Scale invariance in anomalous diffusion

Abstract

Transport in disordered systems has been an active topic of research with emphasis on sublinear behaviour in time of the mean square displacement of a random walker. This dispersive transport has been shown to originate from diffusion on scale-invariant fractal structures or to be due to long-time tails in waiting-time distributions which are scale invariant in time. Another type of diffusion, with the mean square displacement being superlinear in time has become of recent interest. We discuss the different diffusional behaviours and compare their propagators. We describe the Lévy walk scheme for diffusion in the framework of continuous-time random walks with coupled memories. This type of walk describes random motion in space and time in a scaling fashion and may lead to enhanced diffusion. We focus on some properties of enhanced diffusion in one dimension: the mean square displacement, the propagator and the mean number of distinct sites visited.