Generalized Plasma Dispersion Function: One-Solve-All Treatment, Visualizations, and Application to Landau Damping

TL;DR

This paper introduces a unified, efficient numerical method for calculating the plasma dispersion function across various distribution functions, addressing singularities, enabling precise Landau damping analysis, and providing visualizations and verification through simulations.

Contribution

It presents a novel, general approach for computing the plasma dispersion function for diverse distributions, extending beyond Maxwellian, with solutions to singularity issues and visual tools.

Findings

Effective calculation of plasma dispersion for various distributions.

Accurate Landau damping analysis beyond rough approximations.

Validation through linear initial value simulations.

Abstract

A unified, fast, and effective approach is developed for numerical calculation of the well-known plasma dispersion function with extensions from Maxwellian distribution to almost arbitrary distribution functions, such as the $\delta$, flat top, triangular, $\kappa$ or Lorentzian, slowing down, and incomplete Maxwellian distributions. The singularity and analytic continuation problems are also solved generally. Given that the usual conclusion $\gamma\propto\partial f_0/\partial v$ is only a rough approximation when discussing the distribution function effects on Landau damping, this approach provides a useful tool for rigorous calculations of the linear wave and instability properties of plasma for general distribution functions. The results are also verified via a linear initial value simulation approach. Intuitive visualizations of the generalized plasma dispersion function are also provided.