Low Density Density Derivative of the Self-Correlation Function of a Hard Sphere Gas

For the classical gas of hard spheres, exact expressions are derived for [∂Is(r,t)/∂n]n = 0, [∂Is(q,t)/∂n]n = 0, and [∂Ss(q,ω)/∂n]n = 0, the density derivatives of the Van Hove self-correlation functions. The relationships between the direct derivation using the activity expansion, and the derivations based on the generalized kinetic equation and the linearized Boltzmann equation are discussed. Properties of these density derivatives and of the corresponding self-correlation functions, as given by the first two terms of the density expansion, are discussed in detail. The expressions are compared with the hard sphere results of Desai and Nelkin. of Sears and of Mazenko et al.; and also with the predictions of the single relaxation time model and the Langevin diffusion model.