  Within the framework of the homogeneous non-linear Boltzmann equation, we
present a new analytic method, without the intrinsic limitations of existing
methods, for obtaining asymptotic solutions. This method permits extension of
existing results for Maxwell molecules and hard spheres to large classes of
particle interactions, from very hard spheres to softer than Maxwell molecules,
as well as to more general forcing mechanisms, beyond free cooling and white
noise driving. By combining this method with numerical solutions, obtained from
the Direct Simulation Monte Carlo (DSMC) method, we study a broad class of
models relevant for the dynamics of dissipative fluids, including granular
gases. We establish a criterion connecting the stability of the non-equilibrium
steady state to an exponentially bound form for the velocity distribution $F$,
which varies depending on the forcing mechanism. Power laws arise in marginal
stability cases, of which several new cases are reported. Our results provide a
minimal framework for interpreting large classes of experiments on driven
granular gases.
