From the trajectory to the density memory

Chaos, Solitons and Fractals 34 (2007) 19–32 www.elsevier.com/locate/chaos

From the trajectory to the density memory Rasit Cakir a, Arkadii Krokhin a, Paolo Grigolini a

c

a,b,c,*

Center for Nonlinear Science, University of North Texas, P.O. Box 311427, Denton, TX 76203-1427, USA b Dipartimento di Fisica ‘‘E.Fermi’’ – Universita´ di Pisa, Largo Pontecorvo, 3, 56127 Pisa, Italy Istituto dei Processi Chimico Fisici del CNR, Area della Ricerca di Pisa, Via G. Moruzzi, 56124 Pisa, Italy

Abstract In this paper we discuss the connection between trajectory and density memory. The first form of memory is a property of a stochastic trajectory, whose stationary correlation function shows that the fluctuation at a given time depends on the earlier fluctuations. The density memory is a property of a collection of trajectories, whose density time evolution is described by a time convoluted equation showing that the density time evolution depends on its past history. We show that the trajectory memory does not necessarily yields density memory, and that density memory might be compatible with the existence of abrupt jumps resetting to zero the system’s memory. We focus our attention on a time-convoluted diffusion equation, when the memory kernel is an inverse power law with (i) negative and (ii) positive tail. In case (i) there exist both renewal trajectories and trajectories with memory, compatible with this equation. Case (ii), which has eluded so far a convincing interpretation in terms of trajectories, is shown to be compatible only with trajectory memory. Ó 2007 Elsevier Ltd. All rights reserved.

1. Introduction In the modern literature on non-equilibrium statistical mechanics the concept of memory plays a central role. This is made especially evident by the work of Zwanzig [1,2] and Mori [3], and by their time convoluted master and Langevin equations. For a discussion of these memory problems the reader can consult many books and review reports. We limit ourselves to quoting the work of Ref. [4] and the more recent report by Balucani et al. [5]. The time convoluted master equation of Ref. [1,2] was proved [6] to be equivalent to that found by Prigogine and Re´sibois [7], which is at basis of the Subdynamics Theory, as shown in the beautiful book by Balescu [8]. The time convoluted master equations show that the density time evolution depends on its past history, thereby implying that the system under study has memory. Another popular way of introducing memory is that advocated by Mandelbrot [9] with his approach to Fractional Brownian Motion (FBM), meant to be a diffusion process generated by fluctuations with memory. If a fluctuation is positive, also the subsequent fluctuations will be positive. Using the

* Corresponding author. Address: Center for Nonlinear Science, University of North Texas, P.O. Box 311427, Denton, TX 762031427, USA. Tel.: +1 409 565 3294. E-mail addresses: [email protected] (A. Krokhin), [email protected] (P. Grigolini).

0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.01.046

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R. Cakir et al. / Chaos, Solitons and Fractals 34 (2007) 19–32

dynamic approach to FBM [10], we express the position x of the random walker at time t, moving from x = 0 at t = 0, by means of the equation Z t nG ðt0 Þdt0 : ð1Þ xðtÞ ¼ 0

This diffusion process is faster than the ordinary if the fluctuation nðtÞ is persistent, namely, if it tends to keep the same sign for an extended period of time. If the fluctuations are anti-persistent, namely, if a fluctuation in one direction is ordinarily followed by a fluctuation in the opposite direction, the resulting diffusion process is slower than the ordinary. In both cases, the deviation from ordinary diffusion is generated by trajectory memory. In literature it seems to be widely accepted that density and trajectory memory are equivalent. However, there are good reasons to not share this conviction. In the last few years, there has been an increasing interest for the diffusion equations emerging from the perspective of Continuous Time Random Walk (CTRW) [11]. Since the fundamental work of Kenkre et al. [12], it is well known that the CTRW generates, for the time evolution of density, the same time convoluted structure as the methods of Refs. [1,2,7,8]. According to the earlier definitions, this is density memory, and the expected equivalence between trajectory and density memory is clearly violated, if we keep in mind that the CTRW is a process based on jumps, whose occurrence resets to zero the system’s memory. To make this aspect more evident to the readers, in Section 3 we shall show how to derive from CTRW the diffusion equation Z t o o2 dsUðsÞ 2 pðx; t  sÞ: ð2Þ pðx; tÞ ¼ D ox ot 0 Eq. (2) has the earlier mentioned time convoluted structure, thereby implying that the time evolution of the probability density p(x, t) depends on the earlier times t  s: this explains why we call it density memory. What about trajectory memory? As earlier pointed out, there is no such memory here, due to the occurrence of memory erasing jumps. Now, let us point out another aspect conflicting with the naive conviction that trajectory and density memory are equivalent. Let us go back to the popular subject of FBM, or, more precisely, to the dynamic approach to FBM, which is known [13] (see also [14]) to generate the following generalized diffusion equation: Z t  2 o o pðx; tÞ ¼ D dsUðsÞ pðx; tÞ: ð3Þ 2 ot ox 0 Note that the function UðtÞ in this case is the correlation function of n, and it has fat tails, indicating the existence of trajectory memory. Yet, the diffusion equation, does not have the time convoluted structure of Eq. (2). The work of Ref. [15] shows that the emergence of the time-convolutionless structure of Eq. (3), to compare to the time-convolution structure of Eq. (2), is determined by the fact that the stochastic variable nðtÞ is Gaussian in the case of Eq. (3). In the recent work of Ref. [16], Kenkre noticed that, although the two equations might yield the same second moment, the higher moments are not identical and that Eq. (3) cannot produce the transition from merely diffusive to merely wave-like motion, whereas Eq. (2) does. The main purpose of this paper is to shed some light unto the confusion concerning the relations between these two kinds of memory. We shall also solve the very elusive problem of establishing the trajectory properties behind Eq. (2), when this equation generates super-diffusion. The trajectory properties of this equation are well known in the sub-diffusional case [17], but in the super-diffusion case its correct interpretation in terms of CTRW does not exist [18] and there are good reasons [19] to believe that in this case Eq. (2) is incompatible with a CTRW origin. In line with [18,19], we shall find a trajectory memory picture that explains the dynamical origin of Eq. (2) in the super-diffusion case.

