Effects of the tempered aging and its Fokker-Planck equation

Effects of the tempered aging and its Fokker-Planck equation

Weihua Deng    Wanli Wang    Xinchun Tian    Yujiang Wu School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, P.R. China
Abstract

In the renewal processes, if the waiting time probability density function is a tempered power-law distribution, then the process displays a transition dynamics; and the transition time depends on the parameter of the exponential cutoff. In this paper, we discuss the aging effects of the renewal process with the tempered power-law waiting time distribution. By using the aging renewal theory, the -th moment of the number of renewal events in the interval is obtained for both the weakly and strongly aged systems; and the corresponding surviving probabilities are also investigated. We then further analyze the tempered aging continuous time random walk and its Einstein relation, and the mean square displacement is attained. Moreover, the tempered aging diffusion equation is derived.

pacs:
05.40.-a, 05.10.Gg, 02.50.-r, 87.10.Rt
preprint: APS/123-QED

I Introduction

In 1975, Scher and Motroll Scher:1 used the continuous time random walk (CTRW) to study non-Gaussian anomalous diffusion. Nowadays, the CTRW model becomes popular in describing anomalous diffusion and a lot of chemical, physical, and biological processes Naftaly:1 ; Barkai:2 ; Barkai:6 , such as, the transport of electric charge in a complex system, diffusion in a low dimensional chaotic system, and the anomalous diffusion when cooling the mental solid and the twinkling of single quantum dot. In 1996, Monthusyx and Bouchaud introduce a CTRW framework for describing the aging phenomena in glasses Monthusyx:1 . This generalized CTRW is called aging continuous time random walk (ACTRW) in Barkai:1 . The complex dynamical systems displaying aging behaviour are quite extensive, including the fluorescence of single nanocrystals Brokmann:1 , aging effect in a single-particle trajectory averages Schulz:2 .

Research on statistics, based on power-law distributions with a heavy tail, yields many of significant results. Often the power-law distribution doesn’t extend indefinitely, due to the finite life span of particles, the boundedness of physical space. For this reason, in 1994, Mantegna and Stanley omit the large steps to study the truncated Lévy flights Mantegna:1 ; Negrete:1 . While the tempered power-law distribution Rosinski:1 uses a different approach, exponentially tempering the probability of large jumps. Exponential tempering offers technical advantages since the tempered process is still an infinitely divisible Lévy process which makes it convenient to identify the governing equation and compute the transition densities at any scale Baeumera:1 . By tempering, the distribution changes from heavy tail to semi-heavy, and the existence of conventional moments is ensured, which is useful in some practical applications. Recently tempered power-law distributions Meerschaert:1 have been observed for many geophysical processes at various scales Baeumera:1 ; Negrete:1 ; Sokolov:1 ; Meerschaert:2 ; Allegrini:1 , including interplanetary solar-wind velocity and magnetic field fluctuations measured in the alluvial aquifers Bruno:1 .

In this paper, we discuss the aging effects of the renewal processes with exponentially tempered power-law waiting time probability density function (PDF)

(1)

where , is the one side Lévy distribution Feller:1 ; levy:1 , and is generally a small parameter. The semi-heavy tails and scale-free waiting time properties of play a particularly prominent role in diffusion phenomena.


Figure 1: Random variables (r.v.) generated by Eq. (1) with . And the parameter is chosen, respectively, as (a) , (b) , (c) , (d) and , , , 100.

From Eq. (1), it can be noted that if , , while if , . For the random variables generated by Eq. (1), Fig. 1 shows that the maximum and range of fluctuations vary dramatically with the change of . The introduced tempering forces the renewal process to converge from non-Gaussian to Gaussian. But the convergence is very slow, requiring a long time to find the trend. So, with the time passed by, both the non-Gaussian and Gaussian processes can be described.

This paper is organized as follows. In Sec. II, we use the aging renewal theory to obtain the -th moment of the number of renewals within the time interval of the renewal process starting from time zero. The survival probability is, respectively, discussed in weakly and strongly aging system. We then turn to discuss the tempered ACTRW in Sec. III, and the mean square displacement is obtained for the cases and . The numerical simulations confirm the analytical expressions of the mean square displacement. And the propagator function is also numerically obtained. In Sec. IV, we discuss the Einstein relation of the tempered ACTRW. The diffusion equation for the tempered ACTRW is derived in Sec. V, describing the time evolution of the PDF of the position. Finally, we conclude the paper with some remarks.

