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\section{Aside: counting microstates -- thermodynamics versus statistical physics}
This section considers the difference between thermodynamic and statistical entropy: they are not the same quantities, and perhaps researchers are too keen to force them to be so. Not sure yet whether this is relevant to my task.
If you are not careful, answers you get for statistical physics entropy disagree with the thermodynamic result (the classic example is removing the partition between two identical boxes of gas). The resolution is usually to claim that this is evidence for QM: in QM, identical particles are indistinguishable, and this changes the multiplicity by a factor $1/N!$. An unacceptable fudge?
Do we believe this invocation of QM into a classical theory? It seems very {\it ad hoc}.Where else does it appear? What about phase-space quantisation, where a factor of $1/\hbar^3$ accompanies the integrals over phase space? Note that, according to Jaynes (`` The Gibbs Paradox'', 1992), Gibbs himself resolved the paradox without recourse to quantum mechanics. {\bf MORE}.
What happens for smaller $N$ when Stirling's approximation is not applicable?
Does the resolution even work for QM scenario? Are all identical particles in QM indistinguishable?
The macrostate of a system is ``the same'' after permutation of particles -- does this mean physical permutation (where particles follow a trajectory) or non-physical (where particles are switched in your mind)? Does it matter in the derivations? I think so, because deciding whether particles' history / capacity is important (i.e. do their trajectories matter, or just snapshots of the system?), tells us how to count physically different microstates. (Does physically different imply {\it statistical-physically} different?)
A microstate may not be physically accessible from current microstate (though it has the same state vectors) -- should ergodicity go unquestioned? Multiplicity should be the number of ways a microstate can be reached via a {\it physical} pathway.
In classical mechanics, I can label the particles 1, 2, ... If I draw on them, is this different from labeling them in my mind and keeping track with a videocamera? Where does entropy change come from in the latter case, given only I know about the labeling?
