Macroscopic and Microscopic Thermal Equilibrium

TL;DR

This paper investigates how isolated quantum systems reach thermal equilibrium at macroscopic and microscopic levels, clarifying the conditions under which they exhibit thermal properties, especially in systems with many-body localization.

Contribution

It distinguishes between macroscopic and microscopic thermal equilibrium, analyzing their relevance and differences, particularly in systems with many-body localization.

Findings

Most wave functions in an energy shell are in both MATE and MITE.

Systems with many-body localization can be in MATE but not in MITE.

Superpositions of energy eigenfunctions typically exhibit thermal equilibrium properties.

Abstract

We study the nature of and approach to thermal equilibrium in isolated quantum systems. An individual isolated macroscopic quantum system in a pure or mixed state is regarded as being in thermal equilibrium if all macroscopic observables assume rather sharply the values obtained from thermodynamics. Of such a system (or state) we say that it is in macroscopic thermal equilibrium (MATE). A stronger requirement than MATE is that even microscopic observables (i.e., ones referring to a small subsystem) have a probability distribution in agreement with that obtained from the micro-canonical, or equivalently the canonical, ensemble for the whole system. Of such a system we say that it is in microscopic thermal equilibrium (MITE). The distinction between MITE and MATE is particularly relevant for systems with many-body localization (MBL) for which the energy eigenfuctions fail to be in MITE while necessarily most of them, but not all, are in MATE. However, if we consider superpositions of energy eigenfunctions (i.e., typical wave functions $\psi$) in an energy shell, then for generic macroscopic systems, including those with MBL, most $\psi$ are in both MATE and MITE. We explore here the properties of MATE and MITE and compare the two notions, thereby elaborating on ideas introduced in [Goldstein et al., Phys.Rev.Lett. 115: 100402 (2015)].