Ii Tempered aging renewal theory

First, we briefly outline the main ingredients in the CTRW and ACTRW. The standard CTRW assumes that the jumping transitions begin at time , and observation of dynamics starts at . The ACTRW modifies the statistic of time interval for first jump, namely, the waiting time PDF to the first jump is . It describes a CTRW process having the aging time interval , while corresponding to the initial observation time . Aging means that the number of renewals in the time interval depends on the aging time , even when the former is long. Thus generally ACTRW and CTRW exhibit different behaviors.

More concretely, ACTRW describes the following process: a walker is trapped on the origin for time , then jumps to ; the walker is further trapped on for time , and then jumps to a new position; this process is then renewed. Thus, ACTRW process is characterized by a set of waiting times and displacements . Here is the PDF of the first waiting time . In ACTRW process, the random walk starts from the time , therefore may depend on the aging time of the process . The waiting times with are independent and identically distributed (i.i.d.) with a common probability density . And the jump lengths are i.i.d. random variables, described by the probability density .

When , we have , which is just the well known Montroll-Weiss nonequilibrium process. In order to investigate ACTRW, we should first discuss the aging renewal process. In what follows, we suppose that is the probability of the renewal process , where and denotes the number of renewals by time , i.e., is the number of renewals in time interval for a precess starts at the time . Our main work is to discuss the properties of the renewal process Luck:1 ; Schulz:1 in the time interval .

According to the renewal theory developed by Gordèche and Luck Luck:1 ,

(2)

where is the double Laplace transform of the PDF of the first waiting time , and is the Laplace transform of . This paper focuses on taking as Eq. (1), and its Laplace transform () has the asymptotic form

(3)

and if , .

The double Laplace transform of the PDF of reads Barkai:1 , , ,

(4)

For the particular case , from (3), ; and from Eq. (4) we can get , for this case is independent of . Since plays a key role in our discussion, we now derive the analytical formulation of Klafter:1 ,

(5)

where is an incomplete Gamma function. Using the Laplace transform of incomplete Gamma function Oberhettinger:1 , we have,

(6)

where . From the second line of Eq. (5), if , i.e., . Eq. (5) can be given by

(7)

If , i.e., , then there exists . Eq. (5) can be further simplified as

(8)

under the further assumption , i.e., , there exists

(9)

being the same as the one given in Klafter:1 ; Barkai:1 for the power law waiting time, i.e., . When is sufficiently large, Eq. (7) plays a dominant role.


Figure 2: Probability of particles making jumps during the time interval for . The parameters of Eq. (1) are taken as , , and ; and the symboled lines are obtained by averaging trajectories with different . The solid lines from down to up corresponding to the increased are the theoretical results of Eq. (6). When is large, the probability reaches quickly.

Figure 3: Probability of particles making jumps during the time interval . The parameters are taken as , , and ; and the symboled lines are obtained by averaging trajectories with different . The solid lines from down to up corresponding to the increased are the theoretical results of Eq. (6). When is sufficiently small, the distribution is almost the same as pure power law for short times.

In the following, we analyze the asymptotic form of with . From Eq. (2) and Eq. (3), there exists

(10)

For Eq. (10), taking the limit , we have ; then . Using the relation leads to , i.e., all the particles move at time , no aging phenomenon.

Consider the survival probability krusemann:3 , which gives the probability of making no jumps during the interval up to ,

(11)

It is instructive to consider two different limits. If , i.e., , there exists

(12)

For , i.e., , then . Performing double inverse Laplace transform on the above equation results in

(13)

For , Eq. (13) can be simplified as

(14)

It can be noted that is larger than in the parenthesis of Eq. (14), since . For , from Eq. (13) and Eq. (69), we obtain

(15)

being confirmed by FIG. 4, i.e., the lines tend to be close for big .


Figure 4: Time evolution of with different . The parameters are taken as , , and . The lines are obtained by averaging trajectories. The (red) dashed line, , is the fitting result for small and big , which agrees with Eq. (15).

