In statistical mechanics, the distribution of interparticle separations determines the radial distribution function [1].

This Demonstration shows the radial distribution function of a three-dimensional liquid composed of identical hard spheres of diameter , making use of the exact solution of the Percus–Yevick integral equation [2, 3]. You can vary the packing fraction of the liquid, that is, the fraction of the total volume occupied by the spheres themselves. For a packing fraction greater than about 0.49, the corresponding can be regarded as that of a metastable liquid since the true stable phase is a crystal.

The Demonstration also includes three functions directly related to the radial distribution function : the structure factor , the direct correlation function [6], and the bridge function [7].

The exact solution of the Percus–Yevick integral equation for hard spheres was first obtained in [2] and [3]. The entirely analytic forms for the static structure factor were first presented in [4].

[2] M. S. Wertheim, "Exact Solution of the Percus–Yevick Integral Equation for Hard Spheres," Physical Review Letters, 10(8), 1963 pp. 321–323. doi:10.1103/PhysRevLett.10.321.

[3] E. Thiele, "Equation of State for Hard Spheres," The Journal of Chemical Physics, 39(2), 1963 pp. 474–479. doi:10.1063/1.1734272.

[4] N. W. Ashcroft and J. Lekner, "Structure and Resistivity of Liquid Metals," Physical Review, 145(1), 1966 pp. 83–90.