Occurrence of normal and anomalous diffusion in polygonal billiard channels

TL;DR

This study investigates diffusion behaviors in polygonal billiard channels, revealing conditions under which normal or anomalous diffusion occurs, and linking anomalous diffusion to specific periodic orbits and geometric configurations.

Contribution

It provides a comprehensive numerical analysis of diffusion types in polygonal billiard channels, highlighting the role of geometry and orbit families in diffusion anomalies.

Findings

Normal diffusion in finite horizon channels

Asymptotic growth as t log t in infinite horizon cases

Anomalous super-diffusion with power-law growth when parallel scatterers are present

Abstract

From extensive numerical simulations, we find that periodic polygonal billiard channels with angles which are irrational multiples of pi generically exhibit normal diffusion (linear growth of the mean squared displacement) when they have a finite horizon, i.e. when no particle can travel arbitrarily far without colliding. For the infinite horizon case we present numerical tests showing that the mean squared displacement instead grows asymptotically as t log t. When the unit cell contains accessible parallel scatterers, however, we always find anomalous super-diffusion, i.e. power-law growth with an exponent larger than 1. This behavior cannot be accounted for quantitatively by a simple continuous-time random walk model. Instead, we argue that anomalous diffusion correlates with the existence of families of propagating periodic orbits. Finally we show that when a configuration with parallel scatterers is approached there is a crossover from normal to anomalous diffusion, with the diffusion coefficient exhibiting a power-law divergence.