= Single-level grand partition function in the dilute limit
{title2=$\Xi_k=1+x_k+O(x_k^2),\quad x_k=e^{-\beta(\epsilon_k-\mu)}$}
For a single <one-particle state> with $x_k\ll1$, 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.
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