  We report a study of the homogeneous isotropic Boltzmann equation for an open
system. We seek for nonequilibrium steady solutions in presence of forcing and
dissipation. Using the language of weak turbulence theory, we analyze the
possibility to observe Kolmogorov-Zakharov steady distributions. We derive a
differential approximation model and we find that the expected nonequilibrium
steady solutions have always the form of warm cascades. We propose an
analytical prediction for relation between the forcing and dissipation and the
thermodynamic quantities of the system. Specifically, we find that the
temperature of the system is independent of the forcing amplitude and
determined only by the forcing and dissipation scales. Finally, we perform
direct numerical simulations of the Boltzmann equation finding consistent
results with our theoretical predictions.
