Analytic Solution of the Percus-Yevick Equation

The properties of the Percus-Yevick approximate integral equation for the pair distribution function in classical statistical mechanics are examined for the class of pair potentials consisting of a hard core plus a short-range tail. For one-dimensional systems, some elementary theorems of complex variable applied to the Laplace-transformed equations enable one to express the direct correlation function in a very simple form, one which becomes explicit and trivial in the absence of a short-range tail. In the presence of the tail, the direct correlation function satisfies a (coupled) integral equation over a finite domain. The impossibility of a phase transition in one dimension is strongly indicated. Analysis of the case of three dimensions proceeds similarly, but is complicated by the appearance of essential parameters other than the density and compressibility. The character of the direct correlation function is qualitatively unchanged. Principal differences in three dimensions are that a phase transition is no longer prohibited, and the pair distribution function cannot be reasonably expressed as a sum of nth-neighbor contributions.