See for example this question or the answers to this question, neither of which seem to say exactly why the system Hamiltonian eigenstates (and not some other complete basis) are used. It is presumably something that baffles every person learning elementary statistical physics for the first time: why are the "accessible states" always (approximate) eigenstates of the system Hamiltonian? Why are superpositions of these eigenstates not admissible, or other bases entirely?

I cannot seem to find the passage now, but in reading Schlosshauer's textbook on Decoherence I seem to remember him discussing how, in the thermodynamic limit, decoherence's resolution of the problem of the preferred basis (that basis which does not change during entanglement with the environment) implies as a theorem that system eigenstates are selected. Is this generally the underlying reason that, in statistical physics, we only consider system eigenstates? If not, what is said justification or where can I find it?

See for example this question or the answers to this question, neither of which seem to say exactly why the system Hamiltonian eigenstates (and not some other complete basis) are used. It is presumably something that baffles every person learning elementary statistical physics for the first time: why are the "accessible states" always (approximate) eigenstates of the system Hamiltonian? Why are superpositions of these eigenstates not admissible, or other bases entirely?

I cannot seem to find the passage now, but in reading Schlosshauer's textbook on Decoherence I seem to remember him discussing how, in the thermodynamic limit, decoherence's resolution of the problem of the preferred basis (that basis which does not change during entanglement with the environment) implies as a theorem that system eigenstates are selected. Is this generally the underlying reason that, in statistical physics, we only consider system eigenstates? If not, what is said justification or where can I find it?