VlasovMaxwell

The Vlasov-Maxwell system, commonly written as Vlasov-Maxwell equations, is a set of coupled partial