A generalized Oseen theory of two-dimensional Brownian motion

A generalization of the Oseen arguments to a quasi-two-dimensional system of a nonuniformly moving Brownian cylinder yields a remarkable “long time” behavior ø(t) ∼ {1 + 2t/τ}−32t−1 for the velocity autocorrelation function ø(t). The characteristic time τ = 16mbv/(kBT) is generally much greater than the initial relaxation time tR and “long times” means after a few tR. Previous theories have predicted ø(t) ∼ t−1 after a few tR and our result is consistent with this for times much less than τ. On a longer time scale ∼τ our result predicts an ususual behavior ø(t) ∼ t−52 which is sufficient to yield a finite diffusion coefficient from the Green-Kubo formula.