Fokker-Planck and Langevin descriptions of fluctuations in uniform shear flow

Abstract

The Boltzmann description of the preceding paper for tagged particle fluctuations in a nonequilibrium gas is further analyzed in the limit of small mass ratio between the gas and the tagged particles. For a large class of nonequilibrium states the Boltzmann-Lorentz collision operator for the tagged particle distribution is expanded to leading order in the mass ratio, resulting in a Fokker-Planck operator. The drift vector and diffusion tensor are calculated exactly for Maxwell molecules. The Fokker-Planck operator depends on the nonequilibrium state only through the hydrodynamic variables for the fluid. The diffusion tensor is a measure of the “noise” amplitude and is not simply determined from the nonequilibrium temperature; instead, it depends on the fluid stress tensor components as well. For the special case of uniform shear flow, the Fokker-Planck equation is of the linear type and may be solved exactly. The associated set of Langevin equations is also identified and used to describe spatial diffusion in the Lagrangian coordinates of the fluid. The effect of viscous heating on diffusion is discussed and the dependence of the diffusion coefficient on the shear rate is calculated.