Original question:
Consider a physical system in canonical ensemble. The entropy is defined as $$S=-\sum_i p_i\ln p_i,$$ where each $p_i$ corresponds to the probability of each possible microstate in the canonical ensemble, proportional to $\exp(-\beta E)$.
I have heard that, in a reversible adiabatic process, which is a process in which the volume of the system changes very slowly, each $p_i$ is kept at a constant value. I also believe that in reversible adiabatic process, the overall "structure" of how the microstates are arranged, which is decided by a group of parameters, (e.g. $3N$ quantum numbers for $N$ non-interacting distinguishable ideal gas particles), does not change. This means that the degeneracy of each energy does not change.
But this seems to contradict the statement that each $p_i$ is constant: Suppose both the overall "structure" of microstates and each $p_i$ stay unchanged. Then for two microstates 1 and 2, we have $p_1=\exp(-\beta E_1)/Z$, and $p_2=\exp(-\beta E_2)/Z$. The preservation of overall state "structure" means that states 1 and 2 will continue to exist after the reversible adiabatic process, though with different energy values before and after the process. Then if $p_1$ and $p_2$ are kept unchanged, then $\beta (E_1-E_2)$ must not change, which I think is false. Where did I go wrong?
Remark 1:
I found out that for classical ideal gas (Maxwell-Boltzmann distribution), in reversible adiabatic process we do have $$TV^{2/3}=const,$$ so $\beta (E_1-E_2)$ is indeed a constant. However, for ideal Fermi and Bose gases, it seems that this relation does not hold.
Remark 2/Additional question:
I think the only thing we need to know here is that this is an adiabatic process: it's irrelevant which type of ensemble we chose for the system throughout the process. In fact, I think the word "ensemble" has lost its meaning here(see my comment about microcanonical and canonical ensembles). In my opinion, ensemble is only about finding the physical quantities of the (most probable) macrostate for a system at equilibrium: here, equilibrium means every instant of the quasistatic process. It's weird to me that "adiabatic" can connect the distribution of microstates between the ensembles of two different equilibrium states.
I've heard that if the system was in an eigenstate S before the adiabatic process, then after the process it should be in the eigenstate S' corresponding to S, though the energy levels have shifted. However, I don't understand how this correspondence leads to the statement that the overall structure of an ensemble of microstates is preserved during adiabatic process. In fact, I can't understand how an ensemble behaves under a real process. Can anyone explain it to me?
