Macrostate equivalence of two general ensembles and specific relative entropies

TL;DR

This paper explores the relationship between macrostate and measure equivalence in statistical ensembles, establishing conditions under which they imply each other, with applications to quantum thermalization and irreversibility.

Contribution

It demonstrates that measure equivalence implies macrostate equivalence for general states by linking large-deviation functions to relative Renyi entropies, applicable to quantum and classical systems.

Findings

Measure equivalence implies macrostate equivalence via an inequality.

A sufficient condition for thermalization in quantum systems.

Analysis of irreversibility in quantum quenches.

Abstract

The two criteria of ensemble equivalence, i.e. the macrostate equivalence and the measure equivalence, are investigated for a general pair of states. The macrostate equivalence implies the two ensembles are indistinguishable by the measurement of macroscopic quantities obeying the large-deviation principle, and the measure equivalence means that the specific relative entropy of these two states vanishes in the thermodynamic limit. It is shown that the measure equivalence implies the macrostate equivalence for a general pair of states by deriving an inequality connecting the large-deviation rate functions to the specific relative Renyi entropies. The result is applicable to both quantum and classical systems. As applications, a sufficient condition for thermalization, the timescale of quantum dynamics of macrovariables, and the second law with strict irreversibility in a quantum quench are discussed.