Brownian Motion in a Granular Fluid

The Fokker-Planck equation for a heavy particle in a granular fluid is derived from the Liouville equation. The host fluid is assumed to be in its homogeneous cooling state and all interactions are idealized as smooth, inelastic hard spheres. The similarities and differences between the Fokker-Planck equation for elastic and inelastic collisions are discussed in detail. Although the fluctuation-dissipation relation is violated and the reference fluid is time dependent, it is shown that diffusion occurs at long times for a wide class of initial conditions. The results presented here generalize previous results based on the Boltzmann-Lorentz equation to higher densities.