The influence of closure on the behaviour of the Yvon-Born-Green equation for a system of hard rods

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Abstract
The Yvon-Born-Green equation for a system of hard rods is solved using the Kirkwood superposition approximation and the results obtained are compared with those generated from the exact solution to the problem. Whereas the pair correlation function generated using the exact closure is unique over the entire range of densities, multiple solutions of the underlying integral equation appear above a certain density when the full Kirkwood superposition closure is used. The method of complementary variational principles is used to corroborate the findings in the pre-transition regime. The implications of these results are discussed in light of previously reported studies of the Yvon-Born-Green equation on systems of hard discs and hard spheres, where evidence of a fluid-solid transition was found.