The Kirkwood superposition approximation for hard rods at high pressure

We examine the influence of the Kirkwood superposition approximation in the Yvon-Born-Green equation for the pair distribution function of a system of hard rods. We construct a linearization of the difference-differential form of the Yvon-Born-Green equation with Kirkwood superposition. By examining the Laplace transform of the linear equation, we construct the pair distribution functions essentially exactly, and predict a threshold for transition from damped to undamped oscillatory pair distribution functions. We show explicitly the closure for which this linear equation is exactly the Yvon-Born-Green equation. We compare the results to those obtained with the exact closure, and to numerical results obtained from the full nonlinear equation with Kirkwood closure. From these studies, we are able to show that it is the long-range, non-nearest-neighbor coupling, present both in the Kirkwood and linearized closures, that is responsible for the transition to undamped oscillatory pair distribution functions in the dimension d = 1. We speculate on the relevance of this result to the behavior of the Yvon-Born-Green equation in d = 2,3.