Jarzynski Relations for Quantum Systems and Some Applications

TL;DR

This note derives quantum analogues of Jarzynski relations for systems with a time-dependent Hamiltonian starting from a Gibbs state, establishing a quantum Jarzynski equality $\overline{e^{\beta E-\tilde{\beta} E'}}=\frac{Z'(\tilde{\beta})}{Z(\beta)}$ and related inequalities. It then applies these results to two problems: (i) proving a strong entropy-increase law for general compound systems under adiabatic operations, with a statistical derivation based on product Gibbs initial states and Jarzynski-style averages, and (ii) providing a preliminary analysis of heat transfer between two quantum systems at different temperatures, showing nonnegative entropy production and fluctuation-theorem-like symmetry in transition probabilities. The approach yields a concise, self-contained framework that links nonequilibrium quantum dynamics to equilibrium thermodynamics and highlights structure analogous to the classical fluctuation theorem. These results offer rigorous bounds and insights into energy exchange, irreversibility, and entropy production in quantum many-body processes.

Abstract

We derive quantum analogues of Jarzynski's relations, and discuss two applications, namely, a derivation of the law of entropy increase for general compound systems, and a preliminary analysis of heat transfer between two quantum systems at different temperatures. We believe that the derivation of the law of entropy increase is new and of importance.