Density Autocorrelation Function in a Classical Fluid from Initial Correlations

A complete set of time-independent orthogonal phase functions Ψₛ, s=0,1,2,, is generated via the Schmidt process and used to represent the Fourier coefficient R_→k(t) of the time-dependent microscopic density function. The projection of R_→k(t) on Ψ₀ 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.