On the approach to thermal equilibrium of macroscopic quantum systems.

In joint work with J. L. Lebowitz, C. Mastrodonato, and N. Zanghì [2, 3, 4], we considered an isolated, macroscopic quantum system. Let H be a micro-canonical '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+δE. The thermal equilibrium macro-state at energy E corresponds to a subspace Heq of H such that dimHeq/dimH is close to 1. We say that a system with state vector ψ ε H is in thermal equilibrium if ψ is 'close' to Heq. We argue that for 'typical' Hamiltonians, all initial state vectors ψ0 evolve in such a way that ψt is in thermal equilibrium for most times t. This is closely related to von Neumann's quantum ergodic theorem of 1929. [ABSTRACT FROM AUTHOR]