2. The stochastic Liouville approach Let us consider the following diffusion equation: d xðtÞ ¼ nK ðtÞ: dt

ð4Þ

This equation is formally equivalent to Eq. (1), but the stochastic variable nK ðtÞ is not Gaussian. It has the property as X k m jli; nK jli ¼ j0i: ð5Þ nK j0i ¼ l6¼0

We shall see that it is a dichotomous variable.

R. Cakir et al. / Chaos, Solitons and Fractals 34 (2007) 19–32

21

Let us adopt the Liouville formalism and let us write the time evolution of the total probability density qðx; nK ; tÞ as follows:   d d qðx; nK ; tÞ ¼ LT qðx; nK ; tÞ  nK þ LB qðx; nK ; tÞ: ð6Þ dt dx The operator LB is responsible for the time evolution of the probability density of nK, namely, it is a bath operator. Following the stochastic Liouville approach of Kubo [20–22], let us consider the density qðx; nK ; tÞ as a vector state to expand in the basis set of the eigenstates of LB. In the stationary case the bath operator must have an eigenstate j0i with vanishing eigenvalue, LB j0i ¼ 0:

ð7Þ

From within this quantum-like formalism, we have to take into account the out-of-equilibrium properties of the bath, namely we have to consider the non-equilibrium eigenstates jli, with l 6¼ 0, defined by LB jli ¼ ixl jli:

ð8Þ

With no loss of generality, let us assume xl ¼ xl :

ð9Þ

We set p0 ðx; tÞ  h0jqðx; nK ; tÞi

ð10Þ

pl ðx; tÞ  hljqðx; nK ; tÞi:

ð11Þ

and

The projection over the bath eigenstates create reduced densities corresponding to the bath at equilibrium, Eq. (10), and to the bath in an out of equilibrium condition, Eq. (11). By projecting Eq. (6) over the bath equilibrium state and the bath excited states, we obtain X d d p0 ðx; tÞ ¼  h0jnK jli pl ðx; tÞ ð12Þ dt dx l and d d p ðx; tÞ ¼ ixl pl ðx; tÞ  h0jnK jli p0 ðx; tÞ dt l dx with l running from 1 to 1, respectively. The formal solution of Eq. (13) is Z t   pl ðx; tÞ ¼  dt0 exp ixl ðt  t0 Þ hljnK j0ip0 ðx; t0 Þ:

ð13Þ

ð14Þ

0

By plugging Eq. (14) in Eq. (12) we get Z t d o2 dt0 UK ðt  t0 Þ 2 p0 ðx; t0 Þ; p0 ðx; nK ; tÞ ¼ ox dt 0 where UK 

X

eixl t h0jnK jlihljnK j0i:

ð15Þ

ð16Þ

l

Note that within the quantum-like Kubo’s formalism the stochastic variable nK becomes an operator. By moving from the discrete- to the continuous-frequency picture, and using Eq. (9) as well, we rewrite the memory kernel of Eq. (15) under the following form: Z UK ðtÞ ¼ dx cosðxtÞPðxÞ; ð17Þ where PðxÞ ¼ 2h0jnK jxihxjnK j0i:

ð18Þ

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R. Cakir et al. / Chaos, Solitons and Fractals 34 (2007) 19–32

Note that in this case the memory kernel UK ðtÞ is the equilibrium correlation function of the stochastic variable nK. According to the work of Ref. [23] the Generalized Diffusion Equation (GDE) of Eq. (15) is the exact equation of motion for the density p(x, t) if the fourth-order correlation functions of nK fulfills the factorization condition hnk ðt4 Þnk ðt3 Þnk ðt2 Þnk ðt1 Þi ¼ hnk ðt4 Þnk ðt3 Þihnk ðt2 Þnk ðt1 Þi

ð19Þ

and the higher-order correlation functions the analogous factorization conditions, so that at equilibrium the condition 2 n hn2n k i ¼ hn i

