Kinetic equations of interacting Brownian particles

The dynamics of a single Brownian particle and an ensemble of such particles in an equilibrium bath is considered. Starting from the Liouville equation of the system, the corresponding kinetic equations are derived. To that end the simple perturbation method on a small parameter λ = √m/M is worked out (here m and M are the mass of the carrier medium molecule and Brownian particle, respectively). The kinetic equation of a single Brownian particle is obtained in second order of perturbation. It is shown that the kinetic equation of an ensemble of Brownian particles is in principle a nonlocal equation. It does not reduce to the Fokker-Plank equation even if direct particle-particle interactions are absent.