Approach to thermal equilibrium of macroscopic quantum systems.
We consider an isolated macroscopic quantum system. Let H be a microcanonical "energy shell," i.e., a subspace of the system's Hilbert space spanned by the (finitely) many energy eigenstates with energies between E and E+deltaE . The thermal equilibrium macrostate at energy E corresponds to a subspace H(eq) of H such that dim H(eq)/dim H is close to 1. We say that a system with state vector psi is the element of H is in thermal equilibrium if psi is "close" to H(eq). We show that for "typical" Hamiltonians with given eigenvalues, all initial state vectors psi(0) evolve in such a way that psi(t) is in thermal equilibrium for most times t. This result is closely related to von Neumann's quantum ergodic theorem of 1929.
Goldstein
Sheldon
S
Departments of Mathematics and Physics, Rutgers University, Piscataway, New Jersey 08854-8019, USA. oldstein@math.rutgers.edu
Lebowitz
Joel L
JL
Mastrodonato
Christian
C
Tumulka
Roderich
R
Zanghi
Nino
N