The Chapman-Enskog Divergence Problem in Plasma Transport: Structural Limitations and a Practical Regularization Approach

TL;DR

This paper identifies a fundamental divergence in the Chapman-Enskog method for plasma transport coefficients and proposes a regularization using an effective collision frequency to obtain finite, physically consistent results across regimes.

Contribution

It reveals the structural origin of the divergence in Chapman-Enskog expansion and introduces a practical regularization approach based on an effective collision frequency.

Findings

The divergence is linked to the collision operator structure, not just closure artifacts.

The proposed regularization maintains conservation laws and thermodynamic consistency.

The regularized model aligns with exact solutions of a kinetic model, validating its effectiveness.

Abstract

We calculate transport coefficients from the Chapman--Enskog expansion with BGK collision operators, obtaining exactly $\kappa = \frac{5nT}{2m\nu}$, and show that maximum entropy closure yields identical results when applied with the same collision operator. Through structural arguments, we suggest that this $1/\nu$ divergence extends to other local collision operators of the form $\mathcal{L} = \nu\hat{L}$, making the divergence fundamental to the Chapman--Enskog approach rather than a closure artifact. To address this limitation, we propose a phenomenological effective collision frequency $\nu_{\eff} = \nu\sqrt{1 + \Kn^2}$ motivated by gradient-driven decorrelation, where $\Kn$ is the Knudsen number. We verify that this regularization maintains conservation laws and thermodynamic consistency while yielding finite transport coefficients across all collisionality regimes. Comparison with exact solutions of a bounded kinetic model shows similar functional form, providing limited validation of our approach. This work provides explicit calculation of a known divergence problem in kinetic theory and offers one phenomenological regularization method with transparent treatment of mathematical assumptions versus physical approximations.