Expansion of the kinetic hierarchy for a massive particle: The repeated ring and Fokker-Planck equations

The tagged particle BBGKY hierarchy is systematically expanded in inverse powers of the square root of the particle mass. In the Brownian limit, for fixed Knudsen number, the hierarchy reduces to the Brownian limit of the repeated ring equation which itself reduces to the Fokker-Planck equation. The friction coefficient of the Fokker-Planck equation is found to be a functional of the solution of Dorfman, van Beijeren, and McClure's extended Boltzmann equation for a fixed object in a flowing gas. As a consequence, the tagged particle diffusion coefficient calculated in the Brownian limit of the repeated ring equation is valid for all particle sizes.