Abul-Magd A Y, Dietz B, Friedrich T, Richter A
Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt and Faculty of Engineering, Sinai University, El-Arish, Egypt.
Phys Rev E Stat Nonlin Soft Matter Phys. 2008 Apr;77(4 Pt 2):046202. doi: 10.1103/PhysRevE.77.046202. Epub 2008 Apr 3.
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.
本文对两组超导微波台球的本征频率进行了统计分析，一组呈蘑菇状，另一组属于蚶线台球族。这些台球具有混合的规则 - 混沌动力学，但在其经典相空间中结构不同。每个台球的频谱表示为一个时间序列，其中能级顺序起着时间的作用。时间序列分析得出两个最重要的发现。首先，频谱可以由两个不同的弛豫长度来表征。这是基于两个分布函数折叠的超统计方法有效性的前提条件。其次，所谓超统计参数的所得概率密度函数的形状可以用逆卡方分布合理近似。该分布用于计算最近邻间距分布，并在超统计框架内将其与具有混合动力学的台球共振频率的间距分布进行比较。发现所获得的间距分布能很好地描述实验结果，并且与包括贝里和罗布尼克的分布在内的许多其他间距分布具有相同甚至更好的质量。然而，与其他用于理论描述具有混合动力学系统的光谱特性的方法不同，超统计还根据波特 - 托马斯分布的超统计推广提供了本征函数特性的描述。实际上，发现逆卡方参数分布适用于在超统计框架内分析蚶线台球的实验共振强度。
