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

The Fokker–Planck equation for the translational modes of an inelastic system of Brownian particles in a non-equilibrium bath of light particles is derived from first principles of statistical mechanics. The bath and internal modes relax on a time scale that is much shorter than that of the translational modes and they are eliminated using time-dependent projection operators techniques and expansions in several small parameters. These small parameters reflect the difference in masses between the Brownian and bath particles, the weak coupling of the bath to the internal modes, the difference in mean bath and Brownian velocities and the macroscopic gradients of the system. The Fokker–Planck equation is expressed in terms of correlation functions over homogeneous local equilibrium averages and is valid up to second order in the smallness parameters and for times greater than the relaxation time of the fast modes. The non-linear hydrodynamic equations for the translational modes are derived using time-dependent projection operators and the effective Liouvillian obtained from the Fokker–Planck equation. The momentum and energy density hydrodynamic equations are not conserved and present terms which reflect the non-equilibrium nature of the bath and of the internal modes, as well as the irreversible processes occurring in the system.