Velocity Fluctuations of a Hard-Core Brownian Particle

The velocity autocorrelation function $〈v\left(t\right)\mathrm{}v\left(0\right)\mathrm{}〉$ of a hard-sphere Brownian particle moving in an incompressible viscous fluid is calculated on the basis of hydrodynamics. The usually assumed exponential decay at long times is shown to be inconsistent with fluid mechanics. A $\mathrm{slower}\mathrm{decay}\propto t^{-\frac{3}{2}}$ as $t\to \infty$ is derived. This decay has been observed in molecular dynamic computations carried out by Alder and Wainwright.