Mechanism of slow equilibration of isolated quantum systems

TL;DR

This paper investigates the slow process of equilibration in isolated quantum systems, linking microscopic properties to macroscopic timescales and identifying states with exceptionally long equilibration times.

Contribution

It introduces the deviation function as a bound on expectation values, connecting microscopic matrix elements to macroscopic transport timescales and slow equilibration.

Findings

The deviation function bounds expectation values in quantum systems.

Numerical evidence shows initial inhomogeneous states saturate the bound.

Some states exhibit arbitrarily long equilibration times.

Abstract

We discuss the approach toward equilibrium of an isolated quantum system. For a wide class of systems we argue that the time-averaged expectation value of a local operator in any initial state is bounded by the so-called deviation function, which characterizes maximal deviation from the equilibrium for all states with a given value of energy fluctuations. We provide numerical evidence that the bound is approximately saturated by the initial configurations with spatial inhomogeneities at macroscopic level. In this way the deviation function establishes an explicit connection between the macroscopically observed timescales associated with transport and properties of microscopic matrix elements. The form of the deviation function indicates that among the slowest states which saturate the bound there are also those with arbitrarily long equilibration times.