Nonequilibrium distributions from the Fokker-Planck equation: Kappa distributions and Tsallis entropy.

Nonequilibrium systems in chemistry and physics are generally modeled with the Boltzmann, Fokker-Planck, and Master equations. There has been a considerable interest in the nonequilibrium distributions of electrons and ions in space physics in different environments as well as in other systems. An often-used empirical model to characterize these distributions, especially in space physics, is the Kappa distribution. There have been numerous efforts to provide a theoretical basis for the Kappa distribution that include the Fokker-Planck equation with specific drift and diffusion coefficients. Alternatively, the maximization of the Tsallis nonextensive entropy provides the desired Kappa distribution. This paper examines three families of Fokker-Planck equations that provide a steady-state Kappa distribution as well as a myriad of other nonequilibrium distributions. The relationship of these works with analogous studies of distributions with asymptotic high-energy tails is also considered. It is clear that the many different nonequilibrium distribution functions that can occur cannot all be rationalized with Gibbs-Boltzmann statistical mechanics, which uniquely gives equilibrium distributions, or with the Tsallis nonextensive entropy, which gives uniquely the Kappa distribution. The current research is directed towards an improved understanding of the origin of nonequilibrium distributions in several specific systems.