We demonstrate the equilibration of isolated macroscopic quantum systems, prepared in non-equilibrium mixed states with significant population of many energy levels, and observed by instruments with a reasonably bound working range compared to the resolution limit.
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Though real systems are never strictly closed, our present standpoint (also supported by numerical experience) implies that this should not be the pivotal point: Otherwise, standard Quantum Mechanics alone would not be sufficient to model the (entangled) time evolution, and, on the other hand, ESM would not be applicable to strictly closed systems. After including the most relevant perturbations “from outside” into the considered system, it must be possible to theoretically model it as strictly isolated from the rest of the world, with a well-defined Hamiltonian H H , possibly not known in detail but with generic properties
Cited in the paper.
The inequality in ( 3
Cited in the paper.
This is a result of “elementary level counting” and as such not
Cited in the paper.
Since ρ ( t ) \rho(t) is non-negative and Hermitean, ( ψ , ϕ ) := ⟨ ψ | ρ ( t ) + ϵ | ϕ ⟩ (\psi,\phi):=\langle\psi|\rho(t)+\epsilon|\phi\rangle is a well defined scalar product for any ϵ > 0 \epsilon>0 . Hence, Cauchy-Schwarz’s inequality applies and by letting ϵ → 0 \epsilon\to 0 one obtains ( 6
Cited in the paper.
One readily sees that ( 8
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