Scattering Theory in Noncanonical Phase Space: A Drift-Kinetic Collision Operator for Weakly Collisional Plasmas
Naoki Sato, Philip J. Morrison
TL;DR
This work develops a scattering theory for grazing Coulomb collisions in noncanonical phase space and constructs a five-dimensional drift-kinetic collision operator for guiding-center plasmas. The operator is uniquely fixed by the noncanonical Hamiltonian structure and possesses a metriplectic form, ensuring exact conservation laws and an H-theorem that yields generalized equilibria constrained by interior Casimir invariants, notably the magnetic moment $μ$. A comprehensive two-species extension is formulated, including coupled energy and entropy functionals and a two-species H-theorem, which predict self-organized, non-Maxwellian equilibria with spatial inhomogeneity tied to magnetic-field structure. The framework clarifies connections and distinctions with Landau and gyrokinetic theories, showing reduction to Landau in the canonical limit and offering a thermodynamically consistent, computationally efficient kinetic model for large-scale drift-kinetic turbulence, transport, and self-organization in both laboratory and astrophysical plasmas. Prospective extensions to gyrokinetics with finite Larmor radius effects are discussed as a natural but challenging avenue for future work.
Abstract
After developing a scattering theory for grazing collisions in general noncanonical phase spaces, we introduce a guiding center collision operator in five-dimensional phase space designed for plasma regimes characterized by long wavelengths (relative to the Larmor radius), low frequencies (relative to the cyclotron frequency), and weak collisionality (where repeated Coulomb collisions induce cumulatively small changes in particle magnetic moment). The collision operator is fully determined by the noncanonical Hamiltonian structure of guiding center dynamics and exhibits a metriplectic structure, ensuring the conservation of particle number, momentum, energy, and interior Casimir invariants. It also satisfies an H-theorem, allowing for deviations from Maxwell-Boltzmann statistics due to the nontrivial kernel of the noncanonical guiding center Poisson tensor, spanned by the magnetic moment. We propose that this collision operator and its underlying mathematical structure may offer valuable insights into the study of turbulence, transport, and self-organizing phenomena in both laboratory and astrophysical plasmas.