Brownian motion: from kinetics to hydrodynamics
Brownian motion has served as a pilot of studies in diffusion and other transport phenomena for over a century. The foundation of Brownian motion, laid by Einstein, has generally been accepted to be far from being complete since the late 1960s, because it fails to take important hydrodynamic effects into account. The hydrodynamic effects yield a time dependence of the diffusion coefficient, and this extends the ordinary hydrodynamics. However, the time profile of the diffusion coefficient across the kinetic and hydrodynamic regions is still absent, which prohibits a complete description of Brownian motion in the entire course of time. Here we close this gap. We manage to separate the diffusion process into two parts: a kinetic process governed by the kinetics based on molecular chaos approximation and a hydrodynamics process described by linear hydrodynamics. We find the analytical solution of vortex backflow of hydrodynamic modes triggered by a tagged particle. Coupling it to the kinetic process we obtain explicit expressions of the velocity autocorrelation function and the time profile of diffusion coefficient. This leads to an accurate account of both kinetic and hydrodynamic effects. Our theory is applicable for fluid and Brownian particles, even of irregular-shaped objects, in very general environments ranging from dilute gases to dense liquids. The analytical results are in excellent agreement with numerical experiments.