[QUE/SM-04011] SM-PROBLEM

Consider N classical particles of mass M confined to a volume V. Besides the kinetic energy ($\frac{p^2}{2M}$ for each particle) , there is a two particle interaction between the particles given by $$ V(r_{ij})\,=\,\frac{K}{r_{ij}^l} $$ Show that the canonical partition function scales as $$ Z_c(\xi T, \xi^{-3/l} V)\,=\,\xi^{3N(\frac{1}{2}-\frac{1}{l})}Z_c(T,V) $$ where $\xi$ is the scaling parameter. Using the above result show that the internal energy $U$ is given by $$ U\,=\,c_1pV\,+\,c_2 NkT $$ where $$c_1\,=\,\frac{3}{l}\qquad c_2\,=\, \frac{3(l\,-\,2)}{2 l}$$

HSMani