[QUE/SM-04010] SM-Problem

Consider N point particles inside a box of volume V and at temperature T. The energy function is given by $$ H\,=\,\frac{p^2}{2M}\,+\,V(|\vec{r}_i\,-\,\vec{r}_j|) $$ where $$ V(r) \,=\,0 \qquad\rm{for}\,r\,>\,a$$ $$ =\,\infty\qquad \rm{when}\, r\,<\,a $$ Calculate the canonical partition function $Z_c$ and show that $$ Z_c\,=\,\frac{1}{N!}\left(\frac{2\pi m k T}{h^2}\right)^{3/2} Q $$ where $$Q\,=\,V^N\left(1\,-\,\frac{4\pi a^3}{3}\frac{N(N+1)}{2V}\right) $$ with correction of order $V^{(N-2)}$. Evaluate the corrections to $PV\,=\,NkT$ to the next non vanishing order.

HSMani