ð20Þ

applies. This seems to be the natural property of a dichotomous variable. We shall refer throughout to this property as Dichotomous Factorization (DF) property. It is important to point out that in the renewal and non-exponential case this condition is violated even if the stochastic variable has only two values [24]. In this paper we show that the systems with trajectory memory fit the DF property. It is worth making a final remark concerning the form of UK ðtÞ. We have in mind the model of non-Ohmic bath discussed by several authors [25–27]. We select PðxÞ / xg1 , and we find that in the time asymptotic limit UK ðtÞ /

1 tg

for 0 < g < 1

ð21Þ

and UK ðtÞ / 

1 tg

for 1 < g < 2:

ð22Þ

The former state, called sub-Ohmic, is separated from the latter, called super-Ohmic, by the singular condition g ¼ 1, corresponding the a fast relaxation, given by a delta of Dirac. It is worth remarking that a GDE with the form of Eq. (15) was found many years ago by Kenkre and Knox [28]. These authors pointed out that this equation is exactly equivalent to the GDE generated by the adoption of the CTRW method. This is a so important observation as to force us in Section 3 to illustrate this argument for the sake of reader’s convenience.

3. The generalized diffusion equation from the CTRW perspective Let us imagine a sequel of events described by pðnÞ ¼ Mn pð0Þ: The symbol p denotes a vector with infinite components pi fitting the normalization condition 1 X pi ¼ 1:

ð23Þ

ð24Þ

i¼1

We have to focus our attention on the special case when at n = 0 only the site i = 0 is occupied and all the other sites are empty. At each time step the particle makes a transition from the site i = 0 to other sites, depending on the form of the matrix M. In this paper we adopt for M the following specific form: X1 M ðjiihi þ 1j þ ji þ 1ihijÞ: ð25Þ 2 As in Section 2, the choice of this quantum-like formalism is done to avoid the cumbersome matrix notation, and it does not imply any departure from the classical condition. For instance, if at t = 0 only the site i = 0 is occupied 1 Mpð0Þ ¼ Mj0i ¼ ðj1i þ h1jÞ; 2

ð26Þ

which means that the particle at the first step will jump with equal probability from the site i = 0 to the two nearest neighbor sites j1i and j  1i. It should be easy to consider transition matrices with a different form. We make the choice of Eq. (25) for the purpose of deriving a GDE with the same form as that found by Kenkre and Knox [28], namely, Eq. (15). Let us discuss this simple problem from the point of view of an individual trajectory. The position x(n) of the particle after n steps will be

R. Cakir et al. / Chaos, Solitons and Fractals 34 (2007) 19–32

xðnÞ ¼

n X

ni :

23

ð27Þ

i

We assume that the time distance between two consecutive jumps is fixed and equal to tu. The spatial distance between two nearest neighbor sites is fixed and is equal to a. Let us do the experiment with many particles, moving from the same initial site i = 0. The position x(n) will correspond to a given site with index i. We count how many particles are found in this site, we call ni the number of particles occupying this site, and we divide this number by the total number of trajectories, N, and we set the relation ni ¼ pi ðnÞ ¼ hijMn jpð0Þi: ð28Þ N With this equivalence between stochastic trajectories and a probabilistic description in mind, we reach a first conclusion: Eq. (28) represents the probability distribution of this set of random walkers at a given time t = ntu. In this condition we obtain, in accordance with the central limit theorem hx2 ðnÞi1=2 / n1=2

ð29Þ

and the diffusion equation o o2 pðx; nÞ ¼ D 2 pðx; nÞ: ox on

ð30Þ

We make the assumption that tu is the minimum possible time, and that the experimental time t is much larger than tu. Let us now make also the crucial assumption that the time distance between the occurrence of the nth event, occurring at time tn, and the occurrence of the (n + 1)th event, occurring at time tnþ1 , rather than being fixed, fluctuates. The assumption that the time distance between two nearest neighbor events fluctuates is physically plausible. In fact, if the ordinary diffusion generating fluctuations are a consequence of the collision between the particle of interest and the bath molecules, it is reasonable that these collisions do not occur at regular but at erratic times. Thus the quantity sn ¼ tnþ1  tn is a stochastic variable with the distribution density wðsÞ. The vector pðtÞ is related to the vector pð0Þ by means of 1 Z t X pðtÞ ¼ wn ðsÞWðt  sÞMn pð0Þ ds: n¼0

ð31Þ

ð32Þ

0

Note that the function wn ðtÞ denotes the probability density for a sequel of n event to occur, the last of which occurs n ^ exactly at t. Due to the statistical independence of these events, we have that the Laplace transform of wn ðtÞ is ðwðuÞÞ , ^ with wðuÞ denoting the Laplace transform of wðtÞ  w1 ðtÞ, namely the probability density for one event to occur at time t. The function WðtÞ is the probability that no event occurs up to time t, and it is defined by Z 1 dt0 wðt0 Þ: ð33Þ WðtÞ  t

The physical meaning of Eq. (32) is made evident by the following remarks. The specific state pðtÞ is determined by the last of a sequel of collisions occurring prior to time t. The number of collisions is arbitrary, thereby explaining the index n running from 0 to 1. No further event occurs between s and t. This is taken into account by multiplying wn ðsÞ by Wðt  sÞ. It is a straightforward to prove that Eq. (32) is equivalent to Z t o dsUðsÞKpðt  sÞ ð34Þ pðtÞ ¼  ot 0 with K ¼ 1  M:

ð35Þ

The memory kernel UðtÞ is related to wðtÞ in the Laplace domain through ^ uwðuÞ ^ : UðuÞ ¼ ^ 1  wðuÞ

ð36Þ

These results are obtained by comparing the Laplace transform of Eq. (34) to the Laplace transform of Eq. (32).

