Transport properties of Lévy walks: an analysis in terms of multistate processes

Abstract

Continuous time random walks combining diffusive and ballistic regimes are introduced to describe a class of Lévy walks on lattices. By including exponentially-distributed waiting times separating the successive jump events of a walker, we are led to a description of such Lévy walks in terms of multistate processes whose time-evolution is shown to obey a set of coupled delay differential equations. Using simple arguments, we obtain asymptotic solutions to these equations and rederive the scaling laws for the mean squared displacement of such processes. Our calculation includes the computation of all relevant transport coefficients in terms of the parameters of the models.

arXiv comment5 pages, 2 figures. New references added

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• Energy-dependent diffusion in a soft periodic Lorentz gas
• Continuous-time random walk between Lévy-spaced targets in the real line
• Limit theorems for Lévy flights on a 1D Lévy random medium
• Lévy walks on lattices as multi-state processes