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

The Fokker-Planck equation for the translational modes of an inelastic system of Brownian particles in a non-equilibrium bath of light parti cles is derived from first principles of statistical mechanics. The ba th and internal modes relax on a time scale that is much shorter than that of the translational modes and they are eliminated using time-dep endent projection operators techniques and expansions in several small parameters. These small parameters reflect the difference in masses b etween the Brownian and bath particles, the weak coupling of the bath to the internal modes, the difference in mean bath and Brownian veloci ties and the macroscopic gradients of the system. The Fokker-Planck eq uation is expressed in terms of correlation functions over homogeneous local equilibrium averages and is valid up to second order in the sma llness parameters and for times greater than the relaxation time of th e fast modes. The non-linear hydrodynamic equations for the translatio nal modes are derived using time-dependent projection operators and th e effective Liouvillian obtained from the Fokker-Planck equation. The momentum and energy density hydrodynamic equations are not conserved a nd present terms which reflect the non-equilibrium nature of the bath and of the internal modes, as well as the irreversible processes occur ring in the system. (C) 1998 Elsevier Science B.V. All rights reserved .