2014/02/14 by Sheldon Goldstein, Goldstein, Sheldon, Takashi Hara +3 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Mechanics and Applications #Quantum Physics (quant-ph) #Quantum many-body systems #Spectral Theory in Mathematical Physics #Statistical Mechanics (cond-mat.stat-mech) #cond-mat.stat-mech #math-ph #math.MP #quant-ph
paper · pdf · doi:10.48550/arxiv.1402.3380
37 pages, 1 figure. Many improvements have been made in version 2; appendix A is new
openalex publication_date 2014/02/14 · arxiv created 2014/05/16 · arxiv updated 2014/05/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the problem of the approach to equilibrium in a macroscopic quantum system in an abstract setting. We prove that, for a typical choice of "nonequilibrium subspace", any initial state (from the energy shell) thermalizes, and in fact does so very quickly, on the order of the Boltzmann time τ_B:=h/(kBT). This apparently unrealistic, but mathematically rigorous, conclusion has the important physical implication that the moderately slow decay observed in reality is not typical in the present setting. The fact that macroscopic systems approach thermal equilibrium may seem puzzling, for example, because it may seem to conflict with the time-reversibility of the microscopic dynamics. According the present result, what needs to be explained is, not that macroscopic systems approach equilibrium, but that they do so slowly. Mathematically our result is based on an interesting property of the maximum eigenvalue of the Hadamard product of a positive semi-definite matrix and a random projection matrix. The recent exact formula by Collins for the integral with respect to the Haar measure of the unitary group plays an essential role in our proof.