On the Equation of State of the Rigid-Sphere Fluid

A well-known integrodifferential equation due to Yvon, Born and Green, Kirkwood, and Bogolubov (YBGKB) connects the pair and triplet distribution functions g(2) and g(3) of a classical fluid. An expression for g(3) is needed in order that the equation may be solved for g(2). An exact formal expression is known: g(3)(123)=gs(3)(123) exp ∑ n+1∞ρnδn+3(123),where gs(3) is the superposition of the pair distribution functions, and the terms δn+3(123) in the exponent evaluate the correlations between particles fixed at 1, 2, and 3 and n other particles in the fluid. The first term δ4 is known for hard spheres, and we have numerically evaluated δ5 for three special configurations in order to approximate the entire series by the simplest Padé approximant, i.e., ρδ4/ (1—ρδ5/δ4). The YBGKB equation was solved using this approximation for g(3), and the pressure obtained from the contact value of g(2) is in almost perfect agreement with the molecular-dynamics data of Alder and Wainwright up to p/ρkT=6.80, which is the Kirkwood upper limit for stability of a fluid of hard spheres. The results obtained with the use of only δ4 in the expression for the triplet distribution function are also presented. The Kirkwood theory of the fluid—solid transition is briefly discussed with reference to the hard-sphere system.