A generalised diffusion equation corresponding to continuous time random walks with coupling between the waiting time and jump length distributions

Abstract

This work outlines a transiently coupled continuous time random walk framework. The coupling is between the displacement probability density function and the elapsed waiting time, and is of the form 1 − exp (−αt) coupling of this kind generates larger displacements for longer waiting times, however, decouples on longer timescales. Such coupling is proposed to be physically relevant to systems in which diffusion is driven by the development of internal stresses which release and develop cyclically. This article outlines the associated generalised diffusion equation (GDE), which describes the time evolution of the position probability density function, P(x, t). The solution for P(x, t) is obtained using the properties of the Fox H function, both in terms of its transform properties but also its expansion theorems. The second moment and the asymptotic features of the solution are extracted. The relaxation of P(x, t) back to the solution of a decoupled-type GDE is highlighted.