  Random exchange kinetic models are widely employed to describe the
conservative dynamics of large interacting systems. Due to their simplicity and
generality, they are quite popular in several fields, from statistical
mechanics to biophysics and economics. Here we study a version where bounds on
the individual shares of the globally conserved quantity are introduced. We
analytically show that this dynamics allows stationary states with population
inversion, described by Boltzmann statistics at negative absolute temperature.
Their genuine equilibrium nature is verified by checking the detailed balance
condition. An H-theorem is proven: the Boltzmann entropy monotonically
increases during the dynamics. Finally, we provide analytical and numerical
evidence that a large intruder in contact with the system thermalizes,
suggesting a practical way to design a thermal bath at negative temperature.
These results open new research perspectives, creating a bridge between
negative temperature statistical descriptions and kinetic models with bounds.
