Accelerating random walks by disorder

TL;DR

This paper explores how heterogeneous environments influence superdiffusive random walks, revealing that spatial inhomogeneities can accelerate spread, contrary to the usual belief that disorder hampers stochastic processes.

Contribution

It introduces a novel fractional Fokker-Planck equation to analyze the impact of environmental disorder on superdiffusive Lévy flights, highlighting regimes of acceleration and attenuation.

Findings

Disorder can facilitate superdiffusive spread.

Distinct regimes of acceleration and attenuation identified.

Numerical and perturbation analyses support findings.

Abstract

We investigate the dynamic impact of heterogeneous environments on superdiffusive random walks known as L\'evy flights. We devote particular attention to the relative weight of source and target locations on the rates for spatial displacements of the random walk. Unlike ordinary random walks which are slowed down for all values of the relative weight of source and target, non-local superdiffusive processes show distinct regimes of attenuation and acceleration for increased source and target weight, respectively. Consequently, spatial inhomogeneities can facilitate the spread of superdiffusive processes, in contrast to common belief that external disorder generally slows down stochastic processes. Our results are based on a novel type of fractional Fokker-Planck equation which we investigate numerically and by perturbation theory for weak disorder.