Stochastic pathway to anomalous diffusion

Abstract

We present an appraisal of differential-equation models for anomalous diffusion, in which the time evolution of the mean-square displacement is 〈$r^2$(t)〉∼$t^{\gamma }$ with γ≠1. By comparison, continuous-time random walks lead via generalized master equations to an integro-differential picture. Using Lévy walks and a kernel which couples time and space, we obtain a generalized picture for anomalous transport, which provides a unified framework both for dispersive (γ<1) and for enhanced diffusion (γ>1).