On the interpretation of radial distribution functions determined from integral equations

Abstract

Nonlinear integral equations are commonly used as a basis for determining radial distribution functions in equilibrium fluids. We show that a solution to such an equation does not provide sufficient information to fix a unique equilibrium state because no explicit external field is prescribed. It follows that many physically distinct interpretations may be associated with each numerically generated solution. This is illustrated by showing that a family of physically distinct external fields can be associated with the same integral equation. In this context the occurrence of multiple solutions and the occurrence of solutions with long-range oscillations—both formally impossible events in the context of a well-posed problem—can be given a reasonable interpretation, for external fields in which these solutions are in fact appropriate can be identified. This shows, however, that the physical interpretation of numerically generated radial distribution functions is nontrivial. The problems are illustrated explicitly with a simple model integral equation which does exhibit multiple solutions and long-range solutions even though these are absent in the system the integral equation is intended to model.