On the Approach to Thermal Equilibrium of Macroscopic Quantum Systems

TL;DR

This paper demonstrates that for typical quantum Hamiltonians, most initial states of a macroscopic system evolve over time to be close to thermal equilibrium, supporting the quantum ergodic hypothesis.

Contribution

It establishes that most initial states in a macroscopic quantum system evolve towards thermal equilibrium, extending von Neumann's quantum ergodic theorem.

Findings

Most initial states become close to equilibrium for most times

The result applies to typical Hamiltonians with given eigenvalues

Supports the quantum ergodic hypothesis in macroscopic 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.