Integro-differential diffusion equation for continuous time random walk

Integro-differential diffusion equation for continuous time random walk

arXiv:1007.2186v1 [cond-mat.stat-mech] 13 Jul 2010

Kwok Sau Fa1 and K. G. Wang2 1

Departamento de F´ısica, Universidade Estadual de Maring´ a, Av. Colombo 5790, 87020-900, Maring´a-PR, Brazil,∗ and 2

Department of Physics & Space Sciences,

Materials Science and Nanotechnology Institute, Florida Institute of Technology, Melbourne, FL 32901, USA

Abstract In this paper we present an integro-differential diffusion equation for continuous time random walk that is valid for a generic waiting time probability density function. Using this equation we also study diffusion behaviors for a couple of specific waiting time probability density functions such as exponential, and a combination of power law and generalized Mittag-Leffler function. We show that for the case of the exponential waiting time probability density function a normal diffusion is generated and the probability density function is Gaussian distribution. In the case of the combination of a power-law and generalized Mittag-Leffler waiting probability density function we obtain the subdiffusive behavior for all the time regions from small to large times, and probability density function is non-Gaussian distribution. PACS numbers: 02.50.-r, 05.10.Gg, 05.40.-a Keywords: Continuous time random walk; integro-differential diffusion equation; anomalous processes; generalized Mittag-Leffler function

∗

Electronic address: [email protected]

1

I.

INTRODUCTION

Diffusion is a ubiquitous phenomenon, and it is one of the fundamental mechanisms for transport of materials in physical, chemical and biological systems. The well-known example of a diffusion process is the Brownian motion. Diffusion processes are classified according to their mean-square displacements. The normal diffusion is that the meansquared displacement grows linearly with time, and in other situations the processes are said to exhibit anomalous diffusion. Nowadays, there are several approaches to describe anomalous diffusion processes, and they can be applied to many situations of natural systems [1–9]. One of the most interesting features incorporated into these approaches is the memory effect. In particular, the memory effect incorporated into the Langevin approach, referred to as generalized Langevin equation (GLE) [9], can be associated to the retardation of friction and fractal media [10, 11]. Moreover, according to the fluctuation-dissipation theorem [1], the internal friction is directly related to the correlation function of the random force. In many situations, a finite correlated noise is necessary for describing the real systems in equilibrium states. However, in order to describe anomalous diffusion, a nonlocal friction should be employed so that it satisfies the fluctuation-dissipation theorem. For instance, anomalous diffusion processes have been observed in a variety of systems such as bacterial cytoplasm motion [12], conformational fluctuations within a single protein molecule [13] and fluorescence intermittency in single enzymes [14]. These processes have been described by the GLE, and the memory effect has also been shown in the generalized Fokker-Planck equations (GFPE) [9]. The continuous time random walk (CTRW) of Montroll and Weiss [15] has been employed to describe anomalous diffusion [16–22]. The CTRW with a power-law waiting time probability density function (pdf) [23, 24] was also linked to the following fractional Fokker-Planck equation, ∂ρ(x, t) ∂2 1−α =0 Dt Kα 2 ρ(x, t) , ∂t ∂x where 1−α ρ(x, t) 0 Dt

1 ∂ = Γ(α) ∂t

Z

0

t

ρ(x, t1 ) dt1 (t − t1 )1−α

(1)

(2)

is the Riemann-Liouville fractional derivative and Γ(z) is the Gamma function. ρ(x, t)dx is 2

the probability for finding particle in position between x and x + dx at time t. The CTRW model may be described by a set of Langevin equations [4, 25, 26] or an appropriate generalized master equation [19, 27, 28]. The pdf ρ(x, t) obeys the following equation in Fourier-Laplace space [4]:

ρ(k, s) =

1 − g(s) ρ0 (k) , s 1 − ψ(k, s)

(3)

where ρ0 (k) is the Fourier transform of the initial condition ρ0 (x), ψ(x, t) is the jump pdf and g(t) is the waiting time pdf defined by Z ∞ g(t) = ψ(x, t)dx .

