# A Unified Numerically Solvable Framework for Complicated Kinetic Plasma Dispersion Relations

## TL;DR

This paper introduces a comprehensive, numerically solvable framework for kinetic plasma dispersion relations accommodating multiple species, arbitrary drifts, and collisions, applicable to various plasma environments including space, astrophysics, and fusion.

## Contribution

It develops the most general matrix-based numerical method for solving kinetic plasma dispersion relations with multiple species, drifts, and collision effects, extending previous models significantly.

## Key findings

- Provides a unified dispersion relation framework for diverse plasma conditions.
- Enables solving complex plasma wave problems without initial root guesses.
- Applicable to space, astrophysics, fusion, and laser plasma studies.

## Abstract

A unified numerically solvable framework for dispersion relations with arbitrary number of species drifting at arbitrary directions and with Krook collision is derived for linear uniform/homogenous kinetic plasma, which largely extended the standard one [say, T. Stix, {\em Waves in Plasmas}, AIP Press, 1992]. The purpose of this work is to provide a kinetic plasma dispersion relation tool not only the physical model but also the numerical approach be as general/powerful as possible. As a very general application example, we give the final dispersion relations which assume further the equilibrium distribution function be bi-Maxwellian and including parallel drift, two directions of perpendicular drift (i.e., drift across magnetic field), ring beam and loss-cone. Both electromagnetic and electrostatic versions are provided, with also the Darwin (a.k.a., magnetoinductive or magnetostatic) version. The species can be treated either magnetized or unmagnetized. Later, the equations are transformed to the matrix form be solvable by using the powerful matrix algorithm [H. S. Xie and Y. Xiao, Plasma Science and Technology, 18, 2, 97, 2016], which is the first approach can give all the important solutions of a linear kinetic plasma system without requiring initial guess for root finding and thus can be extremely useful to the community. To our knowledge, the present model would be the most comprehensive one in literature for the distribution function constructed bases on Maxwellian, which thus can be applied widely for study waves and instabilities in space, astrophysics, fusion and laser plasma. We limit the present work to non-relativistic case.