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Data-driven construction of a generalized kinetic collision operator from molecular dynamics

Yue Zhao, Joshua W. Burby, Andrew Christlieb, Huan Lei

TL;DR

The paper addresses the inadequacy of classical collision operators like Landau in plasmas with non-negligible correlations by learning a generalized collision operator CM2 directly from molecular dynamics. It defines CM2 as a symmetric, energy-conserving kernel ω = P ( gr^2 r_hat r_hat^T + gs^2 s_hat s_hat^T ) P, with u = v − v' and r, s encoding pair and environment interactions, and trains gr and gs via a weak-form objective using MD data. The key contributions are the demonstration that CM2 captures anisotropic, inhomogeneous energy transfer in the plane perpendicular to the relative velocity, preserves fundamental invariants, and remains accurate in the Γ ~ O(1) regime where Landau falters, while offering reduced computational cost compared to higher-dimensional kinetic models. This MD-informed, physics-preserving framework enables scalable, mesoscopic kinetic modeling for plasmas with stronger correlations and lays the groundwork for extensions to multi-species and inhomogeneous systems.

Abstract

We introduce a data-driven approach to learn a generalized kinetic collision operator directly from molecular dynamics. Unlike the conventional (e.g., Landau) models, the present operator takes an anisotropic form that accounts for a second energy transfer arising from the collective interactions between the pair of collision particles and the environment. Numerical results show that preserving the broadly overlooked anisotropic nature of the collision energy transfer is crucial for predicting the plasma kinetics with non-negligible correlations, where the Landau model shows limitations.

Data-driven construction of a generalized kinetic collision operator from molecular dynamics

TL;DR

The paper addresses the inadequacy of classical collision operators like Landau in plasmas with non-negligible correlations by learning a generalized collision operator CM2 directly from molecular dynamics. It defines CM2 as a symmetric, energy-conserving kernel ω = P ( gr^2 r_hat r_hat^T + gs^2 s_hat s_hat^T ) P, with u = v − v' and r, s encoding pair and environment interactions, and trains gr and gs via a weak-form objective using MD data. The key contributions are the demonstration that CM2 captures anisotropic, inhomogeneous energy transfer in the plane perpendicular to the relative velocity, preserves fundamental invariants, and remains accurate in the Γ ~ O(1) regime where Landau falters, while offering reduced computational cost compared to higher-dimensional kinetic models. This MD-informed, physics-preserving framework enables scalable, mesoscopic kinetic modeling for plasmas with stronger correlations and lays the groundwork for extensions to multi-species and inhomogeneous systems.

Abstract

We introduce a data-driven approach to learn a generalized kinetic collision operator directly from molecular dynamics. Unlike the conventional (e.g., Landau) models, the present operator takes an anisotropic form that accounts for a second energy transfer arising from the collective interactions between the pair of collision particles and the environment. Numerical results show that preserving the broadly overlooked anisotropic nature of the collision energy transfer is crucial for predicting the plasma kinetics with non-negligible correlations, where the Landau model shows limitations.

Paper Structure

This paper contains 12 sections, 2 theorems, 27 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Methods
  3. Numerical results
  4. Summary
  5. Proposition and construction of the generalized collision operator
  6. Training details
  7. The setup of MD simulations
  8. Validation of the high temperature regime
  9. Additional results of the low temperature regime
  10. Encoder functions of the CM2 model
  11. Additional results of the plasma kinetics in the low-temperature regime
  12. Time scaling parameter of the Landau model

Key Result

Proposition A.1

With the collision operator satisfying Eq. eq:conditions, the kinetic model eq:collision, strictly conserves the mass, momentum, and energy, and preserves the frame indifference constraints. Furthermore, it ensures non-negative solution and entropy production, and admits the Maxwellian distribution

Figures (18)

  • Figure 1: The energy transfer magnitude functions $\mathbb{E}_{s}[g_{r}^{2}(u,r,s)]$ and $\mathbb{E}_{s}[g_{s}^{2}(u,r,s)]$ of the present collision model CM2 in Eq. \ref{['eq:NN3']}. The ensemble average is taken over the component $s$ under the equilibrium distribution. Unlike the Landau model, $g_{r}^2 < g_{s}^2$ implies the anisotropic nature of the energy transfer arising from the collective interaction between the pair of particles and the environments.
  • Figure 2: The instantaneous distribution of the radial velocity magnitude with the BKW model as the initial condition predicted by the full MD simulations and the kinetic equation with the Landau, Boltzmann, and the present CM1 and CM2 collision models at (a) $t=0.2$ fs and (b) $t=0.6$ fs.
  • Figure 3: The instantaneous velocity PDF in the $v_{1}- v_{2}$ plane from a trimodal initial distribution predicted by the full MD, the Landau and the present CM2 collision model at $t = 0.2~\text{fs}$ (upper) and $0.6~\text{fs}$ (lower).
  • Figure 4: The instantaneous velocity PDF in the $v_{1}- v_{2}$ plane from an asymmetric double-well distribution predicted by the full MD, the Landau and the present CM2 collision model at $t = 0.2~\text{fs}$ (upper) and $0.6~\text{fs}$ (lower).
  • Figure 5: The instantaneous velocity PDF in the $v_{1}- v_{2}$ plane from a uniform initial distribution predicted by the full MD, the Landau and the CM1 collision model at $t = 2~\text{fs}$ (upper) and $4~\text{fs}$ (lower).
  • ...and 13 more figures

Theorems & Definitions (4)

  • Proposition A.1
  • proof
  • Proposition A.2
  • proof