Time Scales in the Approach to Equilibrium of Macroscopic Quantum Systems

We prove two theorems concerning the time evolution in general isolated quantum systems. The theorems are relevant to the issue of the time scale in the approach to equilibrium. The first theorem shows that there can be pathological situations in which the relaxation takes an extraordinarily long time, while the second theorem shows that one can always choose an equilibrium subspace, the relaxation to which requires only a short time for any initial state.

Figure 1

Left: the basis state $|{\xi }^{(\nu )}⟩$ of ${\mathcal{H}}_1$ is in a narrow energy interval of width $\Delta U/L$. Right: the basis state $|{\xi }^{(\nu )}⟩$ of ${\mathcal{H}}_2$ spreads sparsely over the whole energy range.