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R. Cakir et al. / Chaos, Solitons and Fractals 34 (2007) 19–32

Notice that, as earlier pointed out, with the choice of Eq. (25) we can express the result of Eq. (34) in a form equivalent to that proposed by Kenkre and Knox in Ref. [28]. In fact, if the choice of Eq. (25) applies, we have Kpji ¼ ð1  MÞpji ¼ pi þ

piþ1 pi1 1 o2 þ ¼ pðx; tÞ: 2 ox2 2 2

Thus, we write Eq. (34) as follows: Z t o o2 pðx; tÞ ¼ D dsUðsÞ 2 pðx; t  sÞ; ot ox 0

ð37Þ

ð38Þ

which coincides [29] with Eq. (18) of Ref. [28]. Notice that if we assign the value a to the distance between the nearest neighbor sites of the model here under discussion, the diffusion coefficient D reads D ¼ a2 =2. It is worth mentioning that the GDE of Eq. (38) drives the density of the diffusion process described by d xðtÞ ¼ nR ðtÞ; dt

ð39Þ

where nR ðtÞ is a fluctuation almost always vanishing but in the correspondence of a collision, where it takes the value 1 or 1, according to the coin tossing prescription. It is worth remarking that the diffusion equation of Eq. (38) in the time asymptotic limit yields scaling. This means 1 x ð40Þ pðx; tÞ ¼ d F d t t with d called scaling coefficient, and F(y) being a function of y, different in general from a Gaussian function. This is a formal but rigorous way of assessing that in the time asymptotic condition x / td . The deviation from ordinary diffusion is signaled by d 6¼ 0:5 and/or F(y) departing from the Gaussian form. It is evident that the departure from ordinary diffusion, when it occurs, takes the form of sub-diffusion, namely, with d < 0:5: this is so because the longer the sojourn time in one site, the slower the diffusion process. To make the sojourn in one site as long as possible, we set wðsÞ ¼ ðl  1Þ

T l1 ðs þ T Þl

ð41Þ

with the power index l meeting the condition ð42Þ

1
so that the mean sojourn time in one site is infinite. To evaluate the scaling in the case of Eq. (38) it is convenient to study the Fourier–Laplace transform of pðx; tÞ, with k and u being the variables conjugated to x and t, respectively. We denoted the Fourier–Laplace transform of p(x, t) with ^pðk; uÞ. Its explicit form is ^pðk; uÞ ¼

1 : ^ u þ k DUðuÞ 2

ð43Þ

Using Eq. (36) and the limiting condition u ! 0 of wðsÞ of Eq. (41) when the condition of Eq. (42) applies, ^ wðuÞ ¼ 1  Cð2  lÞðuT Þl1

ð44Þ

we obtain ^pðk; uÞ ¼

1 2 2l

u D u þ kCð2lÞ

:

ð45Þ

To determine the unknown scaling coefficient d, we set k / ud , we plug it into Eq. (45) and we look for the value of d making ^pðud ; uÞ become proportional to 1/u. We get d¼

ðl  1Þ ; 2

which is in fact smaller than 1/2, when condition (42) applies.

ð46Þ

R. Cakir et al. / Chaos, Solitons and Fractals 34 (2007) 19–32

25

4. The fractional derivative argument In this section we plan to address the issue of the relation between the GDE of Eq. (15) and Eq. (38) and the fractional derivative approach to complexity. The adoption of fractional [30,31] rather than integer derivatives is becoming more and more popular to address the study of complex systems. An interesting recent example is given by the work of Ref. [17]. Sokolov and Klafter proposed a form of GDE that we write here with a notation change so as to make it easier for the reader to see the connection with the other results of this paper Z t o 1 d 1 o2 pðx; tÞ ¼ ds pðx; sÞ ð47Þ 2l ot Cð2  lÞ dt 0 ox2 ðt  sÞ with the power index l meeting the condition of (42). The term on the right-hand side of this equation can be expressed by means of the Riemann–Liouville fractional derivative [30,31] defined in general by Z t da 1 dn f ðtÞ ¼ ðt  t0 Þna1 f ðt0 Þ dt0 ; ð48Þ Cðn  aÞ dtn 0 dta with n being the smallest integer exceeding a. With this definition, Eq. (47) reads o d1a o2 pðx; tÞ ¼ 1a 2 pðx; tÞ ot dt ox

