Asymptotic behavior of the radial distribution function

The pair distribution function in a uniform classical fluid is equivalent to the one-body density when one particle is fixed. An implicit relation for this nonuniform density is found by a functional expansion of the difference of chemical potential and external potential about its value for a system of uniform density. A linearization of this expansion, followed by retention of, at most, second derivatives of the inhomogenity, reproduces the Ornstein-Zernicke relations for the asymptotic pair correlation. Linearization alone calls for the sum of internal potential and direct correlation function to vanish asymptotically. This relation is developed for the case of weak long-range forces, resulting in the Debye-Huckel expression for an electron gas, and reproducing the asymptotic correlations of the Kac-Uhlenbeck-Hemmer one-dimensional model. The relation is also shown to follow from the virial expansion for the direct correlation function.