(4)

−∞

Moreover, from the jump pdf we also have the jump length pdf defined by Z ∞ φ(x) = ψ(x, t)dt .

(5)

0

The continuous time random walk model can be simplified through the decoupled jump pdf ψ(k, s) = φ(k)g(s). Furthermore, jump length pdf can be approximated as following [4]: φ(k) ∼ 1 − Dk 2 + O(k 4 ) .

(6)

Different types of the CTRW models are specified through specifying the waiting time pdf. The CTRW model is also connected to a class of Fokker-Planck equations [4, 29]. Furthermore, in the CTRW model, solutions for ρ(x, t) under the condition of long-tailed waiting time pdf can be found in Refs. [30–32]. Substituting Eq.(6) into Eq.(3), we have ρ(k, s) =

1 − g(s) ρ0 (k) . s 1 − (1 − Dk 2 ) g(s)

(7)

When Cakir et al. [33] studied trajectory and density memory, the Eq.(43) in their paper is same as our Eq.(7) with consideration of ρ0 (k) = 1. In their study, the initial condition is that when t = 0, particle is at origin, and therefore ρ0 (k) = 1. However, Eq. (7) can be applied to cases without requirement of ρ0 (k) = 1 that appeared in Cakir et al.’ paper [33]. In particular, the CTRW model can be classified by the characteristic waiting time T and the jump length variance Σ2 defined by T =

Z

∞

tg(t)dt , 0

3

(8)

and 2

Σ =

Z

∞

x2 φ(x)dx .

(9)

−∞

For finite T and Σ2 , the long-time limit corresponds to the Brownian motion [4]. Although Eq. (7) is valid for a finite jump length variance, anomalous diffusion can be produced by Eq.(7) with appropriate choices of waiting time pdf. The aim of this work is: (1) from Eq.(7) to investigate general behavior of particle’s diffusion without any specified waiting time pdf; (2) study the effects of different waiting time pdf’s on the behavior of particle’s diffusion in the framework of CTRW. This paper is organized as follows. In Section 2 we study CTRW under a general waiting time pdf and two specific waiting time pdf’s. In Section 3, we show the exact solutions for pdf ρ(x, t). Finally, conclusions are presented in Section 4.

II.

CONTINUOUS TIME RANDOM WALK AND INTEGRO-DIFFERENTIAL

EQUATION FOR GENERAL WAITING TIME PROBABILITY DENSITY FUNCTION

Eq.(7) was already employed to study diffusion behavior when the form of waiting time pdf g(t) was first specified such as exponential function or a power-law function in the longtime limit [4]. However, for a generic form of g(t) Eq.(7) is not convenient to be used to study diffusion behavior due to the difficulty in the inverse transformations of Fourier and Laplace in Eq.(7). In this paper, we will derive an integro-differential equation for CTRW, which is valid for general waiting time probability density function. The details of derivation can be found in Appendix A. This integro-differential equation for CTRW is written by Z t Z t ∂ρ(x, t) ∂ 2 ρ(x, t1 ) ∂ ∂ρ(x, t1 ) g (t − t1 ) − dt1 = D dt1 . (10) g (t − t1 ) ∂t ∂t1 ∂t 0 ∂x2 0 For the case of g(t) = tα−1 /Γ(α), it is noted that the Caputo fractional derivative appears on the left side of Eq. (10) and the Riemann-Liouville fractional derivative appears on the right side of Eq. (10). The Caputo fractional derivative requires the integrability of the derivative and contains the initial value of the function. Therefore the Caputo fractional derivative is more restrictive than the Riemann-Liouville fractional derivative is. It should be pointed out that the use of these operators may lead to different behaviors including 4

unphysical behavior in different systems [34, 35]. Eq. (10) is derived from a well-defined physical process, and no ambiguities and unphysical behaviors are expected. The left side of Eq. (10) shows the variation of ρ(x, t) with respect to time, which does not depend only on the ordinary derivative operator but also the difference between ordinary and non-local integral operators. Our Eq. (10) is different from the fractional Fokker-Planck Eq. (1) in the structure because the fractional Fokker-Planck Eq. (1) does not have the second term of the left side of Eq. (10). Substituting g(t) = tα−1 /Γ(α) into Eq. (10) cannot directly obtain Eq. (1). In order to exclude the second term of the left side of Eq. (10), the asymptotic behavior for g(t) must be carefully treated. We note that

d hx2 i = dt

Z

∞

x2 −∞

∂ρ(x, t) dx . ∂t

(11)

