Chapter 3 - Phase Space Kappa Distributions With Potential Energy

Abstract

This chapter presents the theory, formulations, and properties of the kappa distributions that describe particle systems characterized by a nonzero potential energy. Among others, we investigate: (1) the phase space kappa distribution of the Hamiltonian given by the sum of the velocity-dependent kinetic energy and the position-dependent potential energy; (2) the derived two marginal distributions of the phase space distribution, that is, the distribution of the velocity (or kinetic energy) after integrating over the position, and the distribution of the position after integrating over the velocity; (3) the mean kinetic energy, that is, the kinetic definition of temperature, in the presence of a positional potential energy, showing that is the same as in the absence of the potential energy; (4) the degeneration of the kappa index in the presence of a potential energy; (5) the generic formulation of the phase space and positional kappa distributions for positive and negative potentials; (6) the specific two main attractive power law potentials, i.e., the positive potential with positive power law exponent (oscillator type) and the negative potential with negative power law exponent (gravitational type); (7) distributions for potentials with both positive and negative values; (8) gravitational potentials, e.g., the linear gravitational potential and the barometric formula, the spherical gravitational potential, the virial theorem, and the Jeans radius; (9) the local distributions, namely, the kappa distribution of velocity, the density, temperature, and thermal pressure, all expressed as functions of the position; and (10) potentials, depending on the velocity, may cause anisotropic velocity kappa distributions. These developments allow the researcher to describe any particle population of space, geophysical, laboratory, or other plasmas that are subject to a nonnegligible potential energy.