Thermal equilibrium of a macroscopic quantum system in a pure state

TL;DR

This paper explores the concept of thermal equilibrium in pure quantum states of macroscopic systems, distinguishing between macroscopic and microscopic equilibrium, and analyzing their relevance in systems with many-body localization.

Contribution

It introduces the distinction between macroscopic thermal equilibrium (MATE) and microscopic thermal equilibrium (MITE) for pure states, highlighting their differences and implications for many-body localized systems.

Findings

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

MITE can fail in systems with many-body localization, while MATE may still hold.

Classical systems typically satisfy MATE but not MITE.

Abstract

We consider the notion of thermal equilibrium for an individual closed macroscopic quantum system in a pure state, i.e., described by a wave function. The macroscopic properties in thermal equilibrium of such a system, determined by its wave function, must be the same as those obtained from thermodynamics, e.g., spatial uniformity of temperature and chemical potential. When this is true we say that the system is in macroscopic thermal equilibrium (MATE). Such a system may however not be in microscopic thermal equilibrium (MITE). The latter requires that the reduced density matrices of small subsystems be close to those obtained from the microcanonical, equivalently the canonical, ensemble for the whole system. The distinction between MITE and MATE is particularly relevant for systems with many-body localization (MBL) for which the energy eigenfunctions fail to be in MITE while necessarily most of them, but not all, are in MATE. We note however that for generic macroscopic systems, including those with MBL, most wave functions in an energy shell are in both MATE and MITE. For a classical macroscopic system, MATE holds for most phase points on the energy surface, but MITE fails to hold for any phase point.