Long-time translation and rotational Brownian motion in two dimensions

The long-time translational and rotational motion of a Brownian particle in two dimensions is studied on the basis of the fluctuation-dissipation theorem and linearized hydrodynamics. The long-time motion follows from the low frequency behavior of the mobility matrix. The coefficient of the long-time tail for the translational motion turns out to be independent of shape and size of the body, in agreement with mode-coupling theory. For rotational Brownian motion the coefficient of the long-time tail is found to depend on the shape of the body. This result is in conflict with a recent prediction from mode-coupling theory, and indicates that the mode-coupling calculation should be revised.This article is dedicated in friendship to Prof. Matthieu Ernst on the occasion of his 60th birthday.