The Kirkwood–Salsburg equations for a bounded stable Kac potential. I. General theory and asymptotic solutions

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Abstract
We derive for Kac potentials of the form γsψ (λx) an expansion, for all the distribution functions, in powers of γs (s=dimension) and prove for z<?cr that the expansion is at least asymptotic. The coefficients in the expansion are shown to be solutions of linear operator equations similar to the Kirkwood–Salsburg equation. We also explicitly obtain a rather simple expression for the coefficients of γs and show that they are given by solving the Ornstein–Zernicke integral equation with the choice of −βψ (y) for the direct correlation function.