Continuous Time Random Walks with Internal Dynamics and Subdiffusive Reaction-Diffusion Equations

TL;DR

This paper develops a generalized master equation for continuous time random walks with internal deterministic dynamics and derives subdiffusive reaction-diffusion equations, expanding the modeling framework for anomalous diffusion and reactions.

Contribution

It introduces a new formulation of the master equation incorporating internal dynamics and derives subdiffusive reaction-diffusion equations using a mean field approach.

Findings

Formulated a generalized master equation for CTRWs with internal dynamics

Derived subdiffusive reaction-diffusion equations from the master equation

Provided examples with advection-diffusion and jump-diffusion schemes

Abstract

We formulate the generalized master equation for a class of continuous time random walks in the presence of a prescribed deterministic evolution between successive transitions. This formulation is exemplified by means of an advection-diffusion and a jump-diffusion scheme. Based on this master equation, we also derive reaction-diffusion equations for subdiffusive chemical species, using a mean field approximation.