A statistical analysis of the eigenfrequencies of two sets of superconducting microwave billiards, one with mushroomlike shape and the other from the family of the Limaçon billiards, is presented. These billiards have mixed regular-chaotic dynamics but different structures in their classical phase spaces. The spectrum of each billiard is represented as a time series where the level order plays the role of time. Two most important findings follow from the time series analysis. First, the spectra can be characterized by two distinct relaxation lengths. This is a prerequisite for the validity of the superstatistical approach, which is based on the folding of two distribution functions. Second, the shape of the resulting probability density function of the so-called superstatistical parameter is reasonably approximated by an inverse chi2 distribution. This distribution is used to compute nearest-neighbor spacing distributions and compare them with those of the resonance frequencies of billiards with mixed dynamics within the framework of superstatistics. The obtained spacing distribution is found to present a good description of the experimental ones and is of the same or even better quality as a number of other spacing distributions, including the one from Berry and Robnik. However, in contrast to other approaches toward a theoretical description of spectral properties of systems with mixed dynamics, superstatistics also provides a description of properties of the eigenfunctions in terms of a superstatistical generalization of the Porter-Thomas distribution. Indeed, the inverse chi2 parameter distribution is found suitable for the analysis of experimental resonance strengths in the Limaçon billiards within the framework of superstatistics.
Using all the available empirical information, we analyze the spacing distributions of low-lying 2 + levels of even-even nuclei. To obtain statistically relevant samples, the nuclei are grouped into classes defined by the ratio R 4/2 of the exitation energies of the first 4 + and 2 + levels. This ratio serves as a measure of collectivity in nuclei. With the help of Bayesian inference, we determine the chaoticity parameter for each class. This parameter is found to vary strongly with R 4/2 and takes particularly small values in nuclei that have one of the dynamical symmetries of the interacting Boson model.
Using Beck and Cohen's superstatistics, we introduce in a systematic way a family of generalised Wishart-Laguerre ensembles of random matrices with Dyson index β = 1,2, and 4. The entries of the data matrix are Gaussian random variables whose variances η fluctuate from one sample to another according to a certain probability density f (η) and a single deformation parameter γ. Three superstatistical classes for f (η) are usually considered: χ 2 -, inverse χ 2 -and log-normaldistributions. While the first class, already considered by two of the authors, leads to a power-law decay of the spectral density, we here introduce and solve exactly a superposition of Wishart-Laguerre ensembles with inverse χ 2 -distribution. The corresponding macroscopic spectral density is given by a γ-deformation of the semicircle and Marčenko-Pastur laws, on a non-compact support with exponential tails. After discussing in detail the validity of Wigner's surmise in the Wishart-Laguerre class, we introduce a generalised γ-dependent surmise with stretched-exponential tails, which well approximates the individual level spacing distribution in the bulk. The analytical results are in excellent agreement with numerical simulations. To illustrate our findings we compare the χ 2 -and inverse χ 2 -class to empirical data from financial covariance matrices.
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