Fokker-Planck equation and non-linear hydrodynamic equations of a system of several Brownian particles in a non-equilibrium bath

The Fokker-Planck equation of a system of several Brownian particles immersed in a non-equilibrium bath of light particles is derived from first principles of statistical mechanics using time-dependent projection operators. The Fokker-Planck equation contains the usual equilibrium streaming and dissipative terms as well as terms reflecting spatial variations in the bath pressure, temperature and velocity. We make use of the effective Liouvillian obtained from the Fokker-Plank equation and of time-dependent projection operators involving properties of local equilibrium distribution functions to derive the exact non-linear hydrodynamic equations of the Brownian particles. The exact equations are 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 bath. The number density hydrodynamic equations for the bath and the Brownian particles remain unchanged from the case of a system of isolated particles, but the momentum and energy density expressions are no longer conserved and contain additional terms accounting for the non-equilibrium 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 hydrodynamic equations for the total densities of the system