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.
Highlights
The idea to model apparently disordered spectra, like those of heavy nuclei, using random matrices was suggested in the mid-50’s by Wigner, and formalized in the early 60’s in the work of Dyson and Mehta [1]-[6]
Bohigas, Giannoni and Schmit (BGS) formulated a conjecture [7] stating that time-reversal invariant quantum systems with classically fully chaotic counterpart have universality properties given by random matrix theory (RMT)
Theoretical support of the BGS conjecture came from the semiclassical theory of spectral rigidity by Berry [15] [16], who showed that universal behaviour in the energy level statistics is due to long classical orbits
Summary
The idea to model apparently disordered spectra, like those of heavy nuclei, using random matrices was suggested in the mid-50’s by Wigner, and formalized in the early 60’s in the work of Dyson and Mehta [1]-[6]. They showed that random matrices of Gaussian orthogonal ensembles (GOE) generate a Wigner-type nearestneighbour level spacing (NNS) distribution [6]. Theoretical support of the BGS conjecture came from the semiclassical theory of spectral rigidity by Berry [15] [16], who showed that universal behaviour in the energy level statistics is due to long classical orbits. Further refinements have been made by Keating and Müller [20]
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