L. Bocquet et J. Piasecki, MICROSCOPIC DERIVATION OF NON-MARKOVIAN THERMALIZATION OF A BROWNIAN PARTICLE, Journal of statistical physics, 87(5-6), 1997, pp. 1005-1035
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.