A variational approach to the statistical mechanics of hard discs and hard spheres

The theory of complementary variational principles is used to obtain analytic approximations to hard disc and hard sphere pair correlation functions in the high density regime. We study the Yvon-Born-Green non-linear integral equation (under the superposition closure) and draw upon earlier theoretical studies of this equation, particularly those which deal with the short range and asymptotic form of the YBG pair correlation function in d-dimensions, to motivate possible choices of the variational trial functions. These functions are optimized in accordance with extremum principles of the theory and the results are compared with existing numerical solutions of the YBG equation for hard discs and hard spheres. It is found that the analytic functions determined via the method of complementary variational principles are in excellent accord with the numerical data, even to the extent of indicating the onset of a fluid-periodic transition in the (very) high density regime. The relevance of this work to the problem of stability in hard disc and hard sphere systems is stressed in the concluding section of the paper.