Misc,
Chaos and Quantum Thermalization
(1994)cite arxiv:cond-mat/9403051Comment: 28 pages in Plain TeX plus 2 uuencoded PS figures (included); minor corrections only, this version will be published in Phys. Rev. E; UCSB-TH-94-10.
DOI: 10.1103/PhysRevE.50.888
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
$O(\hbar/\Delta)t^-1$, 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.
