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

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.