Long-Time Behavior of Typical Pure States from Thermal Equilibrium Ensembles

Abstract

We consider an isolated macroscopic quantum system in a pure state ${\psi }_t$ evolving unitarily in a separable Hilbert space $\mathcal{H}$ and take for granted that different macro states $\nu$ correspond to mutually orthogonal subspaces ${\mathcal{H}}_{\nu }\subset \mathcal{H}$. Let $P_{\nu }$ be the projection to ${\mathcal{H}}_{\nu }$. It was recently shown that for all Hamiltonians with no highly degenerate eigenvalues and gaps most ${\psi }_0\in {\mathcal{H}}_{\mu }$ are such that for most $t\ge 0$, $\Vert P_{\nu }{\psi }_t{\Vert }^2$ is close to a $t$- and ${\psi }_0$-independent value $M_{\mu \nu }$ provided that $M_{\mu \nu }$ is not too small. Here, ``most'' refers to the uniform distribution on the sphere $\mathbb{S}({\mathcal{H}}_{\mu })$. 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 $\rho$ on $\mathcal{H}$, $\mathrm{GAP}(\rho )$ is the most spread out distribution on $\mathbb{S}(\mathcal{H})$ with density matrix $\rho$. We show that also for $\mathrm{GAP}(\rho )$-most ${\psi }_0\in \mathcal{H}$ for most $t\ge 0$, $\Vert P_{\nu }{\psi }_t{\Vert }^2$ is close to a fixed value $M_{\rho P_{\nu }}$ (which must not be too small). Moreover, we prove a generalization for certain operators $B$ instead of $P_{\nu }$ 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.

28 pages LaTex, no figures; v2 numerous improvements throughout the paper; final version

References in corpus (11)

• Long-Time Behavior of Macroscopic Quantum Systems: Commentary Accompanying the English Translation of John von Neumann's 1929 Article on the Quantum Ergodic Theorem
• Generalization of von Neumann's Approach to Thermalization
• Concentration of measure for quantum states with a fixed expectation value
• Typicality of pure states randomly sampled according to the Gaussian adjusted projected measure
• Why are macroscopic experiments reproducible? Imitating the behavior of an ensemble by single pure states
• Time Evolution of Typical Pure States from a Macroscopic Hilbert Subspace
• Nature abhors a vacuum: A simple rigorous example of thermalization in an isolated macroscopic quantum system
• Canonical Typicality For Other Ensembles Than Micro-Canonical
• Typical Macroscopic Long-Time Behavior for Random Hamiltonians
• Macroscopic Irreversibility in Quantum Systems: Free Expansion in a Fermion Chain
• Macroscopic Thermalization for Highly Degenerate Hamiltonians After Slight Perturbation