Deterministic Walks in Quenched Random Environments of Chaotic Maps

TL;DR

This paper studies how particles move chaotically through a fixed random environment modeled by chaotic maps, revealing Gaussian fluctuations around a linear drift and confirming results with high-precision numerics.

Contribution

It provides a rigorous analysis of particle propagation in quenched random media using chaotic maps, including the convergence of fluctuations to a Gaussian distribution.

Findings

Fluctuations around the drift are Gaussian in the scaling limit.

The variance of fluctuations depends on the system.

Numerical simulations confirm analytical predictions.

Abstract

This paper concerns the propagation of particles through a quenched random medium. In the one- and two-dimensional models considered, the local dynamics is given by expanding circle maps and hyperbolic toral automorphisms, respectively. The particle motion in both models is chaotic and found to fluctuate about a linear drift. In the proper scaling limit, the cumulative distribution function of the fluctuations converges to a Gaussian one with system dependent variance while the density function shows no convergence to any function. We have verified our analytical results using extreme precision numerical computations.