Non-Perturbative Solutions to the Vlasov-Boltzmann Equation for Weakly Ionized Plasmas

TL;DR

This paper develops a non-perturbative, renormalization group-based approach to solve the Vlasov-Boltzmann equation, revealing new insights into electron energy distributions in low-temperature plasmas across different regimes.

Contribution

It introduces a novel RG method to derive analytic EEDF solutions valid from kinetic to hydrodynamic regimes, explaining experimental phenomena and plasma self-organization.

Findings

Derived analytic scaling relations for electron heating.

Predicted bimodal and exponential EEDF forms in different regimes.

Established a thermodynamic basis for plasma self-organization.

Abstract

The electron energy distribution function (EEDF) in low-temperature plasmas exhibits features not fully captured by classical collisional models, particularly across the transition from kinetic to hydrodynamic regimes. This work attributes these phenomena to a dynamically broken scale invariance within the Vlasov-Boltzmann equation. By applying renormalization group (RG) techniques directly to the kinetic operator, we derive non-perturbative EEDF solutions valid across a range of collisionality. The formalism yields analytic scaling relations for electron heating and predicts the emergence of distinct EEDF forms - bimodal in the kinetic limit and generalized exponential in the hydrodynamic limit - separated by a critical pressure. It is shown that the stable RG fixed point governing the system's long-time behavior corresponds to a state of minimum entropy production, establishing a thermodynamic basis for plasma self-organization. The theory provides a quantitative explanation for the experimentally observed universal data collapse of rescaled EEDFs and resolves standing issues like Godyak's EEDF metamorphosis.