Theoretical aspects of Hamiltonian kappa distributions

The paper examines the formalism of phase-space kappa distribution of particle Hamiltonian in space and other plasmas that reside in stationary states out of the classical thermal equilibrium. Plasmas in the classical thermal equilibrium have their particle velocities described by Maxwell-Boltzmann distributions, in contrast to space plasmas, which are collisionless particle systems residing in the general thermal equilibrium characterized by kappa distributions. In particular, we show: (i) the full phase-space distribution of the Hamiltonian, and the marginal positional and velocity distributions; (ii) the unique property that characterizes kappa distributions, that is, their invariant functional form for any kinetic or potential degrees of freedom; (iii) the statistical moments of the phase-space, velocity, and positional distributions; (iv) the kinetic definition of temperature, namely, that the second statistical moment of velocities is independent of the presence of the potential energy; (v) the general positional distributions; (vi) the positional dependence - profiles - of thermal variables such as the density, temperature, and pressure; (vii) the Bernoulli integral and the connection of the kappa and polytropic indices.