Thermodynamic origin of kappa distributions

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Kappa distributions describe particle velocities and energies in collisionless particle systems such as space and astrophysical plasmas. While we understand the possible mechanisms that may generate these distributions in these plasmas, the thermodynamic origin of these distributions remains a challenge: Particle systems at thermal equilibrium are known to be described by the Maxwell-Boltzmann distribution, but it is unknown whether this distribution is unique or other distribution functions consistent with thermodynamics may exist. This paper resolves the thermodynamic origin of kappa distributions. For particle systems eventually reaching thermal equilibrium, we show: i) the existence of two thermodynamic integrals characterizing thermal equilibrium, corresponding to two independent intensive thermodynamic quantities, temperature and kappa index (parameter labelling kappa distributions); ii) the adaptation of Sackur-Tetrode entropy for kappa distributions with applications in solar wind plasma; iii) the pseudo-additivity rule of entropy; iv) that the most general, physically meaningful, distribution function that particle systems are stabilized into when reaching thermal equilibrium is the kappa distribution.