MICROSCOPIC DERIVATION OF NON-MARKOVIAN THERMALIZATION OF A BROWNIAN PARTICLE

In this paper, the first microscopic approach to Brownian motion is de veloped in the case where the mass density of the suspending bath is o f the same order of magnitude as that of the Brownian (B) particle. St arting From an extended Boltzmann equation, which describes correctly the interaction with the fluid, we derive systematically via multiple- time-scale analysis a reduced equation controlling the thermalization of the B particle, i.e., the relaxation reward the Maxwell distributio n in velocity space. In contradistinction to the Fokker-Planck equatio n, the derived new evolution equation is nonlocal both in time and in velocity space, owing to correlated recollision events between the flu id and particle B. In the long-time limit, it describes a non-Markovia n generalized Ornstein-Uhlenbeck process. However, in spite of this co mplex dynamical behavior, the Stokes-Einstein law relating the frictio n and diffusion coefficients is shown to remain valid. A microscopic e xpression for the friction coefficient is derived, which acquires the form of the Stokes law in the limit where the mean-free path in the ga s is small compared to the radius of particle B.