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We develop the theoretical basis for the connection of the variety ofanisotropic distributions with the statistical correlations among particlesvelocity components. By examining the most common anisotropic distribution, wederive the correlation coefficient among particle energies, show how thiscorrelation is connected to the effective dimensionality of the velocitydistribution, and derive the connection between anisotropy and adiabaticpolytropic index. Having established the importance of correlation amongparticles in the formulation of anisotropic kappa distributions, we generalizethese distributions within the framework of nonextensive statistical mechanicsand based on the types of homogeneous or heterogeneous correlations among theparticles velocity components. The formulation of the developed generalizeddistributions mediates the main two types of anisotropic kappa distributions,considering (a) equal correlations, or (b) zero correlations, among differentvelocity components. Finally, the developed anisotropic kappa distributions areexpressed in terms of the energy and pitch angle in arbitrary reference frames.

Anisotropic kappa distributions I: Formulation based on particle correlations