Mesoscopic description of reactions under anomalous diffusion: A case study

TL;DR

This paper explores how reaction-diffusion equations can be generalized to account for anomalous diffusion, revealing that only Markovian processes allow for simple decoupling of reaction and diffusion terms, unlike non-Markovian cases.

Contribution

It introduces a framework for reaction-diffusion equations under anomalous diffusion with decoupled step length and waiting time distributions, highlighting the limitations of Markovian assumptions.

Findings

Decoupling of reaction and diffusion occurs only in Markovian cases.

Non-Markovian anomalous diffusion couples reaction and transport properties.

Reaction properties influence the transport operator in non-Markovian anomalous diffusion.

Abstract

Reaction-diffusion equations deliver a versatile tool for the description of reactions in inhomogeneous systems under the assumption that the characteristic reaction scales and the scales of the inhomogeneities in the reactant concentrations separate. In the present work we discuss the possibilities of a generalization of reaction-diffusion equations to the case of anomalous diffusion described by continuous-time random walks with decoupled step length and waiting time probability densities, the first being Gaussian or Levy, the second one being an exponential or a power-law lacking the first moment. We consider a special case of an irreversible or reversible A ->B conversion and show that only in the Markovian case of an exponential waiting time distribution the diffusion- and the reaction-term can be decoupled. In all other cases, the properties of the reaction affect the transport operator, so that the form of the corresponding reaction-anomalous diffusion equations does not closely follow the form of the usual reaction-diffusion equations.