Anisotropic kappa distributions. I. Formulation based on particle correlations

We develop the theoretical basis for the connection of the variety of anisotropic distributions with the statistical correlations among particles’ velocity components. By examining the most common anisotropic distribution function, we derive the correlation coefficient among particle energies, show how this correlation is connected to the effective dimensionality of the velocity distribution, and derive the connection between anisotropy and adiabatic polytropic index. Having established the importance of the correlation among particles in the formulation of anisotropic kappa distributions, we generalize these distributions within the framework of nonextensive statistical mechanics and based on the types of homogeneous or heterogeneous correlations among the particles’ velocity components. The formulation of the developed generalized distributions mediates the main two types of anisotropic kappa distributions that consider either (a) equal correlations, or (b) zero correlations, among different velocity components. Finally, the developed anisotropic kappa distributions are expressed in terms of the energy and pitch angle in arbitrary reference frames.