Action principle for Coulomb collisions in plasmas

TL;DR

This paper develops an action principle framework for Coulomb collisions in plasmas, enabling conservation law derivation and potential for variational numerical methods, despite the absence of a natural Lagrangian.

Contribution

It introduces an Eulerian variational formulation for the Landau-Fokker-Planck equation, linking it to Langevin dynamics and enabling variational integrator development.

Findings

Derived an action principle for Coulomb collisions in plasmas.

Established conservation laws using a generalized energy-momentum tensor.

Connected the action principle to Langevin equations for test particles.

Abstract

In this letter we derive an action principle for Coulomb collisions in plasmas. Although no natural Lagrangian exists for the Landau-Fokker-Planck equation, an Eulerian variational formulation is found considering the system of partial differential equations that couple the distribution function and the Rosenbluth potentials. Exact conservation laws are derived after generalizing the energy-momentum stress tensor for second order Lagrangians and, in the case of a test-particle population in a given plasma background, the action principle is shown to correspond to the Langevin equation for individual particles. Being suitable for discretization, the presented action allows construction of variational integrators. Numerical implementation is left for a future study.