Steady states of a Boltzmann equation for driven granular media

We study a three-dimensional model for driven granular media in which particles interact inelastically while they follow Brownian dynamics in between collisions. A steady Boltzmann-type kinetic equation associated with a pseudo-Maxwellian model is analyzed. Homogeneous steady states are found by a small inelasticity expansion. These states are given by a Maxwellian distribution corrected by the second Sonine polynomial up to third order in the expansion. The resulting correction is a quartic polynomial in velocity space. This result agrees qualitatively with the molecular dynamics simulation in C. Bizon, M. D. Shattuck, J. B. Swift, and H. L. Swinney, Phys. Rev. E 60, 4340 (1999).