Diffusion and escape from polygonal channels: extreme values and geometric effects

TL;DR

This study investigates how polygonal billiard channels exhibit anomalous transport behaviors, such as superdiffusion and exponential displacement tails, influenced by geometric features and rational angles, with implications for experimental observations.

Contribution

It provides a detailed numerical analysis linking polygonal geometry to pseudo-chaotic transport phenomena, highlighting the impact of rational angles on first-passage times.

Findings

Transport shows strong anomalous diffusion with superlinear mean square displacement.

Displacement probability density has exponential tails and ballistic fronts.

Mean first-passage time diverges for channels with rational aperture angles.

Abstract

Polygonal billiards are an example of pseudo-chaotic dynamics, a combination of integrable evolution and sudden jumps due to conical singular points that arise from the corners of the polygons. Such pseudo-chaotic behaviour, often characterised by an algebraic separation of nearby trajectories, is believed to be linked to the wild dependence that particle transport has on the fine details of the billiard table. Here we address this relation through a detailed numerical study of the statistics of displacement in a family of polygonal channel billiards with parallel walls. We show that transport is characterised by strong anomalous diffusion, with a mean square displacement that scales in time faster than linear, and with a probability density of the displacement exhibiting exponential tails and ballistic fronts. In channels of finite length the distribution of first-passage times is characterised by fat tails, with a mean first-passage time that diverges when the aperture angle is rational. These findings have non trivial consequences for a variety of experiments.