On the Ornstein—Zernike relation, the BBGKY hierarchy and closures

We examine the Ornstein-Zernike relation for correlation functions and the Kirkwood hierarchy of integral equations for reduced distribution functions with a view toward obtaining closures. A generalized closure relation is thereby obtained which reduces to the conventional closures that give rise to the Percus-Yevick integral equation, the hypernetted chain integral equation, and the mean spherical approximation integral equation for the pair correlation function for a simple fluid, when different approximations are made to the closure obtained. An integral equation for the pair correlation function is proposed on the basis of the closure mentioned. In the sense that the aforementioned closure is inclusive of the Percus-Yevick, hypernetted chain, and mean spherical approximation integral equation, the new integral equation generalizes the conventional integral equations just mentioned.