Modified Fokker-Planck equation for the motion of Brownian particles in a nonuniform gas

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Abstract
The Brownian-motion Fokker-Planck equation describing the velocity-distribution function of a collection of Brownian particles suspended in a rarefied carrier gas in equilibrium is generalized to the case of a nonuniform suspending gas. The gas-particle collision integral appearing in the Boltzmann equation is expanded in powers of the ratio of masses between gas molecules and particles, and the gas-distribution function is taken as the first approximation to the Chapman-Enskog expression. Also, the characteristic width of the particle-distribution function is assumed to be of the order of its value in equilibrium. The modified collision term obtained involves a new force proportional to the local temperature gradient, and closely related to the Chapman-Enskog thermal-diffusion effect. The contributions from the nonhomogeneity of the gas-velocity field is small, but introduces new "shear forces" proportional to the fluid-particle relative velocity and the local gas traceless symmetrized velocity-gradient tensor. The new equation can be used to describe the non-equilibrium dynamics of gas mixtures with disparity of masses when the heavy species is dilute and the light is not too far from equilibrium.