Machta-Zwanzig regime of anomalous diffusion in infinite-horizon billiards

TL;DR

This paper models anomalous diffusion in infinite-horizon billiards using a Lévy walk framework, deriving transport coefficients that generalize the Machta-Zwanzig approximation for normal diffusion.

Contribution

It introduces a novel Lévy walk model for infinite-horizon billiards, capturing anomalous diffusion with explicit trapping time distributions and probability calculations.

Findings

Derived an effective trapping mechanism for infinite-horizon billiards.

Provided explicit formulas for transport coefficients in the anomalous regime.

Extended the Machta-Zwanzig approximation to systems with anomalous diffusion.

Abstract

We study diffusion on a periodic billiard table with infinite horizon in the limit of narrow corridors. An effective trapping mechanism emerges according to which the process can be modeled by a L\'evy walk combining exponentially-distributed trapping times with free propagation along paths whose precise probabilities we compute. This description yields an approximation of the mean squared displacement of infinite-horizon billiards in terms of two transport coefficients which generalizes to this anomalous regime the Machta-Zwanzig approximation of normal diffusion in finite-horizon billiards [Phys. Rev. Lett. 50, 1959 (1983)].