Density Autocorrelation Function in a Classical Fluid from Initial Correlations

Abstract

A complete set of time-independent orthogonal phase functions $\left\{{\Psi }_s\right\}$, $s=0,1,2,\dots$, is generated via the Schmidt process and used to represent the Fourier coefficient $R_{\overrightarrow{\mathrm{k}}}\mathrm{}\left(t\right)\mathrm{}$ of the time-dependent microscopic density function. The projection of $R_{\overrightarrow{\mathrm{k}}}\mathrm{}\left(t\right)\mathrm{}$ on ${\Psi }_0$ is essentially the density autocorrelation function. The equation of motion of the coefficients of this expansion is found and formally solved to yield the Laplace-Fourier transform of the density autocorrelation function as a ratio of infinite determinants, closely related to Mori's continued-fraction expansion. A non-Markovian memory function is then readily defined in the same terms. These exact results are illustrated by explicit calculations for the ideal gas. Finally, a perturbation expansion of the memory function is developed, leading to practical approximations.