Abstract
We analyse a scheme of transition from the Poissonian statistics for quantum levels to the Gaussian one of random matrix ensembles in the framework of level dynamics discussed by Yukawa. We propose a means of connecting these two limiting statistics by showing a result that Yukawa's parameter γ/β of the exponential family can be efficiently replaced by the ratio <E>/<Q> which reflects directly a degree of the eigenvalue correlations of each sample matrix in the ensemble. On this basis, we discuss a correspondence between the level statistics of a generic quantum system and its classical regular/chaotic dynamics in terms of the semiclassical power spectrum and its second moment formulated by Feingold-Peres and Prosen-Robnik. We also discuss some limiting proceduresN→∞ (infinite limit of the matrix dimension) pertinent to the Gaussian ensembles, and remark about the possibility offractional power law of Brody's type.
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