Chapter 1 - Statistical Background of Kappa Distributions: Connection With Nonextensive Statistical Mechanics

Abstract

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 plasmas, are characterized by a non-Maxwellian behavior, typically described by the so-called kappa distributions. Empirical kappa distributions have become increasingly widespread across space and plasma physics. However, a breakthrough in the field came with the connection of kappa distributions with the solid background of nonextensive statistical mechanics. Understanding the statistical origin of kappa distributions is a cornerstone of further theoretical developments and applications, which, among others, involve (1) the physical meaning of temperature, thermal pressure, and other thermodynamic parameters; (2) the physical meaning of the kappa index and its connection to the degrees of freedom and their correlation; (3) the Sackur–Tetrode entropy for kappa distributions; (4) the multiparticle description of kappa distributions; and (5) the kappa distribution of a Hamiltonian with a nonzero radial or angular potential. With the results provided in this study, the full strength and capability of nonextensive statistical mechanics are available for the space physics community to analyze and understand the kappa-like properties of the various particle and energy distributions observed in space.