t c4 ðtÞ ¼ E exp þ , (18) a;a 2 ðt4 Þa 2 c4 t4 c4 t4 ta4 where c4 is dimensionless. The Laplace transform of Eq. (18) is c4 ðuÞ ¼
1 1 1 1 . aþ 2 1 þ ðt4 uÞ 2 1 þ c4 t4 u
(19)
Hence we find 1 1 1 c4 ðuÞ 1 2 þ ðt4 uÞa þ c4 ðt4 uÞ 1 ½lðkÞ 1 1 ½lðkÞ 1 P4 ðk; uÞ ¼ ¼ u 1 c4 ðuÞ u c4 ðt4 uÞ þ ðt4 uÞa þ 2c4 ðt4 uÞ1þa 2 2 1 1 1 sk s4 k4 2 þ ðt4 uÞa þ c4 ðt4 uÞ 1þ þ ¼ . u ðt4 uÞa 2 24 c4 ðt4 uÞ1a þ 1 þ 2c4 t4 u
ð20Þ
It follows that in the fractional continuum limit lim
a
t4 !0;s!0;s2 =ðt4 Þ ¼K a
P4 ðk; uÞ ¼
ua1 . ua þ K a k 2
(21)
Thus, the fractional Green function Eq. (2) is obtained [22]. So far we discussed two continuum limits, the fractional and the HCL. Hilfer [13] however suggests to consider one more continuum limit, the conventional continuum limit defined on the left-hand side of Eq. (22). We will show that when this conventional continuum limiting transition is applied to the CTRW, we obtain wrong results. For example, using the waiting time PDF, Eqs. (18) and (20), it is straightforward to see that the conventional continuum limit yields lim
t4 !0;s!0;s2 =t4 !Const
1 P4 ðk; uÞ ¼ . u
(22)
Hence in this limit the propagator in Fourier space is independent of k, and the conventional continuum limit implies that the particle is localized on the origin, which is of course wrong. Or slightly more interestingly, from Eq. (23) in Ref. [13], we learn that using example number 3, Eq. (16), the corresponding propagator is Gaussian when the conventional continuum limit is taken lim
t3 !0;s!0;s2 =t3 !Const
P3 ðk; uÞ ¼
1 . u þ K 1 k2
(23)
Therefore, the limiting transition considered leads at least to a Green’s function solution being a valid propagator. However, the result is still wrong, since the mean square displacement of the random walker behaves like hr2 i / ta in the underlying model in the limit of long times, while Eq. (23) leads to a normal behavior hr2 i / t. We see now that it is wrong to manipulate parameters of a waiting time PDF in an arbitrary manner as done in Ref. [13]. The conclusion is simple, we should not use a continuum approximation for the CTRW, which does not yield the correct long-time behavior of hr2 i or more generally of the propagator. We also note that applying the fractional continuum limit to ordinary random walks, for example CTRWs with exponential waiting times, will lead to a non-Gaussian behavior. Of course one never uses the fractional continuum limit for ordinary random walks. Similarly, one should not apply the conventional continuum approximation to long-tailed CTRWs.
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3. What is a continuum limit In this section we consider a method leading to the correct continuum limit of random walk processes. It is the merit of Hilfer’s work that showed us that such limiting transitions are not straightforward in cases when the PDF of waiting times contains several time scales, e.g. t1 ; t2 ; . . . (or similarly many length scales of the PDF of jump length). These waiting times parameters cannot in the general case be treated independently, and hence we discuss a method to overcome this difficulty. The main idea is to re-scale time and space and not the parameters of the waiting time or step length distributions. Space–time rescaling means that we use the transformations r!
r ; f ðcÞ
t!
t c
under which the corresponding waiting time and jump length distributions change as follows: 1 r 1 t lðrÞ ! l ; cðtÞ ! c . f ðcÞ f ðcÞ c c
(24)
(25)
One sees that we may think of rescaling of time and space as of introducing a new (rescaled) random process defined by new jump lengths and waiting times PDFs in Eq. (25). The parameter c will be taken to be small, corresponding to long times i.e., for fixed t the ratio t=c is large. The function f ðcÞ will soon be determined. In Laplace–Fourier space the rescaling is k ! f ðcÞk and u ! cu. After rescaling we have the Montroll–Weiss equation Pc ðk; uÞ ¼
1 cðcuÞ , u½1 lðf ðcÞkÞcðcuÞ
(26)
where the subscript c denoted the propagator of the rescaled process. When c ¼ 1 we have the original process defined by Eq. (4). When c ! 0 and letting f ðcÞ ! 1 we find using cðcuÞ1 AðcuÞa Pc ðk; uÞ
Aua1 . ðs2 k2 =2Þf ðcÞ=ca þ Aua
(27)
Here we assume that higher order terms in the expansion vanish in the limit c ! 0 and f ðcÞ ! 1 when their ratio remains finite (see example below). To complete the continuum limit we demand that the limit exists and is compatible with the behavior of the mean square displacement hr2 ita of the original process. This means that f 2 ðcÞ=ca ¼ 1. Hence we find the continuum limit (without manipulating any arbitrarily chosen PDF parameter) lim Pc ðk; uÞ ¼
c!0
Aua1 . ðs2 k2 =2Þ þ Aua
(28)
which is identical to the fractional diffusion limit. In the case a ¼ 1 (ordinary diffusion) the conventional continuum limit is recovered. To summarize, the continuum limit is found after rescaling of space and time and not as Hilfer suggested after taking one of the parameters t1 ; t2 ; t3 ; t4 , etc. to zero. To see this in even more detail consider our first example Eq. (10) in this case we get Pc ðk; uÞ ’
ua1 ua þ ðs2 f 2 ðcÞk2 =2ðct1 Þa Þ ðs4 f 4 ðcÞk4 =24ðct1 Þa Þ þ
(29)
and the continuum limit with c ! 0 and f ðcÞ ¼ ca is easily seen to lead to Eq. (28). The scaling transformation (change of time and space scales) corresponds in this case to the transformations t1 ! t1 c and s2 ! s2 ca , thus in this example the continuum limit corresponds to changing a particular set of parameters of the waiting time and jump length distributions. More generally the identification of these parameters must be made with care (if at all), following the general limiting procedure we have just discussed.
