The van der Waals limit for classical systems. I. A variational principle

We consider the thermodynamic pressurep(μ, γ) of a classical system of particles with the two-body interaction potentialq(r)+γvK(γr), where υ is the number of space dimensions, γ is a positive parameter, and μ is the chemical potential. The temperature is not shown in the notation. We prove rigorously, for hard-core potentialsq(r) and for a very general class of functionsK(s), that the limit γ→0 of the pressurep(μ, γ) exists and is given by where the limit and the supremum can be interchanged. Here ℛ is a certain class of nonnegative, Riemann integrable functions,D is a cube of volume |D|, anda0(ϱ) is the free energy density of a system withK=0 and density ϱ. A similar result is proved for the free energy.