Statistical origin and properties of kappa distributions

Classical particle systems reside at thermal equilibrium with their velocity distribution function stabilized into a Maxwell distribution. On the contrary, collisionless and correlated particle systems, such as space and astrophysical plasmas, are characterized by a non-Maxwellian behavior, typically described by kappa distributions, or combinations thereof. Empirical kappa distributions have become increasingly widespread across space and plasma physics. A breakthrough in the field came with the connection of kappa distributions to non-extensive statistical mechanics. Understanding the statistical origin of kappa distributions was the cornerstone of further theoretical developments and applications, namely, (i) the concept of temperature; (ii) the physical meaning of the kappa index; (iii) the N-particle description of kappa distributions; and the (iv) the generalization to phase-space kappa distribution of a Hamiltonian with non-zero potential.