ð49Þ

with a ¼ l  1. The authors of Ref. [17] found the interesting result that the same diffusion process can be expressed by means of oa o2 pðx; tÞ ¼ pðx; tÞ ota ox2 provided that use is made of the Caputo fractional derivative, advocated by Mainardi [30,31], defined by Z t da 1 dn f ðtÞ ¼ ðt  t0 Þna1 n f ðt0 Þ dt0 : a Cðn  aÞ 0 dt dt

ð50Þ

ð51Þ

Sokolov and Klafter [17] compare their GDE to the GDE of Kenkre and Knox [28], which is formally identical to Eq. (38), and point out as striking difference between this equation and the equation proposed by Kenkre and Knox [28] the additional time derivative in the front of the integral. Actually, their equation is equivalent to the generalized master equation of Eq. (34) and the additional time derivative has a physical meaning that we plan to properly explain. For this purpose, let us write again Eq. (47) in the slightly different form Z t o 1 d 1 o2 pðx; tÞ ¼ ds pðx; sÞ ð52Þ 2l ot Cð2  lÞ dt 0 ox2 ðT þ t  sÞ with T denoting a time interval that we plan to send to zero at a given phase of this discussion. By differentiating with respect to time we write Eq. (52) in the following form: Z t o 1 o2 ð2  lÞ 1 o2 pðx; tÞ ¼ pðx; tÞ  ds pðx; sÞ: ð53Þ ot Cð2  lÞ ox2 Cð2  lÞ 0 ðT þ t  sÞ3l ox2 This means that we can recover this result from Eq. (34) and so from the form of Kenkre and Knox [28] by adopting for the memory kernel UðtÞ the following expression: DUðtÞ ¼

dðtÞ ð2  lÞ 1  : 2l Cð2  lÞ ðT þ tÞ3l Cð2  lÞT

ð54Þ

In conclusion, the presence of an additional derivative in the front of the integral does not rule out the possibility of establishing a connection with the proposal of Kenkre and Knox [28]. This agrees with the well known fact that the Generalized Master Equation (GME) stemming from the Liouville treatment is equivalent to the GME derived from the adoption of the CTRW [11]. Let us compare the memory kernel of Eq. (54) to the correlation function of Eq. (17). Using the time asymptotic limit of Eq. (22), we conclude that an exact agreement between the Liouville and the CTRW approach is obtained by setting g ¼ 3  l:

ð55Þ

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R. Cakir et al. / Chaos, Solitons and Fractals 34 (2007) 19–32

In conclusion, in the case 1 < g < 2, the generalized diffusion equation produced by the Liouville-like formalism of Section 2, Eq. (15), is compatible with the CTRW perspective, even if, as we shall see in Section 6, in spite of the mathematical equivalence, the CTRW and the Liouville perspective reflect totally different physical processes. In the CTRW case, furthermore, the density memory does not reflect the trajectory memory, which does not exists, while in the Liouville case the density memory is a consequence of the trajectory memory.

5. Auxiliary fluctuation In this section we plan to illustrate the memoryless trajectories behind Eq. (2), namely, the time convoluted diffusion equation generated by the CTRW perspective. The hypothesis of instantaneous collision made in Section 3 is not essential to generate the sub-diffusional process there described. To double check the theoretical arguments of this paper by means of numerical simulation, it is convenient to adopt a fluctuation with the time interval between two consecutive renewal events filled with auxiliary events. We call this kind of fluctuation nA ðtÞ. This section is devoted to illustrating a dynamic model producing this kind of fluctuation. Let us imagine a particle with coordinate y, moving within the interval I  ½0; 1, with the following equation of motion: d y ¼ ay z dt

ð56Þ

with z P 1, and 0 < a  1. When the particle reaches the border, y = 1, is injected back to a new initial condition y0, with uniform probability, it is easy to prove [24,32] that the waiting time distribution is given by (41) with z : ð57Þ l¼ ðz  1Þ Let us now define a fluctuation nA ðtÞ as follows. From time t = 0 to time s1, corresponding to the particle reaching the border, the fluctuation nA ðtÞ is identical to the velocity dy/dt of the particle, with a sign ± depending on the coin tossing prescription, namely, l=ðl1Þ

ay 0 il ; nA ðtÞ ¼  h 1=ðl1Þ a 1  y0 t ðl1Þ

ð58Þ

where y0 is a random number of the interval I. This means that we prepare the system in such a way that at time t = 0 all the systems of the ensemble are at the beginning of the deterministic dynamic process, at the end of which a new back injection will occur. Of course y0 is selected randomly, with uniform distribution within the interval I. Thus, the time of the second back injection changes from system to system, and the probability of the first sojourn time, s1, is given by Eq. (41). From time t ¼ s1 , corresponding to the first jump after preparation to time t ¼ s1 þ s2 , at which the second jump occurs, the fluctuation nðtÞ goes as follows: l=ðl1Þ

ay 1 il ; nA ðtÞ ¼  h 1=ðl1Þ a 1  y1 ðt  s Þ 1 ðl1Þ

ð59Þ

where y1 denotes the initial condition after the first back injection, again selected randomly with uniform probability in the interval I, and so on. Let us focus now our attention on the diffusion process described by d xðtÞ ¼ nA ðtÞ: dt

