A full solution of Brownian motion

Brownian motion has served as a pilot of studies in diffusion and other transport phenomena for over a century. It is known that a full description of Brownian motion in the entire course of time should involve both kinetic and hydrodynamic effects. Nevertheless, a formula accounts for both effects has been established only for the limiting case, namely, when Brownian particles are much bigger and heavier than fluid particles. Moreover, for applications it is important to consider small Brownian particles down to a size smaller than fluid particles. But, unfortunately, in these cases only formulae in the short- or long-time limit are available. In this Letter, we derive a general formula for Brownian motion applicable to a wide spectrum of suspended particle, with sizes from being smaller than the fluid particles to comparable to the usual Brownian particles. Our analytical results are well corroborated by the numerical experiments.