Hamiltonian reduction of Vlasov-Maxwell to a dark slow manifold
arXiv:2104.03092 · doi:10.1017/S0022377821000556
Abstract
We show that nonrelativsitic scaling of the collisionless Vlasov-Maxwell system implies the existence of a formal invariant slow manifold in the infinite-dimensional Vlasov-Maxwell phase space. Vlasov-Maxwell dynamics restricted to the slow manifold recovers the Vlasov-Poisson and Vlasov-Darwin models as low-order approximations, and provides higher-order corrections to the Vlasov-Darwin model more generally. The slow manifold may be interpreted to all orders in perturbation theory as a collection of formal Vlasov-Maxwell solutions that do not excite light waves, and are therefore "dark". We provide a heuristic lower bound for the time interval over which Vlasov-Maxwell solutions initialized optimally-near the slow manifold remain dark. We also show how dynamics on the slow manifold naturally inherit a Hamiltonian structure from the underlying system. After expressing this structure in a simple form, we use it to identify a manifestly Hamiltonian correction to the Vlasov-Darwin model. The derivation of higher-order terms is reduced to computing the corrections of the system Hamiltonian restricted to the slow manifold.
arXiv comment34 pages
References in corpus (12)
- GEMPIC: Geometric ElectroMagnetic Particle-In-Cell Methods
- A multi-dimensional, energy- and charge-conserving, nonlinearly implicit, electromagnetic Vlasov-Darwin particle-in-cell algorithm
- Simulations of electromagnetic effects in high frequency capacitively coupled discharges using the Darwin approximation
- Finite-dimensional collisionless kinetic theory
- Slow manifold reduction for plasma science
- Differential forms for plasma physics
- Magnetohydrodynamic motion of a two-fluid plasma
- Guiding center dynamics as motion on a formal slow manifold in loop space
- Long time estimates for the Vlasov-Maxwell system in the non-relativistic limit
- Casimir invariants and the Jacobi identity in Dirac's theory of constrained Hamiltonian systems
- An integral transform technique for kinetic systems with collisions
- Hamiltonian formulations for perturbed dissipationless plasma equations