A new generalized formulation of dielectric tensor for multi-species plasma with anisotropic drift product-bi-kappa distribution

It is well established that space plasmas often contain particle components with high-energy tails that approximately follow a power-law distribution in velocity space. Such superthermal distributions, particularly those with an excess of fast particles, are more accurately described by generalized Kappa (Lorentzian) distributions rather than Maxwellian distributions. We propose the product-bi-kappa (PBK) distribution as an alternative to the kappa-Maxwellian (KM) and bi-Maxwellian (BM) distributions, leading to a new linear transformation form of the dielectric tensor for multi-component plasmas. This method utilizes the rational form and multi-pole expansion of the plasma dispersion function to convert the dielectric tensor of multi-component PBK plasmas into a linear eigenvalue system in conjunction with Maxwell's equations. Our approach transforms the traditional numerical iteration process for solving the finite roots of the dielectric tensor from initial values into an eigenvalue problem within a new, complete linear system, thus enabling the simultaneous determination of all principal eigenvalues. Constructing the eigenvalue system for multi-component plasmas is crucial for developing new, comprehensive oblique propagation solvers for multi-species anisotropic drift plasmas, including PBK, KM, and BM plasmas. The detailed construction of the linear eigenvalue system is extensively discussed in this work.