Typicality of thermal equilibrium and thermalization in isolated macroscopic quantum systems

TL;DR

This paper develops a general theory for the typicality of thermal equilibrium in isolated quantum systems, showing that pure states generally represent equilibrium and establishing conditions for thermalization.

Contribution

It introduces a framework linking macroscopic observations to quantum state typicality and proves that thermal equilibrium is a typical property of pure states in the microcanonical ensemble.

Findings

Thermal equilibrium represented by pure states is typical in the microcanonical shell.

Large-deviation bounds support the typicality of equilibrium states.

Conditions for thermalization include moderate energy distribution and eigenstate thermalization hypothesis.

Abstract

Based on the view that thermal equilibrium should be characterized through macroscopic observations, we develop a general theory about typicality of thermal equilibrium and the approach to thermal equilibrium in macroscopic quantum systems. We first formulate the notion that a pure state in an isolated quantum system represents thermal equilibrium. Then by assuming, or proving in certain classes of nontrivial models (including that of two bodies in thermal contact), large-deviation type bounds (which we call thermodynamic bounds) for the microcanonical ensemble, we prove that to represent thermal equilibrium is a typical property for pure states in the microcanonical energy shell. We believe that the typicality, along with the empirical success of statistical mechanics, provides a sound justification of equilibrium statistical mechanics. We also establish the approach to thermal equilibrium under two different assumptions; one is that the initial state has a moderate energy distribution, and the other is the energy eigenstate thermalization hypothesis.