Statistical-mechanical theory of Brownian motion -translational motion in an equilibrium fluid

Fluctuations of a large and heavy spherical Brownian particle (B) in an equilibrium fluid are rigorously studied by using two kinds of methods for asymptotic evaluation proposed by Mori; a projection operator method and a scaling method. It is shown that depending on the relative magnitudes of the mass (M) and the mass density (ϱB) of B to those of the bath particles (m and ϱ), there exist three kinds of kinetic processes which are described by three types of linear stochastic equations. One is a Markov kinetic process when m/M « 1 and ϱ/ϱB « 1 , which leads to the Stokes law ξ = 4πηR, where ξ is the friction constant, η the shear viscosity and R the radius of B. The others are non-Markov kinetic processes when m/M « 1 and ϱ/ϱB ≃ 1, and m/M å 1 and ϱ/ϱB « 1, which are characterized by the long time t−32 tail and the time t−12 decay of an average velocity of B, respectively. Next the usual Brownian point heavy particle case is also discussed from our viewpoint and a Markov kinetic process with the friction constant ξ/(1 + D0/DSE) is obtained, where D0 is the bare diffusion constant and DSE the Stokes-Einstein diffusion constant given by kBT/4πηR. Finally, it is shown that the hydrodynamic process of B obeys the diffusion equation whose diffusion constant is given by DSE and D0 + DSE for a large particle and for a point particle, respectively.