The precise time-dependent solution of the Fokker–Planck equation with anomalous diffusion
Introduction
The Fokker–Planck (FP) equation was first applied to the Brownian motion problem [1]. With the equation of motion of a Brownian particle, Langevin equation, and the corresponding FP equation, the probability distribution to find the particle in a given region may be determined by solving the equation. The simplest situation of the Brownian motion is a Brownian particle moving in the medium with friction constant and diffusion constant , and the link between the two constants is , known as the fluctuation–dissipation relation (FDR) [2]. In such a situation the Langevin equation and the FP equation are both linear and the solutions (stationary and time-dependent) are Gaussian distributions or Maxwell–Boltzmann (MB) distributions. But for a general situation when a Brownian particle is moving in a complex medium in which the friction and diffusion coefficients can depend on the variables, the Langevin equation is nonlinear and then solving the corresponding FP equation becomes very complicated. In fact, not much has been known in general about the long-time steady-state solution of an arbitrary FP equation. Only in some special cases if a FDR can be invoked, a steady-state solution is found.
The Brownian motion characterized as a pure diffusion process has a probability distribution that is Gaussian at all times and obeys the Einstein relation at long time, the mean-square displacement , where is a constant, which is called normal diffusion. Anomalous diffusion is random motion having with and therefore there is no constant diffusion coefficient ( may be space/velocity dependent [3], [4], [5], [6], [7], [8], [9]) and the associated probability distribution is non-Gaussian or non-MB/power-law distributions [10], [11], [12], [13], [14], [15], [16], [17]. Many nonlinear FP equations which appear to be some “fractal structure” are frequently constructed to describe the systems which have anomalous diffusion [18], [19], [20], [21], [22], [23]. It is interesting that they found the steady-state solution following a power-law -distribution in nonextensive statistics [24]. However, these fractal FP equations are all “nonstandard” and due to the lack of physically corresponding Langevin equation, the dynamical origins of the power-law distribution and the physical mechanism that leads to such a distribution are unknown.
Non-Gaussian or non-MB/power-law distributions have been noted prevalently in physical, chemical, biological and even social systems. In recent years, theoretical and experimental researches of these distributions have attracted great attention in the various fields of science, such as astronomy and astrophysics [25], [26], [27], [28], [29], plasmas and space physics [10], [14], [30], [31], [32], [33], [34], and reaction rate theory in chemistry [35], [36], [37], [38], [39]. In terms of the above studies, the power-law distributions often link to the complex systems involving long-range interactions, inhomogeneity and non-equilibrium dissipation processes. Information about the dynamical origins of these anomalous distributions is important for the understanding of many different processes in complex systems. This problem may be seeking a solution from the standard FP equation based on the Langevin equation for the dynamics of Brownian motion [16]. We have studied a general position–momentum Brownian motion in an inhomogeneous medium and with a multiplicative noise. The diffusion coefficient and friction coefficient can be position/momentum dependent for a Brownian particle moving in complex medium. By invoking a generalized FDR one could seek the steady-state solutions from the standard FP equations in both Ito’s, Stratonovich’s and Zwanzig’s (or the backward Ito’s) rules, where many different forms of power-law distributions were found [16], [17]. Besides the steady-state solutions, the time-dependent solutions of the FP equations are also important for us to understand the dynamical evolution of the probability distributions in complex systems. However it is not easy to solve a general multivariable FP equation. In this work, we will try to find the time-dependent solution of the standard FP equation with an anomalous diffusion in momentum space.
The paper is organized as follows. In Section 2, we briefly describe the standard FP equation based the Langevin equation for Brownian motion in a complex medium, the generalized FDR and its associated power-law distribution. In Section 3, we solve the time-dependent FP equation with an anomalous diffusion in momentum space. The precise time-dependent analytical solution will be given. In Section 4, numerical studies are made to examine the accuracy and validity of the analytical solution, including a general test and the application to the Ornstein–Uhlenbeck process. Finally in Section 5, we give the conclusion.
Section snippets
The Fokker–Planck equation and power-law distribution
We consider a Brownian particle, with the mass moving in a medium with a friction coefficient as well as a noise , and under a potential field . In the simplest case, the friction and diffusion coefficients may be regarded as constant approximately. But this is not always true. For example, the plasma immerged in a superthermal radiation field would lead to a multiplicative stochastic process in the velocity–space diffusion and therefore the friction and diffusion coefficients are both
Time-dependent solution of the FP equation with anomalous diffusion
Here we focus on the time evolution of the FP equation (3). Before finding the time-dependent solution, we make some simplifying assumptions. We let and the friction coefficient be a constant. Accordingly, the diffusion coefficient in the FDR (4) only becomes a function of . In this case, the probability distribution is a function of (, ), and after integrating FP equation (3) over the position it becomes and the generalized FDR (4) reads
General test
The precise time-evolution analytical solution of FP equation (6) with the FDR (7) is expressed by the sum in (24). In order to examine the validity of the analytical solution, we employ the implicit Runge–Kutta method [46] to do numerical studies of the solution of Eq. (6) with the FDR (7) for , and then compare with our analytical solution (24). Because we can only take the first finite terms in the infinite terms of (24) to calculate the analytical solution (here we have taken the first
Conclusion
Many different forms of the stationary power-law distributions can be generated exactly from the well-known Langevin equation of the Brownian motion and the associated Fokker–Planck (FP) equations under a generalized fluctuation–dissipation relation (FDR), but the time behavior of the equations is still unknown. In this work, we study the time evolution of the probability distribution from the standard FP equation (in Zwanzig’s or backward-Ito’s rule) based on the Langevin equation of the
Acknowledgments
This work is supported by the National Natural Science Foundation of China under Grant No. 11175128 and by the Higher School Specialized Research Fund for Doctoral Program under Grant No. 20110032110058.
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