And for , we have , i.e., for small , , the waiting time is generally long; in a small observation time , we cannot find movement of the particles. Eq. (11) can also be rewritten as

(16)

For the case , i.e., , Eq. (16) yields

(17)

under the further assumption , we have

(18)

where is defined in Eq. (68), i.e., when ,

(19)

being the same as the result given in Luck:1 for the pure power law case ().


Figure 5: Time evolution of with different . The parameters are taken as , , and . The lines are obtained by averaging trajectories. The dashed line with arrow is for the indicator of slope , confirming Eq. (19).

When the aging time is sufficiently long relative to the observation time and is small, the probability of making no jumps during the time interval approaches to one, i.e., the system is completely trapped. On the contrary, if is short, while the observation time is long enough, then the particles are unacted on the aging time. So, at least one jump will be made, namely, the possibility of making no jumps is zero. Indeed, from Eq. (19), it can be easily obtained that when ; and can also be directly obtained from Eq. (11) under the assumption .

From Eq. (4), we can write the double Laplace transform of the PDF of as

(20)

Inserting Eq. (3) into the above equation yields

(21)

From now on, we start to calculate the -th moment of the aging renewal process note:1 , which reads

(22)

For the cases that and or , there exist

(23)

By double inverse Laplace transform we have

(24)

Taking in (22) leads to

(25)

which can be rewritten as

(26)

We will confirm that if both and are large scales, , which is an important result for normal diffusion. For small and , using the Taylor expansion and , from Eq. (26), we have

(27)

Performing double Laplace transform on the above equation yields

(28)

where (see Appendix C).

For the slightly aging system, , i.e., ; performing the double inverse Laplace transform on both sides of (26) yields

(29)

For the special case, , it can be noted that . For , using the asymptotic expansion of Mittag-Leffler function (66), from Eq. (29), we again obtain . In the long time scale, the process converges to the Gaussian process, and then the first moment of the number of renewal events grows linearly with the observation time . For , from Eq. (29) we have

(30)

It can be seen that when the first moment of is not relevant to the aging time . From Eq. (28) and Eq. (30), we can see that plays an important role in our discussion as expected.


Figure 6: Time evolution of the ensemble average of the renewal times with the waiting time PDF Eq. (1) for slight aging. The parameters are taken as , , and . The real lines are for the analytical result Eq. (29) and the other lines are obtained by averaging trajectories.

While for the strongly aging system, , i.e., ; there exists

(31)

which yields

(32)

Following the way used above, for , the term tends to (69). Then we have

(33)

For , there exists Luck:1 ; Klafter:1

(34)

The above results for the first moment of can be summarized as: 1. if or is greater than , then ; 2. for and (i.e. ), ; 3. for and , behaves as .


Figure 7: The relation between the and for according to the trajectories and the theory. The number of particles is , , , , and , , , and . The real lines are for the analytical result Eq. (32) and the other symbol lines are obtained by averaging trajectories.

Figure 8: Time evolution of the ensemble average of the renewal times for strong aging (). The parameters , , and for (a); for (b); for (c); for (d). The lines are obtained by averaging trajectories. It can be seen that grows linearly with time for fixed for all values of , which confirms the analytical result Eq. (32).

Iii Tempered ACTRW

iii.1 Mean squared displacement

After understanding the statistics of the number of renewals, we go further to discuss the tempered ACTRW with the waiting time distribution Eq. (1). The process of ACTRW has been described in Sec. II. This paper focuses on the symmetric random walk, i.e., the distribution of jump lengths ; and is finite. For such a random walk, we denote as the PDF of particles’ position in the decoupled tempered ACTRW with aging time . Then

(35)

where means the probability of jumping steps in the time interval , and the probability of jumping to the position after steps. In the Fourier-Laplace domain,

(36)

Inserting Eq. (4) into Eq. (36) leads to

(37)

By differentiating Eq. (37) two times with respect to and setting , we derive the second order moment of the random walks, i.e.,

(38)

For the mean square displacement, we present the results of the slightly aging and strongly aging system, i.e.,

(39)

Performing double inverse Laplace transform on yields

(40)

where .

