Velocity-Correlation Functions in Two and Three Dimensions: Low Density

The long-time behavior of velocity-correlation functions ${\rho }^{\mathrm{}\left(d\right)\mathrm{}}\mathrm{}\left(t\right)\mathrm{}$ characteristic for self-diffusion, viscosity, and heat conductivity is calculated for a gas of hard disks or hard spheres on the basis of the kinetic theory of dense gases. In $d$ dimensions one finds that ${\rho }^{\mathrm{}\left(d\right)\mathrm{}}\mathrm{}\left(t\right)\mathrm{}$, after an initial exponential decay for a few mean free times $t_0$, exhibits for times up to at least $\sim 40t_0$ a decay $\sim {\alpha }^{\mathrm{}\left(d\right)\mathrm{}}\left(\rho \right){\left(\frac{t_0}{t}\right)}^{\frac{d}{2}}$, where ${\alpha }^{\mathrm{}\left(d\right)\mathrm{}}$ is of the order of ${\rho }^{d-1\mathrm{}}$, $\rho =n{a}^d$ with $n$ the number density, and $a$ the hard-disk or hard-sphere diameter. The ${\alpha }^{\mathrm{}\left(d\right)\mathrm{}}\left(\rho \right)$ are determined by the same dynamical events that are responsible for the divergences in the virial expansion of the transport coefficients. In this paper the ${\alpha }^{\mathrm{}\left(d\right)\mathrm{}}\left(\rho \right)$ are calculated to lowest order in $\rho$. In this order, they are identical to the low-density limit of the ${\alpha }^{\mathrm{}\left(d\right)\mathrm{}}\left(\rho \right)$ that have been obtained by other authors on the basis of hydrodynamical considerations.