Towards deterministic equations for Levy walks: the fractional material derivative

TL;DR

This paper derives a generalized dynamical equation for Levy walks using a fractional material derivative, enhancing the modeling of particles with broad jump distributions, especially in external fields or phase space.

Contribution

It introduces a novel fractional differential equation formulation for Levy walks, extending their analysis beyond traditional continuous time random walk models.

Findings

Derivation of a fractional material derivative for Levy walks.

Applicable to external force fields and phase space dynamics.

Provides a new framework for modeling Levy walk processes.

Abstract

Levy walks are random processes with an underlying spatiotemporal coupling. This coupling penalizes long jumps, and therefore Levy walks give a proper stochastic description for a particle's motion with broad jump length distribution. We derive a generalized dynamical formulation for Levy walks in which the fractional equivalent of the material derivative occurs. Our approach will be useful for the dynamical formulation of Levy walks in an external force field or in phase space for which the description in terms of the continuous time random walk or its corresponding generalized master equation are less well suited.