the contribution of the hard-core interaction, and $$b_2^0=\frac{1}{N} \left[\mathop{\rm Tr}\nolimits_2 e^{-\beta(
{\cal H}_0-2\Delta)}-
\frac{1}{2} \left(\mathop{\rm Tr}\nolimits_1 e^{-\beta({\cal H}_0
-\Delta)}\right)^2\right]
\label{b20}$$ as the effect of the Bose statistics of the vacancies. The combination $$b_2=b_2^0+b_2^{\rm int}
\label{b2tot}$$ yields the total effect.

The statistical effects are trivial to calculate, as the unperturbed grand partition function is given by $$\ln \Xi^0 =-\sum_{\vec{k}} \ln\left(1-e^{-\beta(\varepsilon(\vec{k})
+\Delta
)}\right) = N \sum_{\ell=1}^{\infty} b_{\ell}^0 e^{-\ell
\beta \Delta},
\label{xi0}$$ such that $$b_{\ell}^0=\frac{1}{N\ell} \sum_{\vec{k}} e^{-\ell\beta
\varepsilon(\vec{k})}
=\frac{1}{\ell} b_1(\beta\ell),
\label{bl0}$$ with $b_1(\beta)$ given by (\[b1\]). Thus it suffices to focus our attention on the calculation of $b_2^{\rm int}$. As a general observation we note that the two contributions in (\[b2tot\]) will have opposite signs. To see this more clearly, we go over to a series in the density $n$ of the vacancies, $$n=\frac{1}{N v_0} \langle \sum_i b_i^{\dagger}b_i \rangle =
-\partial p/
\partial \Delta =\frac{1}{v_0} \sum_{\ell=1}^{\infty} \ell b_{\ell}
e^{-\ell \beta
\Delta}.
\label{dens}$$ Eliminating $e^{-\beta\Delta}$ from (\[vircs\]) and (\[dens\]) we obtain a virial expansion for the pressure $$\beta p = n -(b_2/b_1^2) v_0 n^2 + \ldots .
\label{pexp}$$ So at fixed density $n$ the statistical effects lower the pressure as $b_2^0 $ is positive according to (\[bl0\]). The hard-core repulsion can only increase the 