Rigorous Treatment of the Van Der Waals-Maxwell Theory of the Liquid-Vapor Transition

Rigorous upper and lower bounds are obtained for the thermodynamic free-energy density a(ρ, γ) of a classical system of particles with two-body interaction potential q(r) + γνφ(γr) where ν is the number of space dimensions and ρ the density, in terms of the free-energy density a0(ρ) for the corresponding system with φ(x) ≡ 0. When φ(x) belongs to a class of functions, which includes those which are nonpositive and those whose ν-dimensional Fourier transforms are nonnegative, the upper and lower bounds coincide in the limit γ → 0 and limγ → 0 a(ρ, γ) is the maximal convex function of ρ not exceeding a0(ρ) + ½αρ2, where α ≡ ∫ φ(x) dx. The corresponding equation of state is given by Maxwell's equal-area rule applied to the function p0(ρ) + ½αρ2 where p0(ρ) is the pressure for φ(x) ≡ 0. If a0(ρ) + ½αρ2 is not convex the behavior of the limiting free energy indicates a first-order phase transition. These results are easily generalized to lattice gases and thus apply also to Ising spin systems. The two-body distribution function is found, in the limit γ → 0, to be normally identical with that for φ(x) ≡ 0, but if the system has a phase transition it has the form appropriate to a two-phase system. Some of the upper and lower bounds on a(ρ, γ) are simple enough to be useful for finite γ. Also, some of our results remain valid for quantum systems.