Fully Renormalized Kinetic Theory. II. Velocity Autocorrelation

In a previous paper a general formalism was developed for treating the time-dependent correlation functions that arise in the theory of self-diffusion. In this paper this formalism is used in conjunction with the approximation where the two-particle source is approximated by a sum of one-particle sources. This approximation follows from physical arguments and from an analysis of the exact equation for the two-particle source. The resulting expression for the memory function is similar to that found previously by Pomeau and is related to the ring terms studied by Kawasaki and Oppenheim. It is further shown that this correction to the Boltzmann-Enskog memory function can be written in terms of a product of phase-space correlation functions. This theory, to the extent that the hydrodynamical projection onto these correlation functions is dominant, provides a microscopic basis for the various mode-mode coupling theories. The associated long-time behavior of the velocity-autocorrelation function is shown to go as $t^{\frac{-3\mathrm{}}{2}}$ and the coefficient agrees with that found by Dorfman and Cohen for low densities. For higher densities there are differences. It is further demonstrated how one can remove the wave-number cutoffs used in other theories, and the velocity-autocorrelation function is calculated, in a particular approximation, over the complete range of times with no adjustable parameters.