Velocity Correlation Functions in Two and Three Dimensions

Abstract

Velocity correlation functions $\ensuremath{\rho}(t)$ in hard-disk and hard-sphere fluids are discussed on the basis of the kinetic theory of gases. It is found in $d$ dimensions that $\ensuremath{\rho}(t)\ensuremath{\sim}\ensuremath{\alpha}{(\frac{t}{{t}_{0}})}^{\ensuremath{-}\frac{d}{2}}$ with $\ensuremath{\alpha}=O({\ensuremath{\rho}}^{d\ensuremath{-}1})$ (${{t}_{0}}^{\ensuremath{-}1}=\mathrm{collision}\mathrm{frequency}$; $\ensuremath{\rho}=\mathrm{reduced}\mathrm{density}$). Expressions for $\ensuremath{\alpha}$ are given that seem correct at low densities and a good approximation at least up to half the close-packing density. The agreement with Alder and Wainwright's computer results is good.