Binary Collision Expansion for the Kinetic Equation of Time-Displaced Correlation Functions

Abstract

Using the binary collision expansion of the time-displacement operator, a kinetic equation for the Van Hove correlation functions is derived. The method differs from those already in the literature by the fact that equilibrium correlations are taken into account fully. This permits us to recover the exact "short-time" kinetic equations developed recently by various authors, from the lowest-order term in our expansion. Indeed, when the interaction between the particles is due entirely to their hard cores, then our lowest-order term is identical with the short-time kinetic operator, yielding an Enskog-type correction to the low-density equation. In the particular case of a system of hard rods in one dimension we get the exact kinetic equation found by Lebowitz, Percus, and Sykes. For non-hard-core potentials our lowest-order term represents a natural extension of the Boltzmann-Enskog type equation obtained in the case of hard spheres. The resulting kinetic operator can be put, for sufficiently short collision times, into a form similar to the hard-core case, albeit with a velocity-dependent collision diameter.