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For classical billiards we suggest that a matrix of action or length oftrajectories in conjunction with statistical measures, level spacingdistribution and spectral rigidity, can be used to distinguish chaotic fromintegrable systems. As examples of 2D chaotic billiards we considered theBunimovich stadium billiard and the Sinai billiard. In the level spacingdistribution and spectral rigidity we found GOE behaviour consistent withpredictions from random matrix theory. We studied transport properties andcomputed a diffusion coefficient. For the Sinai billiard, we found normaldiffusion, while the stadium billiard showed anomalous diffusion behaviour. Asexample of a 2D integrable billiard we considered the rectangular billiard. Wefound very rigid behaviour with strongly correlated spectra similar to a Diraccomb. These findings present numerical evidence for universality in levelspacing fluctuations to hold in classically integrable systems and inclassically fully chaotic systems.

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