Quasi-elastic solutions to the nonlinear Boltzmann equation for dissipative gases

The solutions of the one-dimensional homogeneous nonlinear Boltzmann equation are studied in the QE-limit (quasi-elastic; infinitesimal dissipation) by a combination of analytical and numerical techniques. Their behaviour at large velocities differs qualitatively from that for higher dimensional systems. In our generic model, a dissipative fluid is maintained in a non-equilibrium steady state by a stochastic or deterministic driving force. The velocity distribution for stochastic driving is regular and, for infinitesimal dissipation, has a stretched exponential tail, with an unusual stretching exponent bQE = 2b, twice as large as the standard one for the corresponding d-dimensional system at finite dissipation. For deterministic driving the behaviour is more subtle and displays singularities, such as multi-peaked velocity distribution functions. We classify the corresponding velocity distributions according to the nature and scaling behaviour of such singularities