Kappa distribution in the presence of a potential energy

The present paper develops the theory and formulations of the kappa distributions that describe particle systems characterized by a nonzero potential energy. As yet, kappa distributions were used for the statistical description of the velocity or kinetic energy of particles but not of the potential energy. With the results provided here, it is straightforward to use the developed kappa distributions to describe any particle population of space plasmas subject to a nonnegligible potential energy. Starting from the kappa distribution of the Hamiltonian function, we develop the distributions that describe either the complete phase space or the marginal spaces of positions and velocities. The study shows, among others: (a) The kappa distributions of velocities that describe space plasmas can be vastly different from the standard formulation of the kappa distribution, because of the presence of a potential energy; the correct formulation should be given by the marginal kappa distribution of velocities by integrating the distribution of the Hamiltonian over the potential energy. (b) The long-standing problem of the divergence of the Boltzmannian exponential distribution for bounded radial potentials is solved using kappa distributions of negative kappa index. (c) Anisotropic distributions of velocities can exist in the presence of a velocity-dependent potential. (d) A variety of applications, including derivations/verifications of the following: (i) the Jeans', the most frequent, and the maximum radii in spherical/linear gravitational potentials; (ii) the Virial theorem for power law potentials; (iii) the generalized barometric formula, (iv) the plasma density profiles in Saturnian magnetosphere, and (v) the average electron magnetic moment in Earth's magnetotail.