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

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.

arXiv comment24 pages, 9 figures, 2 tables. Accepted in the Journal of Statistical Mechanics: Theory and Experiment

References in corpus (18)

• Anomalous transport in the crowded world of biological cells
• Persistence and First-Passage Properties in Non-equilibrium Systems
• Isolating and enhancing the emission of single erbium ions using a silicon nanophotonic cavity
• On the strong anomalous diffusion
• First passages in bounded domains: When is the mean first passage time meaningful?
• First passage and arrival time densities for Lévy flights and the failure of the method of images
• First passages for a search by a swarm of independent random searchers
• Lévy walks and scaling in quenched disordered media
• Bias- and bath-mediated pairing of particles driven through a quiescent medium
• Occurrence of normal and anomalous diffusion in polygonal billiard channels
• Rare events in generalized Lévy Walks and the Big Jump principle
• Escape Rates and Physically Relevant Measures for Billiards with Small Holes
• Knudsen gas in a finite random tube: transport diffusion and first passage properties
• One-Particle and Few-Particle Billiards
• Displacement Autocorrelation Functions for Strong Anomalous Diffusion: A Scaling Form, Universal Behavior, and Corrections to Scaling
• A simple non-chaotic map generating subdiffusive, diffusive and superdiffusive dynamics
• Diffusivity in multiple scattering systems
• What mathematical billiards teach us about statistical physics?