Mode coupling and tagged particle correlation functions: the Stokes-Einstein law

In this paper a mode coupling theory for equilibrium tagged particle correlation functions in simple fluids is developed from molecular considerations. The mode coupling formalism developed in a previous paper [Physica A 181 (1992) 89] to describe equilibrium fluctuations in simple liquids is generalized to include tagged particle correlations. The mode coupling formalism and N ordering approximation scheme used in the previous paper allow a series for the generalized diffusion constant to be obtained which is exact in the thermodynamic limit. The Stokes-Einstein law for a slip Brownian particle is then derived from this series in a systematic fashion. The techniques applied here obviate the need to assume that the simple liquid is incompressible, and allow a self-consistent series of equations for the diffusion constant of the Brownian particle and transport coefficients of the fluid to be obtained which are useful in the study of supercooled liquids.