A Huygens principle for diffusion and anomalous diffusion in spatially extended systems

TL;DR

This paper introduces a universal framework for understanding diffusive and superdiffusive behaviors in chaotic spatially extended systems, highlighting the role of chaos strength, media isotropy, and space dimension.

Contribution

It establishes a nonlinear Huygens principle linking chaos strength, media isotropy, and space dimension to diffusive behaviors in such systems.

Findings

Strong chaos in anisotropic media causes Brownian motion with drift.

Weak chaos in anisotropic media results in superdiffusive Lévy processes.

In isotropic media, strong chaos leads to Brownian motion, with superdiffusion in odd dimensions.

Abstract

We present a universal view on diffusive behaviour in chaotic spatially extended systems for anisotropic and isotropic media. For anisotropic systems, strong chaos leads to diffusive behaviour (Brownian motion with drift) and weak chaos leads to superdiffusive behaviour (L\'evy processes with drift). For isotropic systems, the drift term vanishes and strong chaos again leads to Brownian motion. We establish the existence of a nonlinear Huygens principle for weakly chaotic systems in isotropic media whereby the dynamics behaves diffusively in even space dimension and exhibits superdiffusive behaviour in odd space dimensions.