Thermodynamic self-consistency for hard spheres at low density

We consider the conditions on the triplet-distribution function g ⁽³⁾, and the pair-distribution g⁽²⁾, sufficient that thermodynamic quantities calculated from g⁽²⁾ be state functions. We show that g⁽³⁾ cannot be, even to first order in density, equivalent to the Kirkwood superposition approximation. We obtain a g⁽³⁾ that does yield a thermodynamically self-consistent g⁽²⁾, to second order in density. We show that this g⁽²⁾ differs little from the exact g⁽²⁾, at that order. We also examine the low-density behavior of a partially self-consistent closure in which angular correlations appear explicitly. The latter closure is found to describe three-particle correlations well when two of the particles are in contact. A principal conclusion which follows from these studies is that the criterion of thermodynamic self-consistency is useful as a constraint in constructing closures, although it does not in itself determine a closure.