Title: Brownian motion: from kinetics to hydrodynamics,
Abstract: 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.