Triplet correlations in the hard rod fluid: A test for topological reduction of graph-theoretic corrections to the superposition approximation

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Abstract
Two series expansions for the triplet correlation function, which have been used previously to study three-dimensional liquids, are evaluated in a case where the exact triple correlation function is known, namely, hard rods in one dimension. These series are studied in the context of the Yvon–Born–Green (YBG) integral equation. The coefficients in the f-bond series are evaluated analytically, but the resultant corrections to the superposition approximation are minor. In contrast, the coefficients of the h-bond series, which are calculated numerically, provide an accurate approximation to the triplet correlation function for densities of interest below two-thirds of the close-packed density. The validity of the ‘‘scaling’’ approximation of the h-bond series, which has been used in theories of quantum liquids, is also examined, and these calculations are shown to be relevant to earlier studies of three-dimensional liquids.