Separation of the interaction potential into two parts in statistical mechanics. II. Graph theory for lattice gases and spin systems with application to systems with long-range potentials

The methods developed in a previous paper for treating systems with a pair potential of the form v(r) = q(r) + w(r) are here applied to lattice gases (isomorphic to Ising systems). We chose q(r) to be the "hard-core" potential preventing the multiple occupancy of a lattice site and w(r) the interaction between two particles (or parallel spins) separated by r. The resulting graphical formalism is similar to that obtained by other authors exclusively for spin systems. We are thus able to connect their work with the general Mayer theory as it was originally applied to fluids and also to find new interpretations for some of the quantities appearing in the spin-system expansion. The formalism is then used in the case where w(r) is a "Kac potential" of the form w(r, γ) ∼ γ^ν φ(γ r), where ν is the dimensionality of the space considered and γ^-1 is the range or w, assumed very large. We then obtain systematic expansions in y for the correlation functions and thermodynamic properties of the system. These expansions are, however, invalid inside the two-phase region and near the critical point of the "van der Waals" system; i.e., a system with γ → 0. To remedy this we introduce a new self-consistent type of approximation which is suggested by our graphical analysis of the γ expansion but is applicable also to systems with general interactions w(r), not necessarily parametrized by γ. The spatially asymptotic behavior of the two-body correlation function at the critical point is then discussed using these graphical methods. From the expansion procedures it seems possible to find specific subsets of graphs which will give any desired asymptotic behavior of the two-body correlation function including known exact ones. However, we could find no a priori reason for the retention of these subsets of graphs to the exclusion of all others.