Introduction to special section on origins and properties of kappa distributions: Statistical background and properties of kappa distributions in space plasmas

Empirical kappa distributions provide a straightforward replacement of the Maxwell distribution for systems out of thermal equilibrium such as space plasmas. Kappa distributions have become increasingly widespread across space physics with the number of relevant publications following, remarkably, an exponential growth rate. However, a breakthrough in the field came with the connection of kappa distributions with the framework of nonextensive statistical mechanics. This introductory paper clarifies fundamental physical concepts and provides mathematical formulations of the theory of kappa distributions, which are a consequence of the connection of kappa distributions with a solid statistical background. Among others, the paper presents the existence of a consistent definition of temperature in systems out of thermal equilibrium described by kappa distributions, the physical meaning of the kappa index, and the formulation of the kappa distribution of a Hamiltonian. In addition, the paper examines the most frequent values of kappa indices in space plasmas. Statistical analysis reveals trends between the characteristic values of density, temperature, and kappa index of space plasmas. Finally, understanding the kinetic interpretation of the temperature as the mean kinetic energy, and of the kappa index as the correlation of kinetic energies, helps to develop all the possible formulations of isotropic/anisotropic kappa distributions.