H-theorem at negative temperature: the random exchange model with bounds

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 a globally conserved quantity are introduced. We analytically show that this dynamic allows stationary states with population inversion, described by Boltzmann statistics at negative absolute temperature, if the conserved quantity has the physical meaning of an energy. The proposed model therefore provides a privileged system for the study of thermalization towards a negative temperature state. First, the genuine equilibrium nature of the stationary state is verified by checking the detailed balance condition. Then, an H-theorem is proven, ensuring that such equilibrium condition is reached by a monotonic increase in the Boltzmann entropy. We also 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.