March 11, 2010

Title: Random matrices and classical dynamics: Statistical measure of trajectories in billiards

Abstract: We suggest that a matrix of classical observables, measured along trajectories corresponding to a set of boundary points, in conjunction with statistical tools from random matrix theory can be used to distinguish classical chaotic from integrable systems. As examples of chaotic systems we consider planar billiards, using length of trajectories as observables. In the fully chaotic case we found agreement with predictions from random matrix theory for the Gaussian orthogonal ensemble (GOE) which can be understood in terms of limit theorems such as the Central Limit Theorem. We also consider the 2-D integrable billiard systems. We find a very rigid spectral behaviour with strongly correlated eigenvalues as for a Dirac comb. Finally, we investigate the almost integrable limit of the stadium and Robnik's billiards, which show results close to the Poissonian behaviour generally observed in quantum mechanics for regular systems. Our findings present evidence for universality in spectral fluctuations also to hold in classically integrable systems and in classically fully chaotic systems. While the GOE behaviour in classically chaotic systems corresponds to GOE behaviour in quantum chaos, the fully correlated Dirac comb behaviour in classically integrable systems contrasts the typical uncorrelated Poissonian behaviour in quantum systems, but still remains clearly distinct from GOE's.