Ergodicity and Mixing in Quantum Dynamics

TL;DR

This paper explores quantum ergodicity and mixing, proposing definitions based on energy level structures, and finds a strong link between these properties and quantum chaos, suggesting a potential third class of quantum dynamics.

Contribution

It introduces new definitions of quantum ergodicity and mixing based on energy level degeneracies and relates these to quantum chaos, also proposing a possible third class of quantum dynamics.

Findings

Quantum ergodicity and mixing relate closely to quantum chaos.

Energy level degeneracies influence ergodic properties.

A potential third class of quantum dynamics characterized by maximized entropy.

Abstract

After a brief historical review of ergodicity and mixing in dynamics, particularly in quantum dynamics, we introduce definitions of quantum ergodicity and mixing using the structure of the system's energy levels and spacings. Our definitions are consistent with usual understanding of ergodicity and mixing. Two parameters concerning the degeneracy in energy levels and spacings are introduced. They are computed for right triangular billiards and the results indicate a very close relation between quantum ergodicity (mixing) and quantum chaos. At the end, we argue that, besides ergodicity and mixing, there may exist a third class of quantum dynamics which is characterized by a maximized entropy.