Approach to thermal equilibrium of macroscopic quantum systems

Sheldon Goldstein, Joel L. Lebowitz, Christian Mastrodonato, Roderich Tumulka, Nino Zanghi

Research output: Contribution to journalArticlepeer-review

109 Scopus citations


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+δE. The thermal equilibrium macrostate at energy E corresponds to a subspace Heq of H such that dim Heq /dimH is close to 1. We say that a system with state vector ψH is in thermal equilibrium if ψ is "close" to Heq. We show that for "typical" Hamiltonians with given eigenvalues, all initial state vectors ψ0 evolve in such a way that ψt is in thermal equilibrium for most times t. This result is closely related to von Neumann's quantum ergodic theorem of 1929.

Original languageEnglish (US)
Article number011109
JournalPhysical Review E - Statistical, Nonlinear, and Soft Matter Physics
Issue number1
StatePublished - Jan 7 2010

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Condensed Matter Physics


Dive into the research topics of 'Approach to thermal equilibrium of macroscopic quantum systems'. Together they form a unique fingerprint.

Cite this