Continuous time random walks and Fokker-Planck equation in expanding media

TL;DR

This paper develops a generalized framework for modeling normal and anomalous diffusion in expanding media using continuous time random walks, deriving related Fokker-Planck equations, and analyzing their physical implications.

Contribution

It introduces a comprehensive model linking expansion, diffusion, and external forces, deriving generalized and fractional Fokker-Planck equations, and analyzing their properties in expanding media.

Findings

Derived a generalized Fokker-Planck equation for expanding media.

Showed anomalous diffusion violates Galilei invariance and Einstein relation.

Supported results with numerical simulations.

Abstract

We consider a continuous random walk model for describing normal as well as anomalous diffusion of particles subjected to an external force when these particles diffuse in a uniformly expanding (or contracting) medium. A general equation that relates the probability distribution function (pdf) of finding a particle at a given position and time to the single-step jump length and waiting time pdfs is provided. The equation takes the form of a generalized Fokker-Planck equation when the jump length pdf of the particle has a finite variance. This generalized equation becomes a fractional Fokker-Planck equation in the case of a heavy-tailed waiting time pdf. These equations allow us to study the relationship between expansion, diffusion and external force. We establish the conditions under which the dominant contribution to transport stems from the diffusive transport rather than from the drift due to the medium expansion. We find that anomalous diffusion processes under a constant external force in an expanding medium described by means of our continuous random walk model are not Galilei invariant, violate the generalized Einstein relation, and lead to propagators that are qualitatively different from the ones found in a static medium. Our results are supported by numerical simulations.