ð60Þ

In this case, we expect that the fluctuation nA ðtÞ will produce a subdiffusion process with the anomalous scaling of Eq. (46) with l meeting the condition of (42). Let us explain the intuitive reasons that lead us to make the prediction of Eq. (46). As explained by the arguments of Section 3, the generalized diffusion equation of Eq. (2), is derived from ordinary diffusion, by assuming that the time distance between two consecutive jumps of the random walker is given by the waiting time distribution wðsÞ. When we make the mean sojourn time in one site infinite, the process of ordinary diffusion is turned into a process of sub-diffusion with the anomalous scaling d of Eq. (46). For the attainment of this result it does not matter whether the walker position remains rigorously unchanged in the time interval between two abrupt jumps, or not. The adoption of the fluctuation nA ðtÞ rather than of the idealized picture to which Eq. (2) refers, is useful for the practical purpose of numerical simulation. In fact, it assigns to the time size of the critical events a finite rather than a

R. Cakir et al. / Chaos, Solitons and Fractals 34 (2007) 19–32

27

100 0.026 tμ-1 10

1

0.1 10

100

t

1000

10000

100000

Fig. 1. The time evolution of the second moment of x(t) generated by the diffusion process of Eq. (39). The dashed curve is the numerical result and the solid curve represents the theoretical prediction of Eq. (46). We have adopted the value l ¼ 1:5.

vanishing measure. The idealized condition would be realized by decreasing the time size of the critical events, but increasing at the same time the computational time, which would become infinite to reproduce the idealized condition. We are now in a position to check numerically our theoretical prediction. The result of this numerical simulation is illustrated in Fig. 1. We evaluate the anomalous scaling of the diffusion process generated by the fluctuation nA ðtÞ through the second moment of the probability distribution density, and we see that in the asymptotic limit the numerical scaling becomes identical to the theoretical scaling of Eq. (46). We think that the results of this section should convince the reader that Eq. (2), which is an evident expression of density memory emerges from the time evolution of a bunch of trajectories, with no memory. We have seen, in fact, that after a jump, the quantity nA ðtÞ undergoes a time evolution determined by a random choice of initial condition, with no memory whatsoever of the time evolution of nA ðtÞ prior to the last jump.

6. On the dynamical nature of the memory kernel of the CTRW generalized diffusion equation In this section we plan to show that the memory kernel UðtÞ of the generalized diffusion equation of Eq. (2) is not an equilibrium correlation function. From Eq. (60) we derive Z t nA ðt0 Þdt0 : ð61Þ xðtÞ ¼ 0

This means that, as in the earlier section, all the walkers move from x = 0 at t = 0. The second moment of this diffusion process reads Z t Z t dt0 dt00 hnA ðt0 ÞnA ðt00 Þi: ð62Þ hxðtÞ2 i ¼ 0

0

By differentiating Eq. (62) twice with respect to time, we obtain Z t d2 2 d 2 hx ðtÞi ¼ 2hn ðtÞi þ 2 dt0 hnA ðtÞnA ðt0 Þi: A dt dt2 0

ð63Þ

On the other hand, using Eq. (38), we get d2 2 hx ðtÞi ¼ 2DUðtÞ: dt2 By comparing Eq. (63) to Eq. (64) we arrive at the important equality Z t d dt0 hnA ðtÞnA ðt0 Þi: DUðtÞ ¼ hn2A ðtÞi þ dt 0

ð64Þ

ð65Þ

We want to point out that this relation shows that UðtÞ is not a correlation function. We see, furthermore, that UðtÞ is related to a true correlation function, hnA ðtÞnA ðt0 Þi, by a functional relation, and that the true correlation function is not stationary. As shown in Section 5, we prepare all the systems of the Gibbs ensemble in the condition where

28

R. Cakir et al. / Chaos, Solitons and Fractals 34 (2007) 19–32 l=ðl1Þ

nð0Þ ¼ y 0 . After this initial value, which is equivalent to the condition created by a jump occurring at t = 0, we have to wait a long time to meet again another jump. Let us discuss here the ideal condition where the time interval between two significant fluctuations is really empty. To fill it, we must have recourse to infinitely many other trajectories, which, by chance, will produce significant fluctuations in this empty time interval. Using the book of Feller [33] (see also Ref. [34]) we find that the density of events goes as t2l . Thus, we conclude that C ; t2l

lim hn2A ðtÞi ¼

t!1

ð66Þ

where C is a constant, whose explicit value is of no interest for this discussion. We make also the plausible conjecture that the non-stationary correlation function hnA ðtÞnA ðt0 Þi, with t0 < t, has the following expression lim hnA ðtÞnA ðt0 Þi ¼

t!1

C F ðt  t0 Þ; t2l

ð67Þ

where as we shall see, F(t) must be a function of time that decays quickly enough as to make its Laplace transform, Fb ðuÞ obey the condition 0 < Fb ðuÞ < 1. With easy calculations we prove that the fundamental relation of Eq. (65) yields lim UðtÞ ¼ 

t!1

ð2  lÞC F^ ð0Þ : Dt3l

ð68Þ

To make more convincing the equivalence with the fractional derivative arguments of Section 4, let us rewrite Eq. (66) and Eq. (67) as hn2A ðtÞi ¼

