Effect of the wall on the velocity autocorrelation function and long-time tail of Brownian motion

Brownian motion of a particle situated near a wall bounding the fluid in which it is immersed is affected by the wall. Specifically, it is assumed that an incompressible viscous fluid fills a half-space bounded by a plane wall and that the fluid flow satisfies stick boundary conditions at the wall. The fluctuation-dissipation theorem shows that the velocity autocorrelation function of the Brownian particle can be calculated from the frequency-dependent admittance valid locally. It is shown that the t(-3/2) long-time tail of the velocity relaxation function, valid in bulk fluid, is obliterated and replaced by a t(-5/2) long-time tail of positive amplitude for motions parallel to the wall and by a t(-5/2) long-time tail of negative amplitude for motions perpendicular to the wall. The latter finding is at variance with an earlier calculation by Gotoh and Kaneda.