For n = 1 , we have the one-particle density. For a crystal it is a periodic function with sharp maxima at the lattice sites. For a non-interacting gas, it is independent of the position r 1 and equal to the overall number density, ρ ρ , of the system. To see this first note that U N = ∞ ∞ in the volume occupied by the gas, and 0 everywhere else. The partition function in this case is