VAN-DER-WAALS LIMIT FOR CLASSICAL SYSTEMS .1. A VARIATIONAL PRINCIPLE

We consider the thermodynamic pressure p(μ, γ) of a classical system of particles with the two-body interaction potential q(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 potentials q(r) and for a very general class of functions K(s), that the limit γ→0 of the pressure p(μ, γ) exists and is given by[Figure not available: see fulltext.] 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|, and a0(ρ{variant}) is the free energy density of a system with K=0 and density ρ{variant}. A similar result is proved for the free energy. © 1969 Springer-Verlag.