  We compare the steady state velocity distributions from our three-dimensional
inelastic hard sphere molecular dynamics simulation for homogeneously heated
granular media, with the predictions of a mean field-type Enskog-Boltzmann
equation for inelastic hard spheres [van Noije & Ernst, Gran. Matt. {\bf 1}, 57
(1998)]. Although we find qualitative agreement for all values of density and
inelasticity, the quantitative disagreement approaches $\sim 40%$ at high
inelasticity or density. By contrast the predictions of the pseudo-Maxwell
molecule model [Carrillo, Cercignani & Gamba, Phys. Rev. E, {\bf 62}, 7700
(2000)] are both qualitatively and quantitatively different from those of our
simulation. We also measure short-range and long-range velocity correlations
exhibiting non-zero correlations at contact before the collision, and being
consistent with a slow algebraic decay over a decade in the unit of the
diameter of the particle, proportional to $r^{-(1+\alpha)}$, where $0.2 <
\alpha < 0.3$. The existence of these correlations imply the failure of the
molecular chaos assumption and the mean field approximation, which is
responsible for the quantitative disagreement of the inelastic hard sphere
kinetic theory.
