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

We consider an isolated macroscopic quantum system in a pure state ψt evolving unitarily in a separable Hilbert space H and take for granted that different macro states ν correspond to mutually orthogonal subspaces Hν⊂H. Let Pν be the projection to Hν. It was recently shown that for all Hamiltonians with no highly degenerate eigenvalues and gaps most ψ0∈Hμ are such that for most t ≥ 0, ‖Pνψt‖2 is close to a t-and ψ0-independent value Mμν provided that Mμν 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 ψ0∈H for most t ≥ 0, ‖Pνψt‖2 is close to a fixed value MρPν (which must not be too small). Moreover, we prove a generalization for certain operators B instead of Pν 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.