Density expansion of the correlation function of a hard sphere gas

Abstract

Using the grand canonical ensemble, the classical Van Hove correlation function G(r, t) is expanded in a power series in density. The zero density limit is the ideal gas result. We have derived, for a classical gas of hard spheres, exact expressions for [Formula: see text], the zero density derivative of the correlation function, and its Fourier transforms. These involve only two particle dynamics. The first two terms in the density expansions provide representation of the correlation functions for appropriate ranges of density and correlation function arguments. We also show that the same result can be obtained from generalized kinetic equations. To this order in density, the moment relations and the time derivatives of I(q, t) at t = 0+ are satisfied. Numerical results are compared with those of Mazenko, Wei, and Yip and with those of the Boltzmann equation and they show the expected behavior.