What do we really mean by microstates in statistical physics? After a not so good introduction to statistical mechanics during the school year, I am trying to read Reif's Fundamentals of Statistical and Thermal Physics to learn more.
In Chapter 2.3, Reif outlines the basic postulate of statistical mechanics, but he also makes the comments below.
I'm wondering three things:
(1) The implication from this paragraph seems to be that there is a degree of arbitrariness in terms of deciding on the unperturbed Hamiltonian (ie. sans interaction terms) which will be solved to get the approximate, unperturbed eigenstates Reif mentions. Is there some theorem which says, in the large N limit, that the "choice", if any, is irrelevant?
(2) Just to confirm, is what Reif outlines standard? That is, when we speak of microstates in general, are we referring to these "approximate quantum states"?
(3) When Reif speaks of being in an approximate state and transitions between approximate quantum states, how are those defined? After all, he notes that none of the copies of the system in the ensemble are actually in such a state, but we just say that they are. Taking my hint from his use of "complete set", do we perhaps say that a system is in a given approximate quantum state if that approximate quantum state (a member of the complete basis set) has the largest projection of the actual state thereon? And transitions then occur when the projection along another basis approximate quantum state becomes the largest?

Before discussing this nonequilibrium situation in greater detail, it is worth making a few comments about the nature of the states used in our theory to describe an isolated system of many particles. These states are not rigorously exact quantum states of the perfectly isolated system with all interactions between particles taken into account. It would be prohibitively complicated to attempt any such utterly precise description; nor does one have available sufficiently detailed information about a macroscopic system to make such a precise description of any experimental interest. Instead one describes the system in terms of some complete set of approximate quantum states which take into account substantially all of its predominant dynamical features without being rigorously exact. When the system is known to be in such a state at any one time, it will not remain in this state indefinitely. Instead,there exists a finite probability that the system will at some later time be found in some of the other approximate states accessible to it, the transitions to these other states being caused by the presence of small residual interactions between the particles (interactions not taken into account in defining the approximate quantum states of the system).

 A: I think the term approximate quantum state is misleading. What is meant here is that one uses an approximate Hamiltonian, neglecting small ("residual") interactions, but the microstates that one uses then in the calculations are the exact microstates of this Hamiltonian.
However, if we limited our description to this Hamiltonian only, the system would never come to the thermodynamical equilibrium - it would forever remain in the state where we put it. In the extreme case, if we place it in an eigenstate of the Hamiltonian, it would forever remain in this eigenstate. It is the small interactions, which are negligeable at any given moment, that lead to thermodynamic equilibrium.
This description is fairly standard and, importantly, not specific to quantum statistical physics. E.g., we would never get a Maxwell-Boltzmann distribution for a non-interacting gas, if we simply calculated the classical dynamics of its molecules. For this gas to come to a thermodynamic equilibrium we need to account for the collisions between the gas molecules themselves and the gas molecules and the walls of the container. Nevertheless, we do not need to include these interactions explicitly in the Hamiltonian.
In mathematical terms this amounts to an approximation, but in physical terms the arguments of the statistical mechanics are rigorous. This is an excellent example that physics is more than just first-principles derivations from the basic equations.