C

ð69Þ

ðt þ T Þ2l

and hnA ðtÞnA ðt0 Þi ¼

C ðt þ T Þ2l

F ðt  t0 Þ;

ð70Þ

respectively. With easy algebra, we find DUðtÞ ¼ 

ð2  lÞC F^ ð0Þ ðT þ tÞ

3l

þ

C ðT þ tÞ2l

F ðtÞ:

ð71Þ

By comparison with Eq. (54) we find that a complete equivalence with the fractional derivative perspective is established by assuming C ðT þ tÞ

2l

F ðtÞ ¼

dðtÞ : Cð2  lÞT 2l

ð72Þ

This is equivalent to assuming that the decay of F ðtÞ=ðT þ tÞ2l is significantly faster that 1=t3l . We double check these theoretical remarks by means of the numerical calculation of the non-stationary correlation function hna ðtÞnA ðt0 Þi. In Fig. 2, we plot F ðtÞ and we prove that it is in fact much faster than 1=t3l . 1 F(t)/t2-μ 0.1

1/t3-μ

0.01

0.001

1e-04 10

Fig. 2. F ðtÞ=t

t 2l

100

(dashed) and 1=t

1000 3l

(solid).

R. Cakir et al. / Chaos, Solitons and Fractals 34 (2007) 19–32

29

7. Dynamical origin of the time convoluted diffusion equation In this section we illustrate the form of fluctuation yielding equation (15). To establish this form we have to meet the constraint that the correlation function UK ðtÞ has the time asymptotic form as that discussed in Section 2, and the constraint that the higher-order correlation functions obey the DF property as well. To generate the stochastic process nK ðtÞ we create first the Gaussian fluctuation responsible for the dynamic derivation of FBM. To do that, we adopt the standard procedure illustrated in Ref. [27] X

 ci xi ð0Þ cos xi t þ vi ð0Þx1 ð73Þ nG ðtÞ ¼ i sin xi t : i

This means that the fluctuation nG ðtÞ is derived from the sum of infinitely many oscillator coordinates. Initial positions and velocities of the oscillators are randomly selected from a canonical distribution and c2i / xg1 , thereby making i Gaussian the random variable nðtÞ. It is straightforward to show that the correlation function Un ðtÞ has the following asymptotics [26]: Un ðt ! 1Þ / signð1  gÞ=tg :

ð74Þ

Actually, for computational reasons rather than using Eq. (73) we generate a fluctuation nG ðtÞ with the correlation function of Eq. (17) by means of the algorithm of Ref. [35]. In the numerical simulations we use a discrete time series, tn ¼ n ¼ 1; 2; 3; . . . To generate the random variable neq ðnÞ with correlation function given by Eq. (74) with either positive or negative tail we use the algorithm [35] Z p=2 pffiffiffiffiffiffiffiffiffiffi 1 2 X neq ðnÞ ¼ Z mþn /ðyÞ cosð2myÞ dy: ð75Þ p m¼1 0 2 2 Here Zn is a Gaussian ensemble of random P1 numbers with hZ n i ¼ 0 and hZ n i ¼ n0 , and the function /ðyÞ is determined through its Fourier series, /ðyÞ ¼ 1 þ 2 k¼1 Un ðkÞ cosð2kyÞ. Using algorithm (75), which serves the purpose of generating a fluctuation equivalent to that Eq. (73), it is possible for us to find the fluctuation nK ðtÞ, responsible for the diffusion process of Eq. (39), nK ðtÞ, and thus for the diffusion process described by Eq. (15). Let us see how to derive nK ðtÞ from nG ðtÞ, namely from neq ðtÞ. We study the time distance between two consecutive recrossings of neq ðtÞ ¼ 0. In Fig. 3, we show that the corresponding waiting function is exponential. In the renewal case, the exponential waiting time would generate a master equation with no memory. To generate the master equation of (15), we need trajectory memory. To prove the existence of trajectory memory in the recrossing process, let us study the waiting time correlation function. Fig. 4 shows that these times are correlated. We state that the fluctuation nK ðtÞ is generated by the prescription

nK ðtÞ  signðnG ðtÞÞ:

ð76Þ

We prove first that the correlation function of nK ðtÞ is the same as that of nG ðtÞ. In Fig. 5, we prove that the time asymptotic property of the correlation function of nK ðtÞ are the same as those of the original fluctuation nG ðtÞ. Thus, the

1 0.438506 e

(-0.340206 t)

0.1 Ψ(t) 0.01

0.001

1e-04

1e-05 0

5

10

t

15

20

25

30

Fig. 3. The distribution of time distances between two re-crossings of neq ¼ 0 (dashed line) and the corresponding fitting function (solid line).

