Approach to thermal equilibrium of macroscopic quantum systems

2009/11/09 by Sheldon Goldstein, Joel L. Lebowitz, Christian Mastrodonato +3 · 10 citations
Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Quantum and electron transport phenomena #Quantum many-body systems #cond-mat.stat-mech #quant-ph

paper · pdf · doi:10.1103/physreve.81.011109

published as Phys. Rev. E 81: 011109 (2010) · 19 pages LaTeX, no figures

arxiv created 2009/11/09 · openalex publication_date 2010/01/07 · arxiv updated 2010/04/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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+\ensuremathδE. The thermal equilibrium macrostate at energy E corresponds to a subspace H<sub>eq</sub> of H such that dim H<sub>eq</sub>/dim H is close to 1. We say that a system with state vector \ensuremathψ∊H is in thermal equilibrium if \ensuremathψ is ``close'' to H_eq. We show that for ``typical'' Hamiltonians with given eigenvalues, all initial state vectors \ensuremathψ₀ evolve in such a way that \ensuremathψ_t is in thermal equilibrium for most times t. This result is closely related to von Neumann's quantum ergodic theorem of 1929.

Citations

• From Time-symmetric Microscopic Dynamics to Time-asymmetric Macroscopic Behavior: An Overview
• Chaos and Quantum Thermalization
• Foundation of Statistical Mechanics under experimentally realistic conditions
• Quantum statistical mechanics in a closed system

Cited by

• Equilibration of Small and Large Subsystems in Field Theories and Matrix Models
• Time Scales in the Approach to Equilibrium of Macroscopic Quantum Systems
• Subsystem eigenstate thermalization hypothesis
• Quantum Mechanics in a Time-Asymmetric Universe: On the Nature of the Initial Quantum State
• Pilot-Wave Quantum Theory with a Single Bohm’s Trajectory
• Dynamical Typicality for Initial States with a Preset Measurement Statistics of Several Commuting Observables
• Random State Technology
• Time’s Arrow in a Quantum Universe: On the Status of Statistical Mechanical Probabilities
• Typical fast thermalization processes in closed many-body systems
• Classical Lieb-Robinson Bound for Estimating Equilibration Timescales of Isolated Quantum Systems

Related

• The approach to equilibrium in a macroscopic quantum system for a typical nonequilibrium subspace
• Macroscopic Thermalization for Highly Degenerate Hamiltonians After Slight Perturbation
• The approach to thermal equilibrium and "thermodynamic normality" --- An observation based on the works by Goldstein, Lebowitz, Mastrodonato, Tumulka, and Zanghi in 2009, and by von Neumann in 1929
• Quantum mechanical evolution towards thermal equilibrium
• Heat flows from hot to cold: A simple rigorous example of thermalization in an isolated macroscopic quantum system