VlasovMaxwellSystem

The Vlasov–Maxwell system is a fundamental kinetic model for collisionless plasmas. It describes how the distribution of charged particles in phase space evolves under self-consistent electromagnetic fields. The model treats different species, such as electrons and ions, through distribution functions fs(t,x,v) in six-dimensional phase space and couples them to the electromagnetic fields E(t,x) and B(t,x).

For each species s with charge qs and mass ms, the distribution function fs satisfies the Vlasov equation: ∂t fs + v·∇x fs + (qs/ms) [E + v×B]·∇v fs = 0. The fields obey Maxwell's equations: ∇·E = ρ/ε0, ∇×E = -∂t B, ∇·B = 0, ∇×B = μ0 J + μ0 ε0 ∂t E, where the charge and current densities are ρ(x,t) = ∑s qs ∫ fs dv and J(x,t) = ∑s qs ∫ v fs dv.

Non-relativistic form is common; relativistic versions are used for high-energy plasmas. In some regimes, the electrostatic approximation leads to the Vlasov–Poisson system, and reductions like gyrokinetics or fluid closures are employed to simplify analyses. The full system is posed in six-dimensional phase space and is typically highly challenging both analytically and numerically.

Key properties include conservation of total particle number for each species, energy conservation, and momentum conservation in the absence of external forcing. The system is often solved numerically by particle-in-cell methods or by direct Vlasov solvers on phase-space grids, and serves as a benchmark for plasma theory and simulations.

Applications encompass space and laboratory plasmas, such as solar wind, magnetospheric dynamics, and laser-plasma interactions, where collisionless behavior dominates and kinetic effects are essential.