30

R. Cakir et al. / Chaos, Solitons and Fractals 34 (2007) 19–32 1

0.8

Φτ(t)

0.6

0.4

0.2

0 0

5

10

15 t 20

25

30

35

40

Fig. 4. Correlation function of s  hsi, the time distances between two re-crossings.

-ΦK(t) -ΦG(t)

0.01

10

t

100

Fig. 5. Time asymptotic properties of the correlation function of dichotomous fluctuation (dashed) and original fluctuation (solid).

important constraint of yielding the same correlation function as that playing the role of the memory kernel of Eq. (15) is fulfilled. Finally, with Fig. 6 we prove that the DF property is fulfilled. 10 <ξk(t1=0) ξk(t2)> <ξk(t3=0) ξk(t4=0)> <ξk(t1=0) ξk(t2) ξk(t3=100) ξk(t4=100)> <ξk(t1=0) ξk(t2) ξk(t3=200) ξk(t4=200)>

1

<ξk(t1=0) ξk(t2)> <ξk(t3=0) ξk(t4=2)> <ξk(t1=0) ξk(t2) ξk(t3=100) ξk(t4=102)>

0.1

<ξk(t1=0) ξk(t2) ξk(t3=200) ξk(t4=202)>

0.01

0.001

10

t2

100

Fig. 6. The DF property in Eq. (19) is shown. In the upper part, when t3 ¼ t4 ¼ 100 (empty square) and t3 ¼ t4 ¼ 200 (full square) the plots match to UðtÞ (solid). In the lower part, when t3 ¼ 100; t4 ¼ 102 (empty circle) and t3 ¼ 200; t4 ¼ 202 (full circle) the plots match to UðtÞUð2Þ (dashed).

R. Cakir et al. / Chaos, Solitons and Fractals 34 (2007) 19–32

31

In conclusion, to create the dichotomous fluctuation responsible for the diffusion process described by Eq. (15) we have used the Gauss fluctuation, generated by a non-Ohmic bath, and used in Ref. [36] for the dynamic derivation of FBM. We have proved numerically that the dichotomous signal, obtained by considering only the sign of this Gaussian fluctuation, produces a correlation function with the same fat tail as the memory kernel of Eq. (15). We have also proved that this dichotomous signal yields the DF property that, according to Ref. [23], the fluctuation nK ðtÞ must have to be a proper generator of Eq. (15). It is interesting to notice that if we consider only the sign of the diffusion variable x(t) created by the Gauss fluctuation nG ðtÞ, we generate a renewal fluctuation that is the generator of Eq. (2) in the subdiffusion case. Thus, the trajectory memory can be converted into density memory.

8. Final remarks The time convoluted equation of Eq. (2) is formally equivalent to Eq. (15), and as a consequence, we would be tempted to conclude that these two equations are also physically equivalent. A recent example of this conviction is given by the interesting work of Budini [37]. Budini adopts a quantum mechanical picture that is similar to that illustrated in the classical case of Section 2, insofar as rather than expressing the memory kernel as the sum of many coherent components, he does express it as the sum of infinitely many exponential processes. As a result of this procedure he gets a time convoluted master equation equivalent to the adoption of the CTRW procedure, in the same way as Eq. (2) is formally equivalent to Eq. (15). The condition of sub-diffusion makes it legitimate to claim for this equivalence, with the warning though of considering the results of Section 6 proving that within the Liouville perspective the memory kernel of the time convoluted equation is an equilibrium correlation function, while in the CTRW perspective is not. In the case of super-diffusion, it is impossible to derive the time convoluted diffusion equation of Eq. (15) from within the CTRW perspective. This fits the conclusions of Balescu [18,19]. It is interesting to notice that in this case the memory density reflects correctly the trajectory density. In fact, albeit with help of numerical rather than only by means of theoretical arguments, in Section 7 we have proved that the dichotomous fluctuation nK, with trajectory memory, generates the diffusion process driven by Eq. (15). The search for the dichotomous fluctuation responsible for this kind of diffusion traces back to the work of Ref. [23] and to the earlier work of Ref. [38]. The failure of these earlier attempts is due to the fact that renewal fluctuations, namely memoryless trajectories, have been searched as generators of this diffusion process. To solve this long-standing problem, we had to shed light first of all on the confusion, existing in literature, between density and trajectory memory. The diffusion process described by Eq. (15) turned out to be characterized by both trajectory and density memory. We think that the results of this paper, supplemented by those of Ref. [36], might help us to settle some puzzling problems. As an example of interesting problem to solve, with the help of the results of this paper, we have in mind the statistical analysis of DNA sequences and literary texts [39–42]. The authors of Refs. [39,40] advocate the use of a perspective that, according to the terminology adopted in this paper, is based on the trajectory memory perspective. The authors of Refs. [41,42] for DNA sequences and literary texts, respectively, propose a theoretical perspective with memoryless trajectories. We hope that the results of the present paper might turn out to be useful to find a connection between these two apparently conflicting perspectives.

Acknowledgements We acknowledge Welch for financial support through Grant # B-1577 and ARO for financial support through Grant # W911NF-05-1-0205.

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