On the Approach to Thermal Equilibrium of Macroscopic Quantum Systems

Abstract

We consider 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 + delta E. The thermal equilibrium macro-state 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 in 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.

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References in corpus (4)

• Thermalization and its mechanism for generic isolated quantum systems
• Localization of interacting fermions at high temperature
• Foundation of Statistical Mechanics under experimentally realistic conditions
• From Time-symmetric Microscopic Dynamics to Time-asymmetric Macroscopic Behavior: An Overview

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