The approach to equilibrium in a macroscopic quantum system for a typical nonequilibrium subspace

摘要: We study the problem of approach to equilibrium in a macroscopic quantum system an abstract setting. prove that, for typical choice "nonequilibrium subspace", any initial state (from energy shell) thermalizes, and fact does so very quickly, on order Boltzmann time $\tau__\mathrm{B}:=h/(k_\mathrm{B}T)$. This apparently unrealistic, but mathematically rigorous, conclusion has important physical implication that moderately slow decay observed reality is not present The systems thermal may seem puzzling, example, because it conflict with time-reversibility microscopic dynamics. According result, what needs be explained is, equilibrium, they do slowly. Mathematically our result based interesting property maximum eigenvalue Hadamard product positive semi-definite matrix random projection matrix. The recent exact formula by Collins integral respect Haar measure unitary group plays essential role proof.