Spectral fluctuations of billiards with mixed dynamics: From time series to superstatistics

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 ${\chi }^2$ 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 ${\chi }^2$ parameter distribution is found suitable for the analysis of experimental resonance strengths in the Limaçon billiards within the framework of superstatistics.

Figure 1

Part of the transmission spectrum of the small (upper panel) and large (lower panel) mushroom billiard. The insets show the geometry of the billiards.

Figure 2

Local kurtosis of the spacing intervals of the mushroom (upper panel) and Limaçon (lower panel) billiards.

Figure 3

Autocorrelation of the spacings within spectral intervals for the mushroom (upper panel) and Limaçon (lower panel) billiards.

Figure 4

Estimation of the parameter distribution for the superstatistical description of spectra of the mushroom (upper panel) and two Limaçon (lower panel) billiards. The solid lines represent the ${\chi }^2$, the dotted lines the inverse ${\chi }^2$, and the dashed lines the log-normal distribution.

Figure 5

Experimental NNSDs for the mushroom (upper panel) and Limaçon (lower panel) billiards compared with the superstatistical distributions. The solid lines represent the ${\chi }^2$, the dashed lines the inverse ${\chi }^2$, the dashed lines the log-normal, and the short-dashed lines the Berry-Robnik distribution.

Figure 6

Relation between the tuning parameters $q$ and $\nu$ of the Berry-Robnik and superstatistical NNSDs, respectively.

Figure 7

Superstatistical NNSD $p_{\text{inv}{\chi }^2}\left(0,s\right)$ compared with the semi-Poisson distribution $P_{\text{SP}}$.

Figure 8

Resonance strength distributions in the mixed Limaçon billiards compared with the predictions of different superstatistics. The solid lines represent the ${\chi }^2$ and the dashed lines the inverse ${\chi }^2$ distribution, while the short-dashed line corresponds to the case in which the strengths have a ${\chi }^2$ distribution.