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We compare the steady state velocity distributions from our three-dimensionalinelastic hard sphere molecular dynamics simulation for homogeneously heatedgranular media, with the predictions of a mean field-type Enskog-Boltzmannequation for inelastic hard spheres [van Noije & Ernst, Gran. Matt. {\bf 1}, 57(1998)]. Although we find qualitative agreement for all values of density andinelasticity, the quantitative disagreement approaches $\sim 40%$ at highinelasticity or density. By contrast the predictions of the pseudo-Maxwellmolecule model [Carrillo, Cercignani & Gamba, Phys. Rev. E, {\bf 62}, 7700(2000)] are both qualitatively and quantitatively different from those of oursimulation. We also measure short-range and long-range velocity correlationsexhibiting non-zero correlations at contact before the collision, and beingconsistent with a slow algebraic decay over a decade in the unit of thediameter of the particle, proportional to $r^{-(1+\alpha)}$, where $0.2 <\alpha < 0.3$. The existence of these correlations imply the failure of themolecular chaos assumption and the mean field approximation, which isresponsible for the quantitative disagreement of the inelastic hard spherekinetic theory.Comment: 23 pages, 15 figures, Phys. Rev. E, in pres

Velocity distributions and correlations in homogeneously heated granular media