Fast spectral separation method for kinetic equation with anisotropic non-stationary collision operator retaining micro-model fidelity

TL;DR

This work addresses the limitation of the Landau collision operator in moderately coupled plasmas by learning a generalized, anisotropic, non-stationary kernel directly from molecular-dynamics data. It proposes a fast spectral separation framework that expresses the kernel as a low-rank tensor product of univariate encoders, enabling $O(N \log N)$ evaluation via FFT while preserving conservation laws and the H-theorem. The data-driven encoders are trained with a weak loss against MD trajectories, resulting in a kernel that captures heterogeneous energy transfer and particle correlations beyond the Landau model. Numerical results show improved accuracy over Landau in MD-like scenarios and substantial computational savings, with a structure-preserving discretization ensuring stability and physical fidelity. This approach advances efficient, high-fidelity collisional plasma modeling across a broader range of coupling regimes and lays the groundwork for implicit, inhomogeneous extensions.

Abstract

We present a generalized, data-driven collisional operator for one-component plasmas, learned from molecular dynamics simulations, to extend the collisional kinetic model beyond the weakly coupled regime. The proposed operator features an anisotropic, non-stationary collision kernel that accounts for particle correlations typically neglected in classical Landau formulations. To enable efficient numerical evaluation, we develop a fast spectral separation method that represents the kernel as a low-rank tensor product of univariate basis functions. This formulation admits an $O(N \log N)$ algorithm via fast Fourier transforms and preserves key physical properties, including discrete conservation laws and the H-theorem, through a structure-preserving central difference discretization. Numerical experiments demonstrate that the proposed model accurately captures plasma dynamics in the moderately coupled regime beyond the standard Landau model while maintaining high computational efficiency and structure-preserving properties.

Figures (8)

• Figure 1: Expectations of encoder functions $\mathbb{E}_{s}[g_{r}^2(u, r, s)]$ and $\mathbb{E}_{s}[g_{r}^2(u, r, s)]$ in Eq. \ref{['eq:CM2']}, and $\mathbb{E}_{s}[g_{1}^2(u, r, s)]$ and $\mathbb{E}_{s}[g_{2}^2(u, r, s)]$ in Eq. \ref{['eq:CM2_sep']} over $s$. Unlike the standard Landau collision operator, the present model takes an anisotropic and non-stationary kernel. In particular, $g_r^2 \le g_s^2$ (or equivalently, $g_2^2 \le g_1^2$) due to the collective interactions between the colliding pair and the environment.
• Figure 2: Predictions of the instantaneous velocity distribution on the $v_x$-$v_y$ plane at $t=0.5$ and $1.0$ with the initial distribution taking the GMM model defined by Eq. \ref{['eq:GMM_dist']}. "SS" represents the present collision kernel with the spectral separation structure.
• Figure 3: Predictions of the instantaneous velocity distribution along the radial direction with the initial condition taking the RM distribution defined by Eq. \ref{['eq:RM_dist']}.
• Figure 4: The simulation time per timestep of the direct simulation method and the present spectral separation (SS) method. The dotted lines represent $\mathcal{O}(N)$ and $\mathcal{O}(N^{2})$.
• Figure 5: The error of total mass $M$, momentum $\bm{P}$, and kinetic energy $E$ along with the simulation time. The grid number is $N=128^{3}$, and the initial condition takes the GMM (left) and RM (right) distribution.

Theorems & Definitions (9)

• Proposition 1
• proof
• Corollary 1
• proof
• Remark 1
• Proposition 2
• proof
• Remark 2
• Remark 3