The approach to equilibrium in a macroscopic quantum system for a typical nonequilibrium subspace

TL;DR

This paper proves that in a typical macroscopic quantum system, states rapidly reach thermal equilibrium on the order of the Boltzmann time, highlighting that slow decay observed in reality is non-typical and requires explanation.

Contribution

It provides a rigorous mathematical proof that typical nonequilibrium subspaces lead to rapid thermalization, contrasting with the slow decay observed physically.

Findings

States thermalize quickly on the order of Boltzmann time

Typical nonequilibrium subspaces lead to rapid approach to equilibrium

Slow decay in real systems is non-typical and needs explanation

Abstract

We study the problem of the approach to equilibrium in a macroscopic quantum system in an abstract setting. We prove that, for a typical choice of "nonequilibrium subspace", any initial state (from the energy shell) thermalizes, and in fact does so very quickly, on the order of the Boltzmann time $\tau__\mathrm{B}:=h/(k_\mathrm{B}T)$. This apparently unrealistic, but mathematically rigorous, conclusion has the important physical implication that the moderately slow decay observed in reality is not typical in the present setting. The fact that macroscopic systems approach thermal equilibrium may seem puzzling, for example, because it may seem to conflict with the time-reversibility of the microscopic dynamics. According the present result, what needs to be explained is, not that macroscopic systems approach equilibrium, but that they do so slowly. Mathematically our result is based on an interesting property of the maximum eigenvalue of the Hadamard product of a positive semi-definite matrix and a random projection matrix. The recent exact formula by Collins for the integral with respect to the Haar measure of the unitary group plays an essential role in our proof.