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 i n a nonequilibrium bath of N light spherical particles of mass m is st udied. A Fokker-Planck equation and a generalized Langevin equation fo r an arbitrary function of the position and momentum of the Brownian p article are derived from first principles of statistical mechanics usi ng time-dependent projection operators. These projection operators ref lect the nonequilibrium nature of the bath, which is described by the exact nonequilibrium distribution function of Oppenheim and Levine [Op penheim, L; Levine, R. D. Physica A 1979, 99, 383]. The Fokker-Planck equation is obtained by eliminating the fast bath variables of the sys tem [Van Kampen, N. G.; Oppenheim, I. Physica A 1986, 138, 231], while the Langevin equation is obtained using a projection operator which a verages over these variables [Mazur, P.; Oppenheim, I. Physica 1970, 5 0, 241]. The two methods yield equivalent results, valid to second ord er in the small parameters epsilon = (m/M)(1/2) and lambda, where lamb da is a measure of the magnitude of the macroscopic gradients of the s ystem.