Theory of Atomic Motions in Simple Classical Liquids

This paper is an attempt towards developing a theory of the self-motion of atoms in monatomic classical fluids, based on a simplified Liouville equation and on a knowledge of the interaction potential $V\left(r\right)\mathrm{}$ and the static pair correlation function $g\left(r\right)\mathrm{}$. A nonlinear integral equation for the velocity autocorrelation function $\Phi \mathrm{}\left(t\right)\mathrm{}$ is derived which under certain approximations can be written as $\frac{d\Phi \mathrm{}\left(t\right)\mathrm{}}{\mathrm{dt}}+\int 0^{t}K(t-t^{\prime })\Phi \left(t^{\prime }\right)d{t}^{\prime }=0\mathrm{}$ where the kernel $K\left(t\right)\mathrm{}$ is an implicit function of $\Phi \mathrm{}\left(t\right)\mathrm{}$. An expression for $K\left(t\right)\mathrm{}$ in terms of $V\left(r\right)\mathrm{}$ and $g\left(r\right)\mathrm{}$ has been given. Explicit numerical calculations for the "memory" function $K\left(t\right)\mathrm{}$ have been made for liquid argon at $T=85.9\mathrm{}°$K, and the results have been compared with those obtained by Rahman from his machine computations.