14:24
@Relativisticcucumber there is an element of being mysterious there, but miao miao is not cloaking behind a whole mountain worth of mysteriousness. It should be understandable as it is written.
@SillyGoose but it is already reasonable...
@SillyGoose this is so funny, that miao miao translated it to a friend, and he still managed to catch the joke.
@SillyGoose This is actually a wrong statement. Even in a system that ought not to have BEC, as $T\to0$ the occupation of the ground state ought to $\to N$, simply the case for any system following Bose-Einstein statistics (or Maxwell-Boltzmann, though that does not actually exist IRL). Only Fermi-Dirac systems will behave differently.
@SillyGoose what new rule?
@Relativisticcucumber no, stat therm is a very well-established theory and everything had been reduced to maths. If there is a theory of physics whereby you can ask every question you wish to ask of it, it ought to be stat therm.
@Relativisticcucumber This is actually a pretty bad answer.
Anyway, my answer to the wacky fowl's question would be the following: You have failed to remember an extremely important fact: Thermodynamics is only well-defined and well-established in the "thermodynamic limit". That is, one first starts by proving that, in the thermodynamic limit, microcanonical, canonical, and grand canonical ensembles always agree in all their predictions. In that way, you can then swap questions that you want to ask in micro- or canonical ensembles, for questions in
the grand canonical ensemble, and obtain correct predictions. Since it is vastly easier to derive things in the grand canonical ensemble, it is thus the case that we would swap a fixed $N$ problem for a variable $N$ problem with fixed $\left<N\right>$, safe in the knowledge that, in the thermodynamic limit, the results are necessarily the same.
Remember that, unless we are talking about surface effects themselves, we are always dealing with thermodynamic systems by imagining an infinitely large system, i.e. the universe is just the infinitely large system we are considering, and then mentally carving out a volume precisely the same size as the actual system we are considering. Again, in the thermodynamic limit, the difference between the infinitely large system thusly considered, and the actual system, differ only by surface terms,
and the corrections matter less and less as the system size increases.
(to be fair, that was a good answer for the question it was trying to answer. It is a bad answer for the question the wacky fowl is asking.)