On the time scales in the approach to equilibrium of macroscopic quantum systems

It is likely that the key quantity N ⁡ ( ε ) / d eff N(\varepsilon)/d_{\mathrm{eff}} in [ 14 ] is of order e const . V e^{\mathrm{const.}V} for large V V . See also the remark in [ 8 ]

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Our result is also directly relevant to the approach to the second law of thermodynamics by using the idea of “waiting time”. See H. Tasaki, arXiv:cond-mat/0011321v2

Original

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By f ⁡ ( V ) ∼ g ⁡ ( V ) f(V)\sim g(V) , we mean V − 1 ​ log ⁡ { f ⁡ ( V ) / g ⁡ ( V ) } → 0 V^{-1}\log\{f(V)/g(V)\}\to 0 as V ↑ ∞ V\uparrow\infty

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Consider the simplest situation where one is interested in the behavior of a single macroscopic quantity A ^ \hat{A} , whose equilibrium value is A ¯ \bar{A} . Then one can define ℋ eq {\cal H}_{\mathrm{eq}} as the subspace spanned by the eigenstates of the nonnegative operator ( A ^ − A ¯ ) 2 (\hat{A}-\bar{A})^{2} corresponding to sufficiently small eigenvalues

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Let P ^ ℋ neq \hat{P}_{{\cal H}_{\mathrm{neq}}} be the projection onto ℋ neq {\cal H}_{\mathrm{neq}} , and consider the expectation value ⟨ φ | P ^ ℋ neq | φ ⟩ \langle\varphi|\hat{P}_{{\cal H}_{\mathrm{neq}}}|\varphi\rangle . By taking the uniform average over all normalized | φ ⟩ ∈ ℋ |\varphi\rangle\in{\cal H} , we get ⟨ φ | P ^ ℋ neq | φ ⟩ ¯ = d neq / D ≪ 1 \overline{\langle\varphi|\hat{P}_{{\cal H}_{\mathrm{neq}}}|\varphi\rangle}=d_{\mathrm{neq}}/D\ll 1 . From the standard argument based on the Markov inequality (see, e.g., [ 6 , 7 ] ), we find that ⟨ φ | P ^ ℋ neq | φ ⟩ ≪ 1 \langle\varphi|\hat{P}_{{\cal H}_{\mathrm{neq}}}|\varphi\rangle\ll 1 for a typical | φ ⟩ ∈ ℋ |\varphi\rangle\in{\cal H} . That ⟨ φ | P ^ ℋ neq | φ ⟩ ≪ 1 \langle\varphi|\hat{P}_{{\cal H}_{\mathrm{neq}}}|\varphi\rangle\ll 1 implies | φ ⟩ |\varphi\rangle is very close to ℋ eq {\cal H}_{\mathrm{eq}} and hence represents the equilibrium state

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In the simplest example discussed in [ 19 ] , one can show that the subspace ℋ eq {\cal H}_{\mathrm{eq}} occupies most of ℋ {\cal H} if the microcanonical average of ( A ^ − A ¯ ) 2 (\hat{A}-\bar{A})^{2} is small. If it happens to be the case that ℋ eq {\cal H}_{\mathrm{eq}} does not

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