An extended version of the hypernetted chain equation is presented. It allows for fluid inhomogeneity and for three-body interactions. Consistent expressions for thermodynamic functions are given. It is argued that thermodynamic consistency holds even though statistical mechanical relationships are not obeyed. Singularities in the hypernetted chain equation, related to loss of convexity of the associated free energy functional, prevent a complete description of liquid/vapor coexistence in a Lennard-Jones fluid.
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Schlijper et al. (1993) studied this question.
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