From FIG. (10), we can see the large fluctuations of even if the number of trajectories is 10000. This is because that most of particles are trapped in the initial position for , which is consistent with Eq. (13). This is related to population splitting Cherstvya:1 .

It can be noted that when , the mean squared displacement has no aging effect; while (), it is deeply affected by the aging time . The surprising result is that , when the second order moment of the jump length is finite; the same things happen for the pure power-law waiting time distribution.


Figure 9: The relation between the and the observation time for getting from the trajectories of the particles (dashed line) and Eq. (40) (real line). The parameters , , and the number of trajectories is . With the increase of , the characteristic of changes from Power law to normal diffusion.

Figure 10: The relation between and the observation time for . It can be noted that increases linearly with for fixed and the fluctuations are large because of population splitting. The parameter , , for (a), for (b), for (c), for (d), and the number of the trajectories is .

Figure 11: The relation between and the observation time for various with .

iii.2 Propagator function

In this subsection, we discuss the propagator function of the tempered ACTRW. Omitting the motionless part of Eq. (37), taking as Gaussian, and performing inverse Fourier transform w.r.t. , there exists

(41)

with

For , Eq. (41) can be rewritten as

(42)

From FIG. (12), it can be noted that for small ( or ) the propagator functions display the characteristics of stable distribution; while for large the shows the classical normal behavior.

Figure 12: Propagator functions with , , and . The red lines with empty symbol are obtained by calculating Eq. (42), and the blue dash lines are got from the generated trajectories of the particles.

Figure 13: Strong aging case (contrary to FIG. (12)) with and . The other parameters are same as FIG. (12).

For , Eq. (37) yields,

(43)

Contrary to FIG. (12), FIG. (13) displays the behaviors of the stable distribution for all kinds of .

From the numerical results and the theory we can see that the ‘ stable distribution’ characteristics can be found for small . Both the distributions for and have the sharp peak and the tail of Eq. (42) decays slowly. While for large , the top of the distribution for Eq. (42) is smooth, being different from the case of small . Therefore, depending on the choice of , one can control the behaviors of the propagator.

Iv strong relation between the fluctuation and response

In this section, we discuss the aging from a new point of view. Based on the CTRW model, consider such a process: the particles begin to move at time and undergo unbiased diffusion in the time interval ; then an external field is switched on the system starting from . If the averaged response of the particles depends on , the process is said to exhibit aging. Generally speaking, giving some disturbance to a system, some characteristics (parameters of thermodynamics) of the system will change, being called response Bertin:1 . Under the small disturbance of external field, if the change of the parameter of thermodynamics is proportional to the force of external field, then it is called linear response. It seems important to use drift diffusion to consider aging. Using the method given in Barkai:7 ; Allegrini:3 ; Froemberg:1 ; Shemer:1 , we discuss the tempered aging Einstein relation.

Let us consider a simple example of random walk on a one-dimensional lattice; the length of the lattice is , and the particles can only move to its neighboring sites. Waiting times of different steps of the random walk are considered independent and have the same distribution . Jumps to the right (left) are performed with the probability (). The total time is ; is called aging interval with ; and is called response interval with . Let , where is the displacement performed in the aging time interval and is the displacement performed in the response time interval, and are the step lengths and are the number of events happened in the two time intervals, repectively.

We consider the correlation function which shows the impact between in the aging interval and in the response interval. And define a parameter to show the relation between fluctuation and response Barkai:7 ,

(44)

If , it shows that and are independent with each other. Using the relation and , then can be shown in another way,

(45)

We further introduce , the probability to occur events in the aging interval and events in the response interval. Following the result given in Luck:1 ,

(46)

where if the event inside the parenthesis occurs, and if not. Using double Laplace transform, if ,

(47)

and if ,

Summing , from to leads to

Using the Laplace transform of (3), we have, when ,

(48)

if , there exists

(49)

Following the results given by Luck:1 ,

there exists

(50)

And is the same as Eq. (25). For , there exists

(51)

From Eq. (51), we know that does depend on .

In the following, we further consider the Einstein relation Froemberg:1 ; Shemer:1 for the tempered aging process. Denoting as the first order moment of the displacement under the influence of a force , from Eq. (32) and , we get that for ,