Transport properties of L\'evy walks: an analysis in terms of multistate processes

TL;DR

This paper models Le9vy walks with mixed diffusive and ballistic behaviors using multistate processes and delay differential equations, deriving scaling laws and transport coefficients for their mean squared displacement.

Contribution

It introduces a novel multistate process framework with delay differential equations to analyze Le9vy walks, including asymptotic solutions and transport coefficients.

Findings

Derived asymptotic solutions for Le9vy walk models.

Revealed scaling laws for mean squared displacement.

Calculated transport coefficients from model parameters.

Abstract

Continuous time random walks combining diffusive and ballistic regimes are introduced to describe a class of L\'evy 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 L\'evy 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.