Superstatistics in Random Matrix Theory

TL;DR

This paper reviews superstatistical extensions of random matrix theory (RMT) that incorporate temperature fluctuations or correlations, enhancing RMT's applicability to mixed regular-chaotic quantum systems and comparing results with experimental data.

Contribution

It introduces superstatistical generalizations of RMT, allowing for correlations and non-equilibrium effects, and demonstrates their effectiveness through spectral distribution calculations.

Findings

Superstatistical RMT models match experimental billiard data.

Inclusion of temperature fluctuations improves RMT's description of mixed systems.

Superstatistics extends RMT to non-equilibrium quantum systems.

Abstract

Random matrix theory (RMT) provides a successful model for quantum systems, whose classical counterpart has a chaotic dynamics. It is based on two assumptions: (1) matrix-element independence, and (2) base invariance. Last decade witnessed several attempts to extend RMT to describe quantum systems with mixed regular-chaotic dynamics. Most of the proposed generalizations keep the first assumption and violate the second. Recently, several authors presented other versions of the theory that keep base invariance on the expense of allowing correlations between matrix elements. This is achieved by starting from non-extensive entropies rather than the standard Shannon entropy, or following the basic prescription of the recently suggested concept of superstatistics. The latter concept was introduced as a generalization of equilibrium thermodynamics to describe non-equilibrium systems by allowing the temperature to fluctuate. We here review the superstatistical generalizations of RMT and illustrate their value by calculating the nearest-neighbor-spacing distributions and comparing the results of calculation with experiments on billiards modeling systems in transition from order to chaos.