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I was looking at the derivation(s) of Fermi-Dirac Statistics by means of the "most probable distribution" (I know the correct way is to use ensembles, but my point is related to this derivation) and it usually employs Lagrange multipliers and Stirling's approximation on the factorials of the ocupation numbers "n_i".

So I would say that this is not correct since, even if you assume n_i to be continuous, the value for "n_i" has to be lower than 1 because of Pauli's principle. Then to make the approximation that "log(n_i!) ~ n_i * log(n_i) - n_i" can not be right!

However it is ussualy done that way in most textbooks. What would you suggest as an alternative to derive Fermi-Dirac Statistics most probable distribution?

thanks,

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# Stirling's approximation in Fermi Statistics derivation

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