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)〉\ensuremath{\sim}${t}^{\ensuremath{\gamma}}$ with \ensuremath{\gamma}\ensuremath{\ne}1. By comparison, continuous-time random walks lead via generalized master equations to an integro-differential picture. Using L\'evy 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 (\ensuremath{\gamma}<1) and for enhanced diffusion (\ensuremath{\gamma}>1).