Single-Particle Motion in Simple Classical Liquids

Abstract

We use the memory-function formalism to investigate ${\tilde{G}}_s (\overrightarrow{\mathrm{k}},\overrightarrow{\mathrm{p}},{\overrightarrow{\mathrm{p}}}^{\prime },t)$, the time autocorrelation function for the single-particle density in phase space and $S_s (\overrightarrow{\mathrm{k}},w)$, the incoherent scattering function of Van Hove. Following a recent treatment for coherent neutron scattering by Akcasu and Duderstadt, we derive a formally exact equation for ${\tilde{G}}_s$. If we assume that the relevant memory function is separable and given by ${\tilde{\varphi }}_s (\overrightarrow{\mathrm{k}},\overrightarrow{\mathrm{p}},{\overrightarrow{\mathrm{p}}}^{\prime },t)={\tilde{\varphi }}_s (\overrightarrow{\mathrm{p}},{\overrightarrow{\mathrm{p}}}^{\prime },o)\tilde{\Phi }(\overrightarrow{\mathrm{k}},t)$, then the equation for ${\tilde{G}}_s$ reduces to a generalized Fokker-Planck equation which can be exactly solved. We use two simple forms for $\tilde{\Phi }(\overrightarrow{\mathrm{k}},t)$, a Gaussian and an exponential, to study the analytical solution both numerically and in various limits. We also compare these results with those obtained in the molecular dynamics computations by Nijboer and Rahman and discuss the extent of validity of the separability assumption for the memory function.