Diffusion and Fokker-Planck-Smoluchowski Equations with Generalized Memory Kernel

Abstract

We consider anomalous stochastic processes based on the renewal continuous time random walk model with different forms for the probability density of waiting times between individual jumps. In the corresponding continuum limit we derive the generalized diffusion and Fokker-PlanckSmoluchowski equations with the corresponding memory kernels. We calculate the qth order moments in the unbiased and biased cases, and demonstrate that the generalized Einstein relation for the considered dynamics remains valid. The relaxation of modes in the case of an external harmonic potential and the convergence of the mean squared displacement to the thermal plateau are analyzed.MSC 2010 : Primary 26A33, 33E12; Secondary 34A08, 35R11

Non-Gaussian Distributions to Random Walk in the Context of Memory Kernels

“…In long times, Equations (34) and (35) imply an anomalous behaviour, i.e., x 2 ∼ t α . This long time behaviour occurs to Caputo and Riemann-Liouville operator [38,43]. For the appropriate limit of parameters τ and τ in Equation (34), to τ → 0, we obtain the standard fractional diffusion.…”

“…[38], in which L{K(t)} = ∞ 0 e −st K(t)dt = K L (s) denote the Laplace transform of memory kernel and τ is characteristic time. For K(t) = δ(t), we recover the Brownian motion.…”

“…First, we approach the waiting time on Sandev sense [38]; there are alternative forms in the literature, as discussed in the Section 1. In fact, the choice of different forms to waiting time distribution change how the memory kernel appears on diffusion equation [49].…”

“…The memory kernel as put on Equation (1) was considered on Fokker-Planck equation [38], the memory kernel in Laplacian term was considered in Ref. [36].…”

Generalized Langevin Equation and the Prabhakar Derivative

“…We further introduce a GLE with a tempered regularized Prabhakar friction term and analyze the normalized displacement correlation function in case of harmonic potential. Tempered fractional equations nowadays attract more and more attention due to their application in different systems [7][8][9][10]. This paper is organized as follows.…”