L\'evy walks on lattices as multi-state processes

TL;DR

This paper models Levy walks on lattices as multi-state processes using delay differential equations, deriving exact mean squared displacement and analyzing diffusive and superdiffusive regimes influenced by initial conditions.

Contribution

It introduces a novel multi-state process framework for Levy walks on lattices, providing exact solutions and insights into asymptotic scaling laws and superdiffusive behavior.

Findings

Exact expression for mean squared displacement

Identification of diffusive and superdiffusive regimes

Effect of initial conditions on transport coefficients

Abstract

Continuous-time random walks combining diffusive scattering and ballistic propagation on lattices model a class of L\'evy walks. The assumption that transitions in the scattering phase occur with exponentially-distributed waiting times leads to a description of the process in terms of multiple states, whose distributions evolve according to a set of delay differential equations, amenable to analytic treatment. We obtain an exact expression of the mean squared displacement associated with such processes and discuss the emergence of asymptotic scaling laws in regimes of diffusive and superdiffusive (subballistic) transport, emphasizing, in the latter case, the effect of initial conditions on the transport coefficients. Of particular interest is the case of rare ballistic propagation, in which case a regime of superdiffusion may lurk underneath one of normal diffusion.