Non-extensive RMT Approach to Mixed Regular-Chaotic Dynamics

TL;DR

This paper introduces a non-extensive random matrix theory based on Tsallis entropy to model systems with mixed regular and chaotic dynamics, capturing the transition from Wigner to Poisson statistics.

Contribution

It formulates a non-extensive RMT framework that maintains basis invariance but relaxes matrix element independence, effectively modeling the chaos-to-regular transition.

Findings

Level density calculations match previous Tsallis-based results.

Spacing distributions align with numerical experiments during transition.

Model effectively describes initial chaos to order transition.

Abstract

We apply Tsallis's q-indexed entropy to formulate a non-extensive random matrix theory (RMT), which may be suitable for systems with mixed regular-chaotic dynamics. The joint distribution of the matrix elements is given by folding the corresponding quantity in the conventional random matrix theory by a distribution of the inverse matrix-element variance. It keeps the basis invariance of the standard theory but violates the independence of the matrix elements. We consider the sub-extensive regime of q more than unity in which the transition from the Wigner to the Poisson statistics is expected to start. We calculate the level density for different values of the entropic index. Our results are consistent with an analogous calculation by Tsallis and collaborators. We calculate the spacing distribution for mixed systems with and without time-reversal symmetry. Comparing the result of calculation to a numerical experiment shows that the proposed non-extensive model provides a satisfactory description for the initial stage of the transition from chaos towards the Poisson statistics.