Chaos and quantum thermalization

TLDR

Berry's conjecture, which posits that energy eigenfunctions behave as Gaussian random variables, is expected to hold only for classically chaotic systems. The authors review evidence and demonstrate that neglected effects reinforce Berry's conjecture, then analyze a classically chaotic hard‑sphere gas to show that eigenstates obeying the conjecture yield Maxwell‑Boltzmann, Bose‑Einstein, or Fermi‑Dirac momentum distributions depending on particle symmetry. The study demonstrates that isolated quantum systems thermalize when their energy eigenfunctions satisfy Berry's conjecture—a process termed eigenstate thermalization—reaching equilibrium at least as fast as O(Elzxh/Δ)t⁻¹ without requiring ensemble averaging, thereby providing a solid foundation for quantum statistical mechanics.

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 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 eigenstate thermalization. We show that a generic initial state will approach thermal equilibrium at least as fast as O(\ensuremath{\Elzxh}/\ensuremath{\Delta})${\mathit{t}}^{\mathrm{\ensuremath{-}}1}$, where \ensuremath{\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 sound foundation for quantum statistical mechanics.