Rigorous Statistical Mechanics for Nonuniform Systems

The thermodynamic limit of a classical system with many‐body interactions and under the influence of an external potential is investigated for the canonical ensemble. By scaling the external potential to a sequence of domains which are isotropic expansions of an initial domain confining the system, it is shown that the canonical free energy per particle has an infinite system limit. Moreover, by restricting consideration to internal interactions which have the property that separated groups of particles have negligible mutual attractive energy as the system becomes infinite, it is proven that the free energy per particle limit is precisely the free energy per particle obtained by minimizing the integral ∫[φρ + f(ρ, β)] with respect to all properly normalized functions ρ(r). φ is the external potential; f(ρ, β) is the free energy per unit volume for a uniform system of density ρ and inverse temperature β. The only technical complication is the above‐mentioned restriction on the allowed internal interaction...