Nodal Expansions. III. Exact Integral Equations for Particle Correlation Functions

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Abstract
The density expansions of the pair distribution function and potential of average force are analyzed topologically in terms of cutting points and bifocal points. The analysis leads to conversion of the expansions into series with cluster integrals involving products of the total correlation functions, h(r) = g(r) − 1, at finite density, rather than the usual zero‐density Ursell f‐functions. An integral equation for the pair potential of average force and the pair distribution function is thus obtained. The equation is formally exact and closed in pair space, involving no triplet distributions such as occur in the treatments of Kirkwood and Yvon‐Born‐Green. Solution of the equation also yields directly the Ornstein‐Zernike direct correlation function. Equations for the free energy in terms of the direct correlation function are presented, thus providing a unified and self‐consistent treatment of all thermodynamic properties of a many‐body system. The relation of the new equation to the Ornstein‐Zernike theory of liquids and to phase transitions is discussed. The possibility of derivation for condensed phases is briefly noted. A simple approximation, involving only the convolutory terms in the cluster expansions of correlation functions, is proposed.

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