Velocity Correlation Functions in Two and Three Dimensions

Velocity correlation functions $\rho \mathrm{}\left(t\right)\mathrm{}$ 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 $\rho \mathrm{}\left(t\right)\mathrm{}\sim \alpha {\left(\frac{t}{t_0}\right)}^{-\frac{d}{2}}$ with $\alpha =O\left({\rho }^{d-1\mathrm{}}\right)$ (${t_0}^{-1\mathrm{}}=\mathrm{collision}\mathrm{frequency}$; $\rho =\mathrm{reduced}\mathrm{density}$). Expressions for $\alpha$ 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.