For $\beta t \rightarrow 0$, this ratio goes to $-2$. The second virial coefficient $b_2=b_2^0+b_2^{\rm int}$ thus approaches $-b_2^0$ for high temperatures. This is indicative of fermionic behaviour, which is indeed what one would expect: at high temperatures, the only important contribution to the free energy of the system is the entropy involved in distributing a certain number of hard core particles over the lattice. This is the same as for fermions. For $\beta t \rightarrow \infty$ the behaviour of (\[ratio\]) depends on the behaviour of $F(\zeta)$ for $\zeta\rightarrow 1$. In two dimensions, $F(\zeta)\propto
(1-\zeta)/\ln(1-\zeta)$, which leads to $b_2^{\rm int}/b_2^0$ going to zero like $-1/\ln(\beta t)$ as $\beta t \rightarrow \infty$. The same low-temperature behaviour is found for quantum hard disks [@Schick]. In three dimensions, $F(\zeta) \propto (1-\zeta)^2$, which gives $b_2^{\rm int}/b_2^0 \propto -1/\sqrt{\beta t}$ for $\beta t
\rightarrow
\infty$, just as for quantum hard spheres [@Uhl]. So in both cases, $b_2$ approaches the value for the ideal Bose gas at low temperatures. This shows that when the thermal wavelength (\[lambda\]) exceeds the lattice constant $a(=1)$, the effects of the Bose statistics starts to dominate. Plots of $b_2/b_2^0$ versus $\beta t$ for two and three dimensions are given in figures 