Universality in statistical measures of trajectories in classical billiards: Integrable rectangular versus chaotic Sinai and Bunimovich billiards

TL;DR

This paper demonstrates that statistical measures of trajectories can distinguish between integrable and chaotic classical billiards, revealing universal spectral fluctuation behaviors across different dynamical regimes.

Contribution

It introduces a method using trajectory length matrices and spectral statistics to differentiate integrable and chaotic billiards, providing numerical evidence of universality.

Findings

Chaotic billiards show GOE spectral statistics.

Integrable billiards exhibit highly correlated spectra.

Transport properties differ: normal vs. anomalous diffusion.

Abstract

For classical billiards we suggest that a matrix of action or length of trajectories in conjunction with statistical measures, level spacing distribution and spectral rigidity, can be used to distinguish chaotic from integrable systems. As examples of 2D chaotic billiards we considered the Bunimovich stadium billiard and the Sinai billiard. In the level spacing distribution and spectral rigidity we found GOE behaviour consistent with predictions from random matrix theory. We studied transport properties and computed a diffusion coefficient. For the Sinai billiard, we found normal diffusion, while the stadium billiard showed anomalous diffusion behaviour. As example of a 2D integrable billiard we considered the rectangular billiard. We found very rigid behaviour with strongly correlated spectra similar to a Dirac comb. These findings present numerical evidence for universality in level spacing fluctuations to hold in classically integrable systems and in classically fully chaotic systems.