A diagrammatic formulation of the kinetic theory of fluctuations in equilibrium classical fluids. IV. The short time behavior of the memory function

Abstract

Using a recently developed diagrammatic formulation of the kinetic theory of fluctuations in liquids, we investigate the short time behavior of the memory function for density fluctuations in a classical atomic fluid. At short times, the memory function has a large contribution that is generated by the repulsive part of the interatomic potential. We introduce a small parameter that is a measure of the softness of the repulsive part of the potential. The diagrams in the memory function that contribute to lowest order in that small parameter are identified and summed to give an explicit expression for the dominant contribution to the memory function at short times. The result leads to a theory for fluids with continuous potentials that is similar to the Enskog theory for hard sphere fluids.

A diagrammatic kinetic theory of density fluctuations in simple liquids in the overdamped limit. I. A long time scale theory for high density

“…In previous papers, [19][20][21][22][23][24] we have developed a diagrammatic theory of a hierarchy of time correlation functions. The theory defines retarded propagators χ and χ s for the correlation functions C and C s , respectively, 29 and expresses the C function for positive values of t − t in terms of the t = t value in the following way:…”

“…A Q c1 11 vertex has an internal line, which distinguishes it from a Q c0 11 , which does not. A Q c2 22 vertex has two internal lines that are not topologically equivalent, so one is drawn as a solid line and the other as a dashed line. (In the actual series, each left root is labeled (R, P, t), and each right root is labeled (R , P , t ), but these labels have been deleted from the figure for simplicity.)…”

“…We have formulated the starting point for the theory of the overdamped limit using vertices that have no such symmetry, and the symmetry number of every diagram is 1. Only two symmetric vertices were introduced later in the development, namely, the T R 22 and T H 22 vertices. Because of the other properties of graphs in which these vertices appear, the points in this vertex do not have the symmetry of the vertex itself, so the overall symmetry number of every graph is one.…”

“…The diagrammatic theory had the advantage of being able to represent both correlation functions and their memory functions in a unified way in terms of diagrams that could be ascribed straightforward physical interpretations thanks to the individual elements of the diagrams being relatively simple (even though the full diagrams in general were not). Andersen and co-workers [22][23][24] then used the physical intuition this theory afforded to derive several candidate graphical representations of the short time part of the memory function of the phase space density correlation function that could be evaluated numerically and used to compute the short time behavior of several correlation functions of interest. Each of these representations in turn implied an explicit diagrammatic series for the corresponding long time part of the memory function, but these series were always too complex to evaluate.…”

Diagrammatic Formulation of the Kinetic Theory of Fluctuations in Equilibrium Classical Fluids. V. The Short Time Approximation for the Memory Function

“…[8][9][10] This formally exact series provides a starting point for the derivation of approximate expressions for the memory function. We have derived 11,17 an approximation for the collisional part of the memory function that includes the effect of a repulsive binary collision. We call it the short time approximation, since it contains the dominant contributions at short time for systems whose repulsive forces are very strong.…”