Transport aspects in anomalous diffusion: Lévy walks

Abstract

In this paper we present a combined analytical and numerical study of transport properties of L\'evy walks. Here, within the framework of continuous-time random walks (CTRW's) with coupled memories, we focus on the probability ${P}_{0}$(t) of being at the initial site at time t and on S(t), the mean number of distinct sites visited in time t. We use the connection between ${P}_{0}$(t) and S(t), which are related via their Laplace transform, and we reanalyze our previous findings for 〈${r}^{2}$(t)〉, the mean-squared displacement. Furthermore, S(t) shows, as a function of the memory parameters, a very interesting, nonuniversal, nonmonotonic behavior, which we corroborate by numerical simulations in one dimension. We compare the findings with those for decoupled CTRW's on regular lattices and on fractals.