Analytic Approach to the Theory of Phase Transitions

Abstract

An approximation to the first equation of the Kirkwood coupling parameter hierarchy and other model equations for the singlet distribution function are cast into the standard Hammerstein form of nonlinear integral equation. We give a criterion for the existence and uniqueness of solutions of this equation involving the first negative eigenvalue of the kernel, which allows us to establish temperatures and densities where the solution is unique. Multiple solutions of the nonlinear equation are associated with instability of the single phase and thus signal a phase transition. A necessary condition for the existence of other solutions of small norm is given by a bifurcation equation. These new solutions are associated with the freezing transition, and the periodic singlet density of the solid falls naturally out of the theory. The bifurcation equation can be related to the Kirkwood instability criterion, but, in contrast to this, predicts no transition for a system of hard rods when a model kernel is used. This model, in an approximate numerical calculation, also predicts no transition for a system of hard spheres, in apparent agreement with Meeron's recent suggestion that systems with purely repulsive forces have no phase transitions.