Concentration of quantum equilibration and an estimate of the recurrence time

TL;DR

This paper demonstrates that quantum systems typically stay close to equilibrium with rare deviations exponentially suppressed, providing bounds on recurrence times, especially in many-body systems where these times are doubly exponential.

Contribution

It establishes a quantitative concentration of measure for quantum equilibration and derives bounds on recurrence times, including for many-body and free fermion systems.

Findings

Quantum systems concentrate around equilibrium with exponentially suppressed deviations.

Recurrence times are bounded below, often doubly exponential in system size for many-body systems.

Free fermions exhibit weaker concentration and earlier recurrences.

Abstract

We show that the dynamics of generic quantum systems concentrate around their equilibrium value when measuring at arbitrary times. This means that the probability of finding them away from equilibrium is exponentially suppressed, with a decay rate given by the effective dimension. Our result allows us to place a lower bound on the recurrence time of quantum systems, since recurrences corresponds to the rare events of finding a state away from equilibrium. In many-body systems, this bound is doubly exponential in system size. We also show corresponding results for free fermions, which display a weaker concentration and earlier recurrences.