Anomalous diffusion in random dynamical systems

TL;DR

This paper investigates how combining chaotic and non-chaotic systems randomly can produce anomalous diffusion, characterized by nonlinear mean square displacement, aging, and ergodicity breaking, with implications for various noise types and nonlinear dynamics.

Contribution

It demonstrates that random mixing of simple deterministic walks can generate anomalous diffusion with complex properties like aging and weak ergodicity breaking.

Findings

Anomalous diffusion arises from random mixing of deterministic walks.

The dynamics exhibit aging and weak ergodicity breaking.

Results are robust across different noise types and nonlinear perturbations.

Abstract

Consider a chaotic dynamical system generating Brownian motion-like diffusion. Consider a second, non-chaotic system in which all particles localize. Let a particle experience a random combination of both systems by sampling between them in time. What type of diffusion is exhibited by this {\em random dynamical system}? We show that the resulting dynamics can generate anomalous diffusion, where in contrast to Brownian normal diffusion the mean square displacement of an ensemble of particles increases nonlinearly in time. Randomly mixing simple deterministic walks on the line we find anomalous dynamics characterised by ageing, weak ergodicity breaking, breaking of self-averaging and infinite invariant densities. This result holds for general types of noise and for perturbing nonlinear dynamics in bifurcation scenarios.