Substitute Eq. (10) into (11) yields Z t Z ∞ Z t d hx2 i ∂ ∂ 2 ρ(x, t) d hx2 i g (t − t1 ) = dt1 + D x2 dxdt1 . g (t − t1 ) dt dt1 ∂t 0 ∂x2 −∞ 0

(12)

After integrating the second term of the right side of Eq.(12) by parts twice, we have Z t Z ∞ Z t d hx2 i ∂ d hx2 i g (t − t1 ) = dt1 + 2D ρ(x, t)dxdt1 , (13) g (t − t1 ) dt dt1 ∂t 0 −∞ 0 where we also consider that limx→±∞ ρ(x, t) and it decreases faster than 1/x. The normalR∞ ization of the pdf ρ(x, t) requires that −∞ ρ(x, t)dx = 1. Finally, Eq.(13) can be rewritten

as

d hx2 i = dt

Z

0

t

∂ d hx2 i dt1 + 2D g (t − t1 ) dt1 ∂t

Employing the Laplace transform in Eq. (14) we obtain

Z

t

0

2 hx2 i0 2Dg(s) x L= + s s [1 − g(s)]

g (t − t1 ) dt1 .

(14)

(15)

in Laplace space. We have shown that from Eq. (10) even without the previous knowledge of the waiting time pdf, the second moment of displacement i.e. the diffusion behavior can also be obtained in Eq. (15). This is one of advantages of Eq.(10). Therefore, Eq.(15) is very general and valid for any waiting time pdf. In fact, different diffusion behaviors can be found when substituting different waiting time pdf’s into Eq.(15). Considering that the waiting time pdf must be positive and normalized, 5

there exist only a few simple functions that can be used as waiting time pdf’s. Now we study diffusion behaviors in two specific cases of waiting time pdf’s. First case: g1 (t) = λe−λt . In this case the characteristic waiting time is finite. The Laplace transform of g1 (t) is given by λ . λ+s

g1 (s) =

(16)

Substituting Eq. (16) into Eq. (15) we obtain

2 2 x = x 0 + 2Dλt .

(17)

Eq.(17) shows the normal diffusion behavior. It has also shown that the generalized diffusion equation (10) can reduce to describe the normal diffusion process when the waiting time pdf is a simple exponential function. Second case: g2 (t) = λα tα−1 Eα,α (−λα tα ). Eµ,ν (y) is the generalized Mittag-Leffler function defined by [36, 37] Eµ,ν (y) =

∞ X n=0

yn , µ > 0, ν > 0 . Γ (ν + µn)

(18)

We restrict the parameter α to 0 < α ≤ 1 because Eα,α (−λα tα ), in this interval, is completely monotone for t > 0 [36]. We note that g2 (t) can also be written as g2 (t) = −

d Eα,1 (−λα tα ). dt

This waiting time pdf has been used to link the CTRW model and fractional master equation [38]. It has also been adopted as subordination function [39, 40]. Fulger et al. [41] incorporated this waiting time pdf into Monte-carlo simulation of in uncoupled continuous-time random walks with a L´evy α-stable distribution of jumps. The L´evy α-stable distribution of jumps used in [41] has a diverging jump length variance in contrast to the finite jump length variance given by Eq. (6) in this paper. The waiting time pdf g2 (t) has the following asymptotic behavior g2 (t) ∼ −

1 λα

t1+α Γ (−α)

,

(19)

which has a power-law behavior. The characteristic waiting time is divergent. The Laplace transform of g2 (t) is given by g2 (s) = 6

λα . λα + sα

(20)

Substituting Eq. (20) into Eq. (15) we obtain 2Dλα α t . Γ (1 + α)

2 2 x = x 0+

(21)

This result is the same as that one obtained in [4] for long-time limit using approximation of a long-tailed waiting time pdf. However, it should be noted that Eq. (21) describes the subdiffusive processes for the all time regions rather than only for long-time or short time region. This point is different from the case of long-tailed waiting time pdf in Ref. [4].

