Bilinear Hydrodynamics and the Stokes-Einstein Law

The autocorrelation function of the density of a tagged particle is studied using the Mori formalism. The variables used are the collective conserved variables, the tagged-particle density, and bilinear products thereof. The case of point particles is considered in two dimensions, and, in three dimensions, self-diffusion by a particle of arbitrary size is treated. It is found that the bilinear-hydrodynamic approach automatically separates the self-diffusion coefficient of the tagged particle into a nonhydrodynamic part, and a hydrodynamic part which resembles the Stokes-Einstein law. In two dimensions, it is found that the mean-square displacement of a particle increases as $t\ln t$, and that certain natural redefinitions of the diffusion and friction coefficients leave Einstein's law invariant. In three dimensions, for a large particle, the Stokes-Einstein law is reproduced. The relation between the well-known $t^{\left(\frac{-3\mathrm{}}{2}\right)}$ "tails" on correlation functions, and the Stokes-Einstein law, is discussed.