Transformation of the Maxwell-Boltzmann distribution to a power law

Power-law (PL) distribution functions (DF) are prevalent in highly diverse systems. The systems range in size from nanometer to mega light years, in complexity from dust grains to living organisms, and characterize the distribution of various events in nature and in various human activities. To gain some insight on why PL DF are so prevalent, we explore the conditions leading to the formation of a PL DF in a simple system of colliding hard sphere. We follow the time evolution of the energy DF through direct Monte Carlo simulations. In statistical equilibrium, the DF evolves into the Maxwell-Boltzmann DF. A transition to a PL DF occurs when the following occurs: (1) The system is initially far from equilibrium. For example, a mix of light and heavy particles with the same velocity. (2) The system dynamics is scale free, which holds in the intermediate asymptotic regime, far from the initial and the final equilibrium states. The scale-free dynamics leads to a DF which evolves in a self-similar form. (3) The system is open with a scale-free boundary condition. For example, a constant injection of particles far from equilibrium. The DF PL index is set by the time dependence of the self-similar DF and by the boundary condition. The PL index is independent of the self-similar DF form. Conditions (1)-(3) are common in a great variety of systems, which may explain why PL DF are so prevalent in nature.