Mesoscopic description of reactions for anomalous diffusion:: a case study

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 in terms of 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 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 the corresponding reaction-anomalous diffusion equations does not closely follow the form of the usual reaction diffusion equations.