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  • On spectral statistics of classically integrable systems
    Robnik, Marko, 1954- ; Veble, Gregor
    This work is an extensive study of the spectral statistics of three representative classically integrable systems, namely rectangle, torus and circle billiards. We analyse the E(k, L) statistics and ... focus on the related level spacing distribution P(S) and the delta statistics [Delta](L). The agreement with the Poissonian model is typically found to be perfect up-to the outer (unfolded) energy scale L < L[sub](max), beyond which the saturation is observed, in agreement with Berry's dynamical theory of the spectral rigidity, where L[sub](max) [to] [infinity] as h [to] 0. The untypical systems are those, where for example P(S) is not a smooth distribution but a sum of the delta functions, due to the "granularity" of the energy scale, for example in the rectangle with the rational squared sides ratio. However, even there we find resonable trend towards Poissonian statistics for large ranges L but L < L[sub](max). We describe teoretically and numerically the broadening of the delta spikes when the rectangular billiard is slightly distorted away from a rational to an irrational shape and find excellent agreement. Also, in irrational rectangle billiards we show and explain the existence of large functions, by one order of magnitude bigger than the statistical ones, whose origin is in the closeness to rome rational billiard shape. These fluctuacions and their amplitude are independent of the energy if the bin size shrinks inversely with energy. Funally we tested the mode distribution (i.e. distribution of the reduced mode fluctuacion number W) and found that it was not Poissonian, in agreement with Steiner's conjecture, and in fact follows the prediction by Bleher et al., that its tail behaves as exp(-W[sup]4). The general reason for non-universal behaviour of P(W) is that at largest energy scales we are always in the saturation regime L > L[sub](max).
    Vir: Journal of physics. A, Mathematical and general. - ISSN 0305-4470 (Vol. 31, 1998, str. 4669-4704)
    Vrsta gradiva - članek, sestavni del
    Leto - 1998
    Jezik - angleški
    COBISS.SI-ID - 42662657

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