Series solution of the HNC and PY equations: The simple chain recursion way

The function tHNC(1,2) that appears in the HNC equation when written in the form tHNC(1,2) =ln gHNC(1,2)+βu (1,2) can be analyzed by graphical expansion and topological reduction to obtain a series solution of the form tHNC(1,2) =A1(1,2) +A2(1,2)+⋅⋅⋅, where each An is graphically represented by a simple chain sum. A similar analysis yields a solution of the same form for the function tPY(1,2) that appears in the PY equation when written in the form tPY(1,2) =gPY(1,2) eβu(1,2)−1. The graphs in a given An are formed recursively from those of lower n. Truncation of the series at different n values provides an infinite number of well-defined approximations to the pair correlation function g (1,2). The series method also represents an efficient computational method for solving the HNC and PY equations that offers some advantages over methods involving the iterative or variational solution of the integral equations. The whole procedure is readily adapted to systems with long-range forces, systems with several species of particles, and to systems suitably related to some reference system with known correlation functions.