Kwok Sau Fa

Anomalous diffusion: Fractional Fokker–Planck equation and its solutions

We analyze a linear fractional Fokker–Planck equation for the case of an external force F(x)∝x|x|α−1 and diffusion coefficient D(x)∝|x|−θ (α,θ∈R). We also discuss the connection of the solutions found here with the Fox functions and the nonextensive statistics based on the Tsallis entropy.

Fractional Langevin equation and Riemann-Liouville fractional derivative

Abstract.In this present work we consider a fractional Langevin equation with Riemann-Liouville fractional time derivative which modifies the classical Newtonian force, nonlocal dissipative force, and long-time correlation. We investigate the first two moments, variances and position and velocity correlation functions of this system. We also compare them with the results obtained from the same fractional Langevin equation which uses the Caputo fractional derivative.

Exact solution of the Fokker-Planck equation for a broad class of diffusion coefficients.

We consider the Langevin equation with multiplicative noise term which depends on time and space. The corresponding Fokker-Planck equation in Stratonovich approach is investigated. Its formal solution is obtained for an arbitrary multiplicative noise term given by g(x, t) = D(x)T (t), and the behaviors of probability distributions, for some specific functions of D(x), are analyzed. In particular, for D(x) ∼ |x| −θ/2 , the physical solutions for the probability distribution in the Ito, Stratonovich and postpoint discretization approaches can be obtained and analyzed.

Bose–Einstein and Fermi–Dirac distributions in nonextensive Tsallis statistics: an exact study

Generalized Bose–Einstein and Fermi–Dirac distributions are analyzed in nonextensive Tsallis statistics by considering the normalized constraints in the effective temperature approach. These distributions are worked in D-dimension by employing a general density of states g(e)∝eD−1(D=D/2+D/n and D>0). Thermodynamic functions such as internal energy and average number of particles are also obtained in this context.

Nonlinear fractional diffusion equation: Exact results

The nonlinear fractional diffusion equation ∂tρ=r1−d∂rμ′{rd−1D(r,t;ρ)∂rμρν}−r1−d∂r{rd−1F(r,t)ρ}+α¯(t)ρ is studied by considering the diffusion coefficient D(r,t;ρ)=D(t)r−θργ and the external force F(r,t)=−k1(t)r+kαrα. In addition, a rich class of diffusive processes, including normal and anomalous ones, is obtained from the study present in this work.

Integrodifferential diffusion equation for continuous-time random walk

In this paper, we present an integrodifferential 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.

Solutions for a fractional nonlinear diffusion equation: Spatial time dependent diffusion coefficient and external forces

We analyze a generalized diffusion equation which extends some known equations such as the fractional diffusion equation and the porous medium equation. We start our investigation by considering the linear case and the nonlinear case afterward. The linear case is discussed taking fractional time and spatial derivatives into account in a unified approach. We also discuss the modifications that emerge by employing simple drifts and the diffusion coefficient given by D(x,t)=D(t)|x|−θ. For the nonlinear case, we study scaling behavior of the time in connection with the asymptotic behavior for the solution of the nonlinear fractional diffusion equation.

Analysis of the relativistic Brownian motion in momentum space

We investigate the relativistic Brownian motion in the context of Fokker-Planck equation. Due to the multiplicative noise term of the corresponding relativistic Langevin equation many Fokker-Planck equations can be generated. Here, we only consider the Ito, Stratonovich and Hanggi-Klimontovich approaches. We analyze the behaviors of the second moment of momentum in terms of temperature. We show that the second moment increases with the temperature T for all three approaches. Also, we present differential equations for more complicated averages of the momentum. In a specific case, in the Ito approach, we can obtain an analytical solution of the temporal evolution of an average of the momentum. We present approximate solutions for the probability density for all three cases.

A continuous time random walk model with multiple characteristic times

In this paper we consider a continuous time random walk (CTRW) model with a decoupled jump pdf. Further, we consider an approximate jump length pdf; for the waiting time pdf we do not use any approximation and we employ a function which depends on multiple characteristic times given by a sum of exponential functions. This waiting time pdf can reproduce power-law behavior for intermediate times. Using this specific waiting time probability density, we analyze the behavior of the second moment generated by the CTRW model. It is known that the waiting time pdf given by an exponential function generates a normal diffusion process, but for our waiting time pdf the second moment can give an anomalous diffusion process for intermediate times, and the normal diffusion process is maintained for the long-time limit. We note that systems which present subdiffusive behavior for intermediate times but reach normal diffusion at large times have been observed in biology.

q-Gaussian trial function and Bose–Einstein condensation

We use q-Gaussian trial wave function, based on Tsallis statistics, to study, variationally, the ground state and vortex state energies of Bose–Einstein condensation of dilute ultracold alkali atoms in anisotropic traps. This trial function works well when compared with the results available by purely numerical techniques, for both kinds of interactions among the atoms: attractive (7Li) and repulsive (87Rb).