Functional derivative relations for a finite non-uniform molecular fluid in the canonical ensemble

Abstract

Functional derivative relations are developed for a finite non-uniform molecular fluid in the canonical ensemble. The total correlation function h(12) satisfies a normalization condition that renders the corresponding response kernel singular. Thus the inverse response kernel does not exist, and the direct correlation function cannot be defined in terms of h(12) by the Ornstein-Zernike equation. This difficulty is circumvented by subtracting the finite volume correction term from h(12) to obtain a modified total correlation function ĥ(12). A simple derivation of the form of this correction term is given; the result confirms and generalizes an earlier result of Lebowitz and Percus. The Ornstein-Zernike equation defines the direct correlation function in terms of ĥ(12). The functional derivative relations then assume the familiar forms appropriate to an infinite system or in the grand canonical ensemble, but with h(12) replaced by ĥ(12) and with the external potential φ(1) replaced by φ(1) - μ, where μ is the chemical potential. The relation of this development to the use of the grand canonical ensemble is briefly discussed.