III.

EXACT SOLUTION FOR PROBABILITY DENSITY

Generally speaking, using Eq.(3) we can calculate the pdf in the framework of CTRW model. However, it is hard to obtain the exact pdf from Eq.(3). Now, we can obtain the exact solutions for ρ(x, t) for the two waiting time pdf’s in Section 2 from Eq.(7). In the following, we consider the initial condition as ρ0 (k) = 1. After we do Fourier inverse for ρ(k, s) in Eq.(7), we have 1 − g(s) ρ(x, s) = 2πDsg(s)

Z

∞

−∞

eikx k2 +

1−g(s) Dg(s)

dk .

(22)

We note that the denominator can have three different poles: 1 − g(s) = 0 gives trivial result, (1 − g(s))/g(s) < 0 gives the poles on the real axis and (1 − g(s))/g(s) > 0 gives the poles on the imaginary axis. For the waiting time pdf’s given in the previous section we can restrict to the case of the poles on the imaginary axis. Thus, the solution for ρ(x, s) is given by ρ(x, s) =

1 √

2π Ds

s

1 − g(s) − √|x| e D g(s)

q

1−g(s) g(s)

.

(23)

Before obtaining the exact solutions to ρ(x, t) we can calculate the second moment in Laplace space which yields

2 x L=

2Dg(s) . s [1 − g(s)]

(24)

This result is similar to (15), except the term of initial value hx2 i0 . This indirectly shows that Eq.(23) is correct. For the case of g1 (t) = λe−λt the solution for ρ(x, t) is given by ρ1 (x, t) = √

1 4πDλt

7

x2

e− 4λDt .

(25)

The pdf shows the Gaussian shape and it is exactly the solution of the normal diffusion equation [2, 4]. For the case of g2 (t) = λα tα−1 Eα,α (−λα tα ) the solution for ρ(x, t) is given by n

 2 ∞ X x2 1 (−1)n
ρ2 (x, t) = √ . Dλα tα 4Dλα tα n=0 n!Γ 1 − α 1+n 2

(26)

This result is exactly the solution of the fractional diffusion equation (1). The pdf (26) presents pronounced cusps in x-coordinate for different times. This form is different from the Gaussian shape (25) which presents smooth shape in the same coordinate [4]. Eq. (26) can also be expressed in terms of the Wright function Φη,δ (y) [42] in the following form: 1 ρ2 (x, t) = √ Φ− α2 ,1− α2 4Dλα tα where Φβ,δ (y) =

∞ X n=0

IV.

yn , n!Γ (βn + δ)

  |x| −√ , α Dλα t 2

(27)

β > −1, δ ∈ C .

(28)

CONCLUSION

In this work we have investigated the CTRW model with the decoupled jump pdf. We have derived the integro-differential equation (10). From Eq.(10) we have derived diffusion behavior that is explicitly related to the waiting time pdf. Eq.(10) is more flexible to be used than Eq.(7) because using Eq.(10) we can avoid Laplace and Fourier transformations and their inverse operations, for instance, Eq. (14) is ready to be used for calculating the second moment numerically. Both Eqs.(10) and (15) are very general and valid for any waiting time pdf. To our knowledge, Eqs.(10) and (15) have not been published elsewhere. In fact, Eq. (7) or Eq. (10) describes random walk model without external force. However, interesting physical problems involve external force. Generalization of Eq. (10) to include the effect of external force is a natural step. In addition, possible generalization of Eq. (10), e.g., using the master equation approach [4], to include the effect of external force would be more convenient and appropriate than Eq. (7). Using Eq. (10) we have studied diffusion behaviors for two cases: (1) in the case of g1 (t) = λe−λt , a normal diffusion or normal Brownian motion is found; (2) in the case of g2 (t) = λα tα−1 Eα,α (−λα tα ) we have discovered a subdiffusive behavior for the all time 8

regions rather than only for long-time or short time regions that appears in the literature. We have also obtained the exact solution for pdf’s in the two cases. For the first case, the exact solution of pdf is Gaussian. For the second case, the exact solution of pdf is non-Gaussian, but it can reduce to Gaussian pdf when α = 1.

