Separation of the Interaction Potential into Two Parts in Treating Many-Body Systems. I. General Theory and Applications to Simple Fluids with Short-Range and Long-Range Forces

Systematic methods are developed for investigating the correlation functions and thermodynamic properties of a classical system of particles interacting via a pair potential v(r) = q(r) + w(r). The method is then applied to the case in which w(r) is a ``Kac potential'' w(r, γ) = γvφ(γr) (v the dimensionality of the space) whose range γ−1 is very long compared to the range of q(r). Our work is related closely to the work of Kac, Uhlenbeck, and Hemmer. The main new feature of our method is the separation of the correlations, e. g., the two-particle Ursell function ℱ(r), into a short-range part ℱs(r, γ) and a long-range part ℱL(y, γ), y ≡ γ r; r the distance between the particles. The two parts of ℱ are defined in terms of their representation by graphs with density (or fugacity) vertices and K-and φ-bonds, K(r) = e−-βq − 1, Φ = −βw. A resummation of these graphs then yields a simple graphical representation for the long-range part of the correlation functions in terms of graphs with φ-bonds and ``hypervertices'' made up of the short-range part of the correlations. This representation is then used in this paper to make separate expansions of ℱs(r, γ) and ℱL(y, γ) and through them of the thermodynamic parameters in powers of γ. Explicit calculations of the Helmboltz free energy is carried out to a higher order in γ than done previously by Hemmer and it is shown how to carry out the calculation, in principle, to any order. The general method is further applied (in separate articles) to lattice gases, plasmas, and to the special problem of critical phenomena.