A Diagrammatic Kinetic Theory of Density Fluctuations in Simple Liquids in the Overdamped Limit. I. A Long Time Scale Theory for High Density

Abstract

Starting with a formally exact diagrammatic kinetic theory for the equilibrium correlation functions of particle density and current fluctuations for a monatomic liquid, we develop a theory for high density liquids whose interatomic potential has a strongly repulsive short ranged part. We assume that interparticle collisions via this short ranged part of the potential are sufficient to randomize the velocities of the particles on a very small time scale compared with the fundamental time scale defined as the particle diameter divided by the mean thermal velocity. When this is the case, the graphical theory suggests that both the particle current correlation functions and the memory function of the particle density correlation function evolve on two distinct time scales, the very short time scale just mentioned and another that is much longer than the fundamental time scale. The diagrams that describe the motion on each of these time scales are identified. When the two time scales are very different, a dramatic simplification of the diagrammatic theory at long times takes place. We identify an irreducible memory function and a more basic function, which we call the irreducible memory kernel. This latter function evolves on the longer time scale only and determines the time dependence of the density and current correlation functions of interest at long times. In the following paper, a simple one-loop approximation for the irreducible memory kernel is used to calculate correlation functions for a Lennard-Jones fluid at high density and a variety of temperatures.