Acknowledgments

K.S.F. acknowledges partial financial support from the Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico (CNPq), Brazilian agency. K.G.W. is pleased to acknowledge partial financial support received from the Materials World Network Program and Metallic Materials and Nanostructures Program of National Science Foundation, Washington, DC, under Grants No. DMR-0710484.

Appendix A: Integro-differential equation

In order to obtain the integro-differential equation (10) from Eq. (7) we employ the Fourier and Laplace transforms of the following expressions:  2  ∂ ρ(x, t) F = −k 2 ρ(k, t) , ∂x2 and the convolution theorem for Laplace transform Z t  L g (t − t1 ) ρ(x, t1 )dt1 = g(s)ρ(x, s).

(A1)

(A2)

0

Firstly, we use the inverse Fourier transform and Eq. (A1) in Eq. (7), and we find sρ(x, s) − sg(s)ρ(x, s) − Dsg(s)

∂ 2 ρ(x, s) = [1 − g(s)] ρ(x, 0) . ∂x2

(A3)

After both sides of Eq. (A3) are divided by s, we reorganize it as follows: 1 1 ∂ 2 ρ(x, s) ρ(x, s) − ρ(x, 0) − g(s)ρ(x, s) + g(s)ρ(x, 0) = Dg(s) . s s ∂x2

(A4)

Using Eq.(A2) and applying the inverse Laplace transform in Eq.(A4) we have Z t Z t Z t ∂ 2 ρ(x, t1 ) dt1 . ρ(x, t)−ρ(x, 0)− g (t − t1 ) ρ(x, t1 )dt1 +ρ(x, 0) g (t1 ) dt1 = D g (t − t1 ) ∂x2 0 0 0 (A5) 9

Using the Leibnitz theorem for integral, we have the following relationship: Z t  Z t ∂ ∂ g (t − t1 ) ρ(x, t1 )dt1 . g (t − t1 ) ρ(x, t1 )dt1 = g(0)ρ(x, t) + ∂t 0 0 ∂t The integral on the right side of Eq.(A6) can be expressed as Z t Z t ∂ ∂ g (t − t1 ) ρ(x, t1 )dt1 = − g (t − t1 ) ρ(x, t1 )dt1 . 0 ∂t1 0 ∂t

(A6)

(A7)

The integral on the right side of Eq.(A7) can be integrated by parts, and Eq.(A7) can be further written as Z t Z t ∂ ∂ g (t − t1 ) ρ(x, t1 )dt1 = −g(0)ρ(x, t) + g(t)ρ(x, 0) + ρ(x, t1 )dt1 . (A8) g (t − t1 ) ∂t1 0 ∂t 0 Substituting Eq.(A8) into Eq.(A6), Eq.(A6) can be rewritten as Z t  Z t ∂ ∂ g (t − t1 ) ρ(x, t1 )dt1 = g(t)ρ(x, 0) + ρ(x, t1 )dt1 . g (t − t1 ) ∂t 0 ∂t1 0

(A9)

Firstly employing the operator ∂/∂t to Eq.(A5) and then substituting Eq.(A9) into the equation, finally we have Z t Z t ∂ρ(x, t) ∂ ∂ 2 ρ(x, t1 ) ∂ρ(x, t1 ) − dt1 = D g (t − t1 ) dt1 . g (t − t1 ) ∂t ∂t1 ∂t 0 ∂x2 0

(A10)

