The density of a nonuniform system in the thermodynamic limit

Abstract

We discuss the existence, continuity, and other properties of the canonical and grand canonical density distributions in the thermodynamic limit for nonuniform classical mechanical systems. For an external potential φ defined on a domain Λ the free energy per unit volume for fixed temperature is given by F(ρ₀,φ) = min ∫_Λ[ρ(x)φ(x) + f(ρ(x))]dx/V(Λ) where the minimum is over all density distributions satisfying the restriction of fixed average density ρ₀, and f(ρ₀,β) is the free energy per unit volume in the thermodynamic limit when φ = 0. We prove that if φ is not constant over any region of finite volume then the density distribution which minimizes is unique, and also that the density is the functional derivative of F(ρ₀,φ) with respect to φ. We also show that the density distribution of an infinite nonuniform system is the limit of density distributions associated with finite systems of increasing size.