Long-time behavior of typical pure states from thermal equilibrium ensembles

We consider an isolated macroscopic quantum system in a pure state ψ t evolving unitarily in a separable Hilbert space H and take for granted that different macro states ν correspond to mutually orthogonal subspaces H ν ⊂ H. Let P ν be the projection to Hν. It was recently shown that for all Hamiltonians with no highly degenerate eigenvalues and gaps most ψ 0 ∈ H μ are such that for most t ≥ 0, ‖P ν ψ t‖2 is close to a t-and ψ 0-independent value M μν provided that M μν is not too small. Here, “most” refers to the uniform distribution on the sphere S ( Hμ ). In the present work, we generalize this result from the uniform distribution, corresponding to the micro-canonical ensemble, to the much more general class of Gaussian adjusted projected (GAP) measures. For any density matrix ρ on H, GAP(ρ) is the most spread out distribution on S ( H ) with density matrix ρ. We show that also for GAP(ρ)-most ψ 0 ∈ H for most t ≥ 0, ‖P ν ψ t‖2 is close to a fixed value Mρ P ν (which must not be too small). Moreover, we prove a generalization for certain operators B instead of P ν and for finite times. Since certain GAP measures are quantum analogs of the (grand-)canonical ensemble, our result expresses a version of equivalence of ensembles.