[1] R. Kubo, M. Toda and N. Hashitsume, Statistical Physics II: Nonequilibrium Statistical Mechanics (Springer, Berlin,1985). [2] H. Risken, The Fokker-Planck Equation (Springer, Berlin,1996). [3] J. P. Bouchaud and A. Georges, Phys. Rep. 195,127 (1990). [4] R. Metzler and J. Klafter, Phys. Rep. 339, 1 (2000). [5] R. Metzler and J. Klafter, J. Phys. A 37, R161 (2004). [6] G. M. Zaslavsky, Phys. Rep. 371, 461 (2002). [7] U. Balucani, M. H. Lee and V. Tognetti, Phys. Rep. 373, 409 (2003). [8] B. J. West, E. L. Geneston and P. Grigolini, Phys. Rep. 468, 1 (2008). [9] K. G. Wang and M. Tokuyama, Physica A 265, 341 (1999). [10] B. J. West and S. Picozzi, Phys. Rev. E 65, 037106 (2002). [11] S. Picozzi and B. J. West, Phys. Rev. E 66, 046118 (2002).

10

[12] I. Golding and E. C. Cox, Phys. Rev. Lett. 96, 098102 (2006). [13] S. C. Kou and X. S. Xie, Phys. Rev. Lett. 93, 180603 (2004); W. Min, G. Luo, B. J. Cherayil, S. C. Kou and X. S. Xie, Phys. Rev. Lett. 94, 198302 (2005). [14] S. Chaudhury, S. C. Kou and B. J. Cherayil, J. Phys. Chem. 111, 2377 (2007). [15] E. W. Montroll and G. H. Weiss, J. Math. Phys. 6, 167 (1965). [16] H. Scher and M. Lax, Phys. Rev. B 7, 4502 (1973). [17] M. F. Shlesinger, J. Stat. Phys. 10, 421 (1974). [18] H. Scher and E. Montroll, Phys. Rev. B 12, 2455 (1975). [19] J. Klafter and R. Silbey, Phys. Rev. Lett 44, 55 (1980). [20] M. Kotulski, J. Stat. Phys. 81, 777 (1995). [21] E. Barkai and J. Klafter, Phys. Rev. Lett 79, 2245 (1997). [22] W. T. Coffey, D. S. F. Crothers, D. Holland and S. V. Titov, J. Mol. Liq. 114, 165 (2004). [23] V. Balakrishnan, Physica A 132, 569 (1985). [24] E. Barkai, R. Metzler and J. Klafter, Phys. Rev. E 61, 132 (2000). [25] H. C. Fogedby, Phys. Rev. Lett. 73, 2517 (1994). [26] H. C. Fogedby, Phys. Rev. E 50, 1657 (1994). [27] J. Klafter, A. Blumen and M. F. Shlesinger, Phys. Rev. A 35, 3081 (1987). [28] B. Berkowitz, A. Cortis, M. Dentz and H. Scher, Rev. Geophysics 44, RG2003, (2006). [29] A. Baule and R. Friedrich, EPL 77, 10002 (2007). [30] K. G. Wang and C. W. Lung, Phys. Lett. A 153, 423 (1991). [31] H. Weissman, G. H. Weiss and S. Havlin, J. Stat. Phys. 57, 301 (1989). [32] S. Havlin and G. H. Weiss, J. Stat. Phys. 58, 1267 (1990). [33] R. Cakir, A. Krokhin and P. Grigolini, Chaos, Solitons & Fractals 34, 19 (2007). [34] Kwok Sau Fa, The European Phys. J. E 24, 139 (2007). [35] Y. E. Ryabov, Phys. Rev. E 68, 030102(R) (2003). [36] A. Carpinteri, and F. Mainardi, Fractals and Fractional Calculus in Continuum Mechanics (Springer, Wien, 1997), pp. 223-276. [37] A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill, USA, 1955), Vol. III. [38] R. Hilfer and L. Anton, Phys. Rev. E 51, R848 (1995). [39] G. Ascolani, M. Bologna and P. Grigolini, Physica A 388, 2727 (2009).

11

[40] R. Gorenflo and F. Mainardi, J. Comp. Applied Math. 229, 400 (2009). [41] D. Fulger, E. Scalas and G. Germano, Phys. Rev. E 77, 021122 (2008). [42] F. Mainardi and G. Pagnini, Applied Math. Comp. 141, 51 (2003).

12