Warm cascade states in a forced-dissipated Boltzmann gas of hard spheres

We study the homogeneous isotropic Boltzmann equation for an open system. For the case of a hard spheres gas, we look for nonequilibrium steady solutions in the presence of forcing and dissipation. Using the language of weak turbulence theory, we analyze the possibility of observing the Kolmogorov-Zakharov steady distributions, i.e. solutions characterized by constant fluxes of conserved quantities. 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 the 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.