A Fractional Diffusion Equation for an n-Dimensional Correlated Levy Walk

TL;DR

This paper derives a fractional diffusion equation for n-dimensional correlated Levy walks, revealing new dynamical mechanisms and providing a theoretical framework for understanding superdiffusive processes in complex systems.

Contribution

It introduces a novel derivation of a fractional diffusion equation for correlated Levy walks using a fractional Klein-Kramers equation and a Cattaneo approximation.

Findings

Derived a fractional diffusion equation for correlated Levy walks

Identified measurable quantities related to diffusion mechanisms

Provided solutions for superdiffusive Levy walk behavior

Abstract

Levy walks define a fundamental concept in random walk theory which allows one to model diffusive spreading that is faster than Brownian motion. They have many applications across different disciplines. However, so far the derivation of a diffusion equation for an n-dimensional correlated Levy walk remained elusive. Starting from a fractional Klein-Kramers equation here we use a moment method combined with a Cattaneo approximation to derive a fractional diffusion equation for superdiffusive short range auto-correlated Levy walks in the large time limit, and solve it. Our derivation discloses different dynamical mechanisms leading to correlated Levy walk diffusion in terms of quantities that can be measured experimentally.