Validity of the one-dimensional dissipative Boltzmann equation for point particles up to the clustering regime

We study stationary states of a one-dimensional gas of granular point-like particles not subject to gravity between two walls at temperatures T- and T+, with T- < T+. Depending on the normalized temperature difference Delta = (T+ - T-)/(T+ + T-) the system may be completely fluidized, or in a mixed state in which a cluster coexists with the fluidized gas. We devise and explain in detail a method for integrating the one-dimensional dissipative Boltzmann equation in the test-particle limit for the stationary case. We then apply this method to test the equation's validity up to the clustering regime, by comparing with results from microscopic Newtonian molecular dynamics. There is very good agreement, with the one-particle phase space density function presenting highly non-Gaussian features, and a discontinuity that corresponds to the test-particle limit. We conclude that Boltzmann's equation is valid at least everywhere in the control parameter space where the system has no cluster. The behavior of the system in its fluid phase is dominated by characteristic lines which resemble trajectories of particles subjected to a force which attracts them to a fixed point. If this point is in the physical region a cluster forms, if not then the system remains fluid.