Kinetic Theory of a Dense Gas: Triple-Collision Memory Function

We study the phase-space density-correlation function $S(\overrightarrow{\mathrm{r}}t; \overrightarrow{\mathrm{p}}{\overrightarrow{\mathrm{p}}}^{\prime })$ for a dense classical gas with repulsive interaction using the language of memory functions. We derive the kinetic equation for $S$ which is valid at all wavelengths and frequencies but limited to second order in the density (triple collisions). This model equation is, on the one hand, an extension of the earlier work of Mazenko to the next order in density and, on the other hand, an extension to arbitrary wavelengths and frequencies of some suggested generalizations of the linearized Boltzmann equation. The memory function for this kinetic equation is shown to be compatible with symmetry properties, sum rules, and the conservation laws. As an illustration of the hydrodynamics, we calculate the shear viscosity and show that the term linear in density agrees with an earlier calculation by Kawasaki and Oppenheim. We also give the analogous kinetic equation for the single-particle correlation function.