Fully renormalized kinetic theory. III. Density fluctuations

Abstract

The fully renormalized kinetic theory (FRKT) previously discussed by the author in the case of self-diffusion is extended to the case of density fluctuations. In the theory techniques are developed for calculating classical phase-space time-dependent correlation functions. The method centers around the development of an exact expression for the memory function associated with the phase-space-correlation function. This exact expression is written in a compact and symmetric form that is convenient for making approximations. It is shown explicitly how one can make contact with the Boltzmann-Enskog approximation for the memory function which is valid for moderate densities, as well as the terms which lead to the much-discussed ${t}^{\ensuremath{-}\frac{2}{3}}$ long-time behavior. The most striking difference between self-diffusion and the case of density fluctuation arises in the region of the critical point. It is shown in the theory how to make contact with the mode-mode coupling results of Kawasaki. From a formal point of view the development of the FRKT is carried forward three important steps in this paper. It is shown how to relate the memory function to the two-particle source function in a more powerful and direct method than the projection-operator approach developed in FRKT I. After a simple rearrangement, the memory function is written in a far more symmetric form than in FRKT I, and finally it is shown how the ideas of connectedness and cumulants can be successfully introduced into the theory.