Fokker−Planck Equation and Langevin Equation for One Brownian Particle in a Nonequilibrium Bath

The Brownian motion of a large spherical particle of mass M immersed in a nonequilibrium bath of N light spherical particles of mass m is studied. A Fokker?Planck equation and a generalized Langevin equation for an arbitrary function of the position and momentum of the Brownian particle are derived from first principles of statistical mechanics using time-dependent projection operators. These projection operators reflect the nonequilibrium nature of the bath, which is described by the exact nonequilibrium distribution function of Oppenheim and Levine [Oppenheim, I.; Levine, R. D. Physica A1979, 99, 383]. The Fokker?Planck equation is obtained by eliminating the fast bath variables of the system [Van Kampen, N. G.; Oppenheim, I. Physica A1986, 138, 231], while the Langevin equation is obtained using a projection operator which averages over these variables [Mazur, P.; Oppenheim, I. Physica1970, 50, 241]. The two methods yield equivalent results, valid to second order in the small parameters ε = (m/M)^1/2 and λ, where λ is a measure of the magnitude of the macroscopic gradients of the system.