Asymptotic densities of planar L\'{e}vy walks: a non-isotropic case

TL;DR

This paper analyzes the asymptotic density profiles of two-dimensional Levy walks in the XY-model, deriving solutions for the ballistic regime and confirming them with numerical simulations.

Contribution

It extends Levy walk models to a non-isotropic 2D case, providing analytical solutions for asymptotic densities in the XY-model.

Findings

Derived asymptotic density profiles for 2D Levy walks.

Obtained exact expressions for specific parameters.

Confirmed analytical results with numerical simulations.

Abstract

L\'{e}vy walks are a particular type of continuous-time random walks which results in a super-diffusive spreading of an initially localized packet. The original one-dimensional model has a simple schematization that is based on starting a new unidirectional motion event either in the positive or in the negative direction. We consider two-dimensional generalization of L\'{e}vy walks in the form of the so-called XY-model. It describes a particle moving with a constant velocity along one of the four basic directions and randomly switching between them when starting a new motion event. We address the ballistic regime and derive solutions for the asymptotic density profiles. The solutions have a form of first-order integrals which can be evaluated numerically. For specific values of parameters we derive an exact expression. The analytic results are in perfect agreement with the results of finite-time numerical samplings.