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

Abstract

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.

Wigner surmise for mixed symmetry classes in random matrix theory

“…It has been shown in numerous studies that this is indeed the case. For example, billiards are showcases for the interplay of chaos and integrability, and certain billiards exhibit Poisson-GOE transitions [11][12][13][14]. A transition between GOE and GUE behavior takes place in the spectrum of a kicked top [15] or kicked rotor [16] when time-reversal symmetry is gradually broken.…”

Generalized superstatistics of nonequilibrium Markovian systems

“…with the value A ij (K) being the same for both the paths P α i,j , P β i,j due to (39). In other words, the quantity A ij (K) is the characteristics of the edge (i, j) with respect to the channel K rather than to its trees individually.…”

“…[8]). In particular, it has been applied to Lagrangian [9][10][11][12][13][14] and Eulerian turbulence [15][16][17][18], defect turbulence [19], atmospheric turbulence [20,21], large energy asymptotics [22], fat tails observed in granular media and hydrodynamic turbulence [23], cosmic ray statistics [24], statistics of solar flares [25], hadronization of quark matter [26], small-world networks [27,28], multi-components self-gravitating systems and collisionless stellar systems [29][30][31], distribution of dark energy in the universe [32], transitions between regular-chaotic dynamics [33][34][35], particle ensembles with fractional reactions [36], analysis of time series [37][38][39], brain activity [40], train delays on the British railway network [41], headway distribution in traffic flow [42], metastasis and cancer dynamics [43], stock markets and financial systems [44][45][46][47], social networks [48] and random networks [49]. It should be also noted that the superstatistics notion seems to have a longer history, at least, its precursors can be foun...…”

Generalized statistical mechanics for superstatistical systems

“…More generally, one can prove a superstatistical generalization of fluctuation theorems [4], develop a variational principle for the large-energy asymptotics of general superstatistics [3], proceed to generalized entropies for general superstatistics [5,40,43], let the q-values in equation (2.1) fluctuate as well [7] and prove superstatistical versions of a central limit theorem [8]. There are also relations with fractional reaction equations [45], random matrix theory [20,21,35], networks [36] and path integrals [9]. Very useful for practical applications is a superstatistical approach to time-series analysis [2,24,27].…”

“…In general, the superstatistical parameter b need not be an inverse temperature but can be an effective parameter in a stochastic differential equation, a volatility in finance or just a local variance parameter extracted from some experimental time series. There are interesting applications in hydrodynamic turbulence [2,19,29,30], defect turbulence [17], cosmic rays [25] and other scattering processes in high-energy physics [31,32], solar flares [18], share price fluctuations [26,27,33,34], random matrix theory [20,21,35], random networks [36], multiplicative-noise stochastic processes [37], wind velocity fluctuations [23,24], hydroclimatic fluctuations [22], the statistics of train departure delays [38] and survival statistics of cancer patients [39]. Maximum entropy principles can be generalized in a suitable way to yield the relevant probability distributions that characterize the various important universality classes in superstatistics [5,[40][41][42][43].…”