Grand canonical partition function ($\mathcal{Z}$)

The grand canonical partition function is

For one independent fermion level, occupations 0 and 1 form a Bernoulli distribution and give $\mathrm{Var}n=\bar{n}(1-\bar{n})$. For one ideal boson level in contact with a reservoir, its geometric distribution gives $\mathrm{Var}n=\bar{n}(1+\bar{n})$. Both approach variance equal to mean at small occupation. Pauli exclusion suppresses fluctuations of nearly full fermion states, whereas large boson occupation gives variance of order mean squared. Fixed-total-number constraints change condensate fluctuation statements.

For a single one-particle state with $x_k\ll 1$, the empty and singly occupied contributions dominate. A fermion state gives ${\Xi }_k=1+x_k$, a boson state gives $(1-x_k{)}^{-1}$, and Maxwell–Boltzmann statistics give $e^{x_k}$. Thus all have the same leading dilute response, $\ln {\Xi }_k=x_k+O(x_k^2)$. This approximation does not impose an exclusion principle on classical particles.