Driven inelastic Maxwell models with high energy tails

The solutions of the homogeneous nonlinear Boltzmann equation for inelastic Maxwell models, when driven by different types of thermostats, show, in general, overpopulated high energy tails of the form approximately exp(-ac), with power law tails and Gaussian tails as border line cases. The results are compared with those for inelastic hard spheres, and a comprehensive picture of the long time behavior in freely cooling and driven inelastic systems is presented.