Quantum Macrostates, Equivalence of Ensembles and an H-Theorem

TL;DR

This paper explores the behavior of quantum macrostates before the thermodynamic limit, analyzing fluctuations, ensemble equivalence, and an H-theorem through a law of large numbers and entropy considerations.

Contribution

It introduces a framework connecting quantum macrostates, ensemble equivalence, and an H-theorem, extending von Neumann's ideas to finite systems.

Findings

Macroscopic averages may not commute before the thermodynamic limit.

A law of large numbers for quantum states concentrates on macroscopic values.

Deviations are governed by entropy, linking fluctuations to thermodynamic properties.

Abstract

Before the thermodynamic limit, macroscopic averages need not commute for a quantum system. As a consequence, aspects of macroscopic fluctuations or of constrained equilibrium require a careful analysis, when dealing with several observables. We propose an implementation of ideas that go back to John von Neumann's writing about the macroscopic measurement. We apply our scheme to the relation between macroscopic autonomy and an H-theorem, and to the problem of equivalence of ensembles. In particular, we show how the latter is related to the asymptotic equipartition theorem. The main point of departure is an expression of a law of large numbers for a sequence of states that start to concentrate, as the size of the system gets larger, on the macroscopic values for the different macroscopic observables. Deviations from that law are governed by the entropy.