Stochastically perturbed billiards: fingerprints of chaos and universality classes

TL;DR

This paper investigates how stochastic perturbations affect billiard systems, revealing distinct behaviors in integrable versus chaotic tables and introducing new statistical properties.

Contribution

It demonstrates that stochastic perturbations preserve chaos features but fundamentally alter integrable billiards, linking them to Evans stochastic billiards with unique boundary measures.

Findings

Chaotic billiards retain their key statistical features under stochastic perturbations.

Integrable billiards exhibit non-uniform boundary measures when perturbed.

Tiny stochastic perturbations cause integrable billiards to behave like Evans stochastic billiards.

Abstract

Billiards tables - a minimal model for particles moving in a confined region - are known to present classical (and quantum) different features according to their shape, ranging from strongly chaotic to integrable dynamics. Here we consider the role of a stochastic perturbation of the elastic reflection law, and show that while chaotic billiards maintain their key statistical feature, the behaviour for integrable billiard tables is completely different: it can be linked, for tiny perturbations, to Evans stochastic billiard, where at each collision the reflected angle is a uniformly distributed stochastic variable on $(-\pi/2,\pi/2$). The resulting spatial stationary measure has peculiar aspects, like being typically non uniform along the boundary, differently from any chaotic billiard table.