Fractional Fokker-Planck Equation for Ultraslow Kinetics

TL;DR

This paper introduces a distributed-order fractional Fokker-Planck equation to model ultraslow diffusion characterized by logarithmic mean squared displacement growth and exponential position distributions, linking it to continuous-time random walks.

Contribution

The paper develops a novel distributed-order fractional Fokker-Planck framework to describe ultraslow kinetics and connects it with continuous-time random walk models.

Findings

Describes ultraslow diffusion with logarithmic MSD growth

Shows solutions have double-sided exponential distributions

Connects fractional equations with random walk schemes

Abstract

Several classes of physical systems exhibit ultraslow diffusion for which the mean squared displacement at long times grows as a power of the logarithm of time ("strong anomaly") and share the interesting property that the probability distribution of particle's position at long times is a double-sided exponential. We show that such behaviors can be adequately described by a distributed-order fractional Fokker-Planck equations with a power-law weighting-function. We discuss the equations and the properties of their solutions, and connect this description with a scheme based on continuous-time random walks.