  The granular gas is a paradigm for understanding the effects of inelastic
interactions in granular materials. Kinetic theory provides a general
theoretical framework for describing the granular gas. Its central result is
that the tail of the velocity distribution of a driven granular gas is a
stretched exponential that, counterintuitively, decays slower than that of the
corresponding elastic gas in equilibrium. However, a derivation of this result
starting from a microscopic model is lacking. Here, we obtain analytical
results for a microscopic model for a granular gas where particles with
two-dimensional velocities are driven homogeneously and isotropically by
reducing the velocities by a factor and adding a stochastic noise. We find two
universal regimes. For generic physically relevant driving, we find that the
tail of the velocity distribution is a Gaussian with additional logarithmic
corrections. Thus, the velocity distribution decays faster than the
corresponding equilibrium gas. The second universal regime is less generic and
corresponds to the scenario described by kinetic theory. Here, the velocity
distribution is shown to decay as an exponential with additional logarithmic
corrections, in contradiction to the predictions of the phenomenological
kinetic theory, necessitating a re-examination of its basic assumptions.
