Past exam of the thermal and statistical physics course of the University of Cambridge / 2021 / Section B / 5 / b

For $N$ indistinguishable classical particles the canonical partition function is an integral over phase space,The factor $1/N!$ removes overcounting under particle permutations, and $h^{3N}$ makes the classical phase-space measure dimensionless. The pair potential depends only on positions, so all momentum integrations separate. The three-dimensional Gaussian integral for each particle givesThe grand canonical partition function is $\Xi ={\sum }_{N=0}^{\infty }{e}^{\beta \mu N}{Z}_N$, where chemical potential $\mu$ controls the mean number of particles. ThereforeFor $N=0$, the empty products and configurational integral equal one. The factor $1/2$ in the interaction prevents double counting of unordered particle pairs.