Velocity Correlation and Relative Diffusion in Simple Liquids

A method of calculating the velocity correlation function in monatomic classical liquids is developed essentially from first principles. The interaction between an atom in a liquid and the density fluctuation around it is formulated in the framework of a generalized Langevin equation. It is shown that the memory function can be rigorously expressed in terms of the interatomic potential v(r), static pair correlation function g(r) and a Green function of a certain linear operator L. With the use of sum-rule arguments and an approximation on the static structure of liquid, the operator L is shown to be reduced to a (non-Markoffian) Smoluchowski operator which has been used in studying relative diffusion in liquid. This gives a prescription for calculating the velocity autocorrelation function in terms of v(r) and g(r). Frequency spectrum and the memory function are calculated numerically both for soft core and long-range-oscillatory systems. The results have been compared with those obtained by machine computations.