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4. Discussion The fractional continuum limit, when applied to the CTRW, yields the propagator of the fractional diffusion equation. Hilfer’s continuum limit differs from the fractional continuum limit, in some cases it gives a propagator which is equal to zero, which is not a propagator at all and in other cases it gives a Gaussian propagator which is wrong. The long-time behavior of the CTRW is well investigated [15,16]. The rigorous limit theorems in this field [17,18] show that CTRWs, in the limit of long times, reach an asymptotic behavior described by one-sided Le´vy distributions e.g. [5]. This asymptotic behavior is identical to the behavior modeled based on the fractional time diffusion equation, hence the fractional continuum limit yields the correct propagator. Since all CTRWs, with power law waiting time PDFs approach a unique limiting behavior described by the fractional diffusion equation, the fractional diffusion equation has a general validity. Finally, we note that a Physical system is not a continuum, hence all continuum limits are approximations. The main idea behind the fractional and conventional continuum limits is that they capture the correct asymptotic behavior of the underlying random walk. Unlike the approach in Ref. [13], the fractional continuum limit should be applied to CTRWs with long-tailed waiting times, the conventional continuum limit theorem should be applied to random walks with Gaussian behavior, and HCL should never be applied. If one does not know how to find the continuum limit in a particular situation, then the full scheme of Section 3 should be used. Acknowledgment EB thank the Israel Science Foundation and the center of complexity in Jerusalem, for financial support. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
W.R. Schneider, W. Wyss, J. Math. Phys. 30 (1989) 134. R. Metzler, J. Klafter, Phys. Rep. 339 (2000) 1. G.M. Zaslavsky, Phys. Report 371 (2002) 461. I.M. Sokolov, J. Klafter, A. Blumen Phys. Today 55 (2002) 48. E. Barkai, Phys. Rev. E 63 (2001) 046118. V. Balakrishnan, Physica A 132 (1985) 569. A. Compte, Phys. Rev. E 53 (1996) 4191. A.I. Saichev, G.M. Zaslavsky, Chaos 7 (1997) 753. E. Barkai, R. Metzler, J. Klafter, Phys. Rev. E 61 (2000) 132. V.V. Uchaikin, Int. J. Theor. Phys. 39 (2000) 2087. I.M. Sokolov, Phys. Rev. E 63 (2001) 056111. R. Gorenflo, A. Vivoli, F. Mainardi, Nonlinear Dyn. 38 (2004) 101. R. Hilfer, Physica A 329 (2003) 35. E. Barkai, Chemical Physics 284 (2002) 13. J.K.E. Tunaley, J. Stat. Phys. 11 (1974) 397. M.F. Shlesinger, J. Klafter, Y.M. Wong, J. Stat. Phys. 27 (1982) 499. M. Kotulski, J. Stat. Phys. 81 (1995) 777. M.M. Meerschaert, H.P. Scheffler, J. Appl. Probab. 41 (2004) 623. R. Hilfer, L. Anton, Phys. Rev. E 51 (1995) R848. A general theme in Hilfer’s work is the relation between the Mittag–Leffler waiting time PDF and fractional kinetic equations. For example in Ref. [21] Hilfer claims that using the Mittag–Leffler function for the CTRW process, the CTRW Green function is identical to the fractional Green function at all times. However, the characteristic function of jump lengths used in Ref. [21] is lðkÞ ¼ 1 k2 =2 which is not a characteristic function at all, namely, the PDF of jump lengths become negative. Of course one could choose lðkÞ1 k2 =2 for k ! 0, however, then the fractional framework is valid only in the long time limit. Thus, in this sense Mittag–Leffler waiting times are similar to all other longed-tailed waiting times PDFs, as we show in the manuscript. Namely Mittag–Leffler waiting time PDFs may yield a fractional behavior only in the limit of long times. Note that Mittag–Leffler functions are known as a natural generalization of the exponential function in the context of anomalous diffusion [2]. [21] R. Hilfer, Fractals 3 (1995) 211. [22] A minor issue, here K a ¼ s2 =ðt4 Þa and not K a ¼ s2 =2ðt4 Þa , since c4 ðuÞ ¼ 1 ðt4 uÞa =2 . . . and not c4 ðuÞ ¼ 1 ðt4 uÞa . . . :
