FOKKER-PLANCK EQUATION AND NONLINEAR HYDRODYNAMIC EQUATIONS OF A SYSTEM OF SEVERAL BROWNIAN PARTICLES IN A NONEQUILIBRIUM BATH

The Fokker-Planck equation of a system of several Brownian particles i mmersed in a nonequilibrium bath of light particles is derived from fi rst principles of statistical mechanics using time-dependent projectio n operators. The Fokker-Planck equation contains the usual equilibrium streaming and dissipative terms as well as terms reflecting spatial v ariations in the bath pressure, temperature and velocity. We make use of the effective Liouvillian obtained from the Fokker-Planck equation and of time-dependent projection operators involving properties of loc al equilibrium distribution functions to derive the exact non-linear h ydrodynamic equations of the Brownian particles. The exact equations a re simplified using the fact that the thermodynamic forces vary slowly on a molecular timescale and the resulting local transport equations are expressed in terms of homogeneous local equilibrium averages. The non-equilibrium conditional distribution for the bath is obtained from the Fokker-Planck equation using time-dependent projection operators and is used to derive the non-linear hydrodynamic equations of the bat h. The number density hydrodynamic equations for the bath and the Brow nian particles remain unchanged from the case of a system of isolated particles, but the momentum and energy density expressions are no long er conserved and contain additional terms accounting for the non-equil ibrium nature of the bath and for the irreversible processes occurring in the system. The non-linear hydrodynamic equations for the bath and Brownian densities are combined to yield the conserved non-linear hyd rodynamic equations for the total densities of the system.