Past exam of the chemistry course of the year 4 of the natural sciences course of the University of Cambridge / 2022 / Paper 1 / 7 / d / Solution

For a fixed-$N,V,T$ canonical ensemble, sample the normalized Boltzmann distribution in phase space:After integrating out the momenta for an ordinary separable Hamiltonian mechanics, configurational sampling uses $\pi (\mathbf{r})\propto e^{-\beta U(\mathbf{r})}$. The correct equilibrium distribution is ensemble-dependent; the Boltzmann form is the canonical answer, rather than a universal prescription for every possible ensemble.