= Exchange contribution to the second virial coefficient
{title2=$B_2^{\mathrm{ex}}=-\eta\lambda^3/2^{5/2}$}
For a single internal species of noninteracting particles with quadratic dispersion in three dimensions, expand the <grand canonical partition function> at small <quantum gas fugacity>. With $\eta=+1$ for <bosons> and $\eta=-1$ for <fermions>, $n\lambda^3=z+\eta z^2/2^{3/2}+O(z^3)$ and $\beta p\lambda^3=z+\eta z^2/2^{5/2}+O(z^3)$. Eliminating $z$ gives $p=nk_BT[1-\eta n\lambda^3/2^{5/2}+\cdots]$. Bosonic bunching lowers the <pressure> at fixed density; fermionic exclusion raises it. This correction needs no interaction potential. For $g$ equally populated internal species, the coefficient in terms of total <number density> is reduced by $g$. In the dilute one-species <limit> the equal-time <pair detection correlation> is $g^{(2)}(r)=1+\eta e^{-2\pi r^2/\lambda^2}$. The exchange contribution to the two-particle cluster <integral> is therefore $-\tfrac12\int[g^{(2)}(r)-1]d^3r=-\eta\lambda^3/2^{5/2}$, providing a direct pair-counting check of the coefficient.
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