Chaos and Quantum Thermalization

TL;DR

This paper demonstrates that quantum systems with chaotic classical counterparts tend to thermalize through eigenstate properties, providing a new foundation for quantum statistical mechanics based on Berry's conjecture and eigenstate thermalization.

Contribution

It establishes that eigenstate thermalization occurs in chaotic quantum systems, linking Berry's conjecture to thermal distributions without ensemble averaging, and offers a new basis for quantum statistical mechanics.

Findings

Eigenstates in chaotic systems predict thermal distributions.

Thermalization occurs at a rate proportional to 5/, independent of ensemble averaging.

Evidence supports Berry's conjecture as a foundation for quantum thermalization.

Abstract

We show that a bounded, isolated quantum system of many particles in a specific initial state will approach thermal equilibrium if the energy eigenfunctions which are superposed to form that state obey {\it Berry's conjecture}. Berry's conjecture is expected to hold only if the corresponding classical system is chaotic, and essentially states that the energy eigenfunctions behave as if they were gaussian random variables. We review the existing evidence, and show that previously neglected effects substantially strengthen the case for Berry's conjecture. We study a rarefied hard-sphere gas as an explicit example of a many-body system which is known to be classically chaotic, and show that an energy eigenstate which obeys Berry's conjecture predicts a Maxwell--Boltzmann, Bose--Einstein, or Fermi--Dirac distribution for the momentum of each constituent particle, depending on whether the wave functions are taken to be nonsymmetric, completely symmetric, or completely antisymmetric functions of the positions of the particles. We call this phenomenon {\it eigenstate thermalization}. We show that a generic initial state will approach thermal equilibrium at least as fast as $O(\hbar/\Delta)t^{-1}$, where $\Delta$ is the uncertainty in the total energy of the gas. This result holds for an individual initial state; in contrast to the classical theory, no averaging over an ensemble of initial states is needed. We argue that these results constitute a new foundation for quantum statistical mechanics.