Mesoscopic description of reactions for anomalous diffusion: a case study

Abstract

Reaction–diffusion equations deliver a versatile tool for the description of reactions ininhomogeneous systems under the assumption that the characteristic reaction scales andthe scales of the inhomogeneities in the reactant concentrations separate. In the presentwork, we discuss the possibilities of a generalization of reaction–diffusion equations to thecase of anomalous diffusion described in terms of continuous-time random walkswith decoupled step length and waiting time probability densities, the first beingGaussian or Lévy, the second one being an exponential or a power law lacking the firstmoment. We consider a special case of an irreversible or reversible conversion and show that only in the Markovian case of an exponential waiting timedistribution can the diffusion term and the reaction term be decoupled. In all other cases,the properties of the reaction affect the transport operator, so the form of thecorresponding reaction–anomalous diffusion equations does not closely follow the form ofthe usual reaction–diffusion equations.