Formulae of Kappa Distributions: Toolbox

The chapter presents all the formulae of kappa distributions necessary to describe particle populations out of thermal equilibrium in plasmas and beyond. In summary, we first provide the isotropic and anisotropic distributions in the absence of a potential energy, while then we continue with the formulae of kappa distributions in the presence of a potential energy; we present the formulation of multiparticle and multispecies kappa distributions, and the generalized kappa distributions based on non-Euclidean norms or under the discrete description of energy. These formulations and their guidelines constitute the most updated "toolbox" of useful and statistically well-grounded equations for future analyses that seek to apply kappa distributions in data analysis, simulations, modeling, theory, and other work in space, geophysical, laboratory, or other plasmas or related particle systems.