A new geometrically based integral equation hierarchy for hard-sphere systems

Abstract

A new hierarchy of integral equations for the background correlation functions of hard spheres is derived using geometrical and physical arguments. The theory is distinct from, but has links with, other theories and results, including scaled-particle theory, zero-separation theorems, and the Born-Green-Yvon hierarchy. Three closure approximations for the hierarchy are proposed and their results examined for the equation of state. The zeroth-order approximation leads to a simple algebraic equation for the compressibility factor as a function of density and gives exact second and third virial coefficients. The first-order approximation also gives an exact fourth virial coefficient, and the second-order approximation gives an exact fifth virial coefficient. The second-order approximation yields compressibility factors much more accurate than any other available first-principles theory. Furthermore, at the highest fluid densities it is essentially as accurate as the Carnahan-Starling equation of state, and at medium densities it is more accurate.