L\'evy flights as an emergent phenomenon in a spatially extended system

TL;DR

This paper demonstrates that the Landau-Lifshitz-Gilbert equation, a stochastic PDE driven solely by Gaussian noise, can exhibit emergent superdiffusive behavior characterized by Le9vy-like jumps in a spatially extended system.

Contribution

It provides the first explicit example of a spatially extended system showing anomalous diffusion without Le9vy noise, revealing emergent superdiffusive behavior.

Findings

The Landau-Lifshitz-Gilbert equation exhibits superdiffusive behavior.

Emergent anomalous diffusion arises from front interface widening events.

The mechanism is identified via reduction to stochastic differential equations.

Abstract

Anomalous diffusion and L\'evy flights, which are characterized by the occurrence of random discrete jumps of all scales, have been observed in a plethora of natural and engineered systems, ranging from the motion of molecules to climate signals. Mathematicians have recently unveiled mechanisms to generate anomalous diffusion, both stochastically and deterministically. However, there exists to the best of our knowledge no explicit example of a spatially extended system which exhibits anomalous diffusion without being explicitly driven by L\'evy noise. We show here that the Landau-Lifshitz-Gilbert equation, a stochastic partial differential equation (SPDE), despite only driven by Gaussian white noise, exhibits superdiffusive behaviour. The anomalous diffusion is an entirely emergent behaviour and manifests itself in jumps in the location of its travelling front solution. Using a collective coordinate approach we reduce the SPDE to a set of stochastic differential equations (SDEs) driven by Gaussian white noise. This allows us to identify the mechanism giving rise to the anomalous diffusion as random widening events of the front interface.