Superstatistics in random matrix theory

TL;DR

This paper extends random matrix theory using superstatistics to model systems with mixed regular and chaotic dynamics, providing analytical spectral characteristics that depend on the degree of chaos.

Contribution

It introduces a basis-invariant superstatistical ensemble where matrix elements are correlated, extending traditional random matrix models for mixed systems.

Findings

Analytical expressions for level density and spacing distributions derived.

Results match for small deviations from chaos, indicating superstatistics's applicability.

Superstatistics is effective mainly in early transition stages from chaos to regularity.

Abstract

Using the superstatistics method, we propose an extension of the random matrix theory to cover systems with mixed regular-chaotic dynamics. Unlike most of the other works in this direction, the ensembles of the proposed approach are basis invariant but the matrix elements are not statistically independent. Spectral characteristics of the mixed systems are expressed by averaging the corresponding quantities in the standard random-matrix theory over the fluctuations of the inverse variance of the matrix elements. We obtain analytical expressions for the level density and the nearest-neighbor-spacing distributions for four different inverse-variance distributions. The resulting expressions agree with each others for small departures from chaos, measured by an effective non-extensivity parameter. Our results suggest, among other things, that superstatistics is suited only for the initial stage of transition from chaos to regularity.