Relaxation model for a homogeneous plasma with spherically symmetric velocity space

TL;DR

This work develops a nonequilibrium relaxation framework for homogeneous plasmas by deriving transport (moment) equations from the Vlasov-Fokker-Planck equation with the Fokker-Planck-Rosenbluth operator under spherical symmetry. The authors adopt a Maxwellian mixture model (MMM) and King function expansion (KFE) within the finitely distinguishable independent features (FDIF) hypothesis to obtain closed-form, higher-order transport equations expressed via Gauss hypergeometric functions ${}_2F_1$, linking fully nonlinear dynamics to tractable closures. The framework recovers classical limits, including the two-temperature thermal equilibrium model, the zeroth-order Braginskii heat transfer model, and the thermodynamic equilibrium model, while enabling adaptive, moment-convergent descriptions of nonlinear kinetic effects. This approach provides a robust, high-order, conservation-respecting tool for simulating nonlinear transport in fusion plasmas and offers a foundation for extending to axisymmetric velocity spaces and multi-scale kinetic phenomena.

Abstract

We derive the transport equations from the Vlasov-Fokker-Planck equation when the velocity space is spherically symmetric. The Shkarofsky's form of Fokker-Planck-Rosenbluth collision operator is employed in the Vlasov-Fokker-Planck equation. A closed-form relaxation model for homogeneous plasmas could be presented in terms of Gauss hypergeometric2F1 functions. This has been accomplished based on the Maxwellian mixture model. Furthermore, we demonstrate that classic models such as two-temperature thermal equilibrium model and thermodynamic equilibrium model are special cases of our relaxation model and the zeroth-order Braginskii heat transfer model can also be derived. The present relaxation model is a nonequilibrium model based on the hypothesis that the plasmas system possesses finitely distinguishable independent features, without relying on the conventional near-equilibrium assumption.

Figures (2)

• Figure 1: Illustration of the velocity distribution functions multiplied by a factor $(1+\hat{u}_a^2)$ for ${N_{K_a}} \equiv 1$ and various normalized average velocity $\hat{u}_a$.
• Figure 2: Flowchart of the kinetic moment-closed model for plasma when velocity space exhibits spherical symmetry without shell structure