A closed quantum system giving ergodicity

TL;DR

This paper investigates how closed quantum systems with many degrees of freedom can exhibit ergodic behavior consistent with quantum statistical mechanics when small perturbations are applied, especially in the large N limit.

Contribution

It demonstrates that small banded random matrix perturbations restore quantum statistical mechanics in large closed systems, with deviations diminishing exponentially as N increases.

Findings

Uncoupled systems can violate statistical mechanical rules.

Small banded random matrix perturbations recover expected quantum statistics.

Deviations from microcanonical distribution decrease exponentially with system size.

Abstract

[This is the unpublished supplemental information from 1989 to the paper: J.M. Deutsch, "Quantum statistical mechanics in a closed system." Phys. Rev. A, 43(4), 2046 (1991).] A closed quantum mechanical system does not necessarily give time averages in accordance with the microcanonical distribution. This question is investigated if the number of degrees of freedom N is large. For systems where the different degrees of freedom are uncoupled, experimental situations are discussed that show a violation of the usual statistical mechanical rules. It is shown that by applying a finite but very small perturbation to such systems, the results of quantum statistical mechanics can indeed be recovered. The form of the perturbation is that of a banded random matrix, which has been used previously to describe strongly chaotic systems in the semiclassical limit. The properties of energy eigenfunctions for this perturbed system are also discussed, and deviations from the microcanonical result are shown to become exponentially small in the limit of large N.