Positive Commutators in Non-Equilibrium Statistical Mechanics

TL;DR

This paper extends the positive commutator method from zero temperature to positive temperatures in non-equilibrium quantum statistical mechanics, providing an alternative proof of the stability of equilibrium states in large quantum systems.

Contribution

It generalizes the positive commutator technique to positive temperatures and offers a new proof of the Return to Equilibrium property.

Findings

Extended positive commutator method to non-equilibrium settings

Proved asymptotic stability of equilibrium states

Demonstrated convergence to equilibrium after perturbations

Abstract

The method of positive commutators, developed for zero temperature problems over the last twenty years, has been an essential tool in the spectral analysis of Hamiltonians in quantum mechanics. We extend this method to positive temperatures, i.e. to non-equilibrium quantum statistical mechanics. We use the positive commutator technique to give an alternative proof of a fundamental property of a certain class of large quantum systems, called {\it Return to Equilibrium}. This property says that equilibrium states are (asymptotically) stable: if a system is slightly perturbed from its equilibrium state, then it converges back to that equilibrium state as time goes to infinity.