Single-level grand partition function in the dilute limit (${\Xi }_k=1+x_k+O(x_k^2),x_k=e^{-\beta ({ϵ}_k-\mu )}$)

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.