Energy-momentum-conserving stochastic differential equations and algorithms for nonlinear Landau-Fokker-Planck equation

TL;DR

This paper introduces new stochastic differential equations that exactly conserve energy and momentum for Coulomb collisions in plasmas, improving the physical fidelity of simulations.

Contribution

The authors derive novel SDEs with exact conservation laws and develop numerical algorithms that preserve these laws in plasma collision simulations.

Findings

New SDEs with exact energy-momentum conservation for Coulomb collisions.

Algorithms that preserve discrete conservation laws in plasma simulations.

Reduced computational complexity techniques for efficient simulations.

Abstract

Coulomb collision is a fundamental diffusion process in plasmas that can be described by the Landau-Fokker-Planck (LFP) equation or the stochastic differential equation (SDE). While energy and momentum are conserved exactly in the LFP equation, they are conserved only on average by the conventional corresponding SDEs, suggesting that the underlying stochastic process may not be well-defined by such SDEs. In this study, we derive new SDEs with exact energy-momentum conservation for the Coulomb collision by factorizing the collective effect of field particles into individual particles and enforcing Newton's third law. These SDEs, when interpreted in the Stratonovich sense, have a particularly simple form that represents pure diffusion between particles without drag. Numerical algorithms that preserve discrete conservation laws are developed and benchmarked in various relaxation processes. Techniques to reduce computational complexity are also discussed.