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To derive Bose-Einstein and Fermi-Dirac distribution, we need to apply grand canonical ensemble:$Z(z,V,T)=\displaystyle\sum_{N=0}^{\infty}[z^N\sideset{}{'}\sum\limits_{\{n_j\}}e^{-\beta\sum\limits_{j}n_j\epsilon_j}]$. There is a constraint $\sideset{}{'}\sum\limits_{\{n_j\}}$ for quantum particles(bosons and fermions) in grand canonical ensemble:$\sum\limits_{j}n_j=N$, but why is there no such a constraint for classical particels?

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    $\begingroup$ Quite on the contrary, the grand canonical ensemble means that the total number of particles is left unconstrained and we specify the dual complementary variable, the chemical potential. That's true for Maxwell-Boltzmann, Fermi-Dirac, and Bose-Einstein - or another - distribution. I also have a trouble with your first sentence. $\endgroup$ – Luboš Motl Apr 6 '13 at 15:05
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For fermions, there is a constraint that each occupation number $n_i$ can only be either 0 or 1 because of the Pauli exclusion principle; no two fermions can occupy the same quantum state, but for bosons, there is no such constraint on the occupation numbers. For classical particles, namely those for which energy levels aren't quantized, there isn't a well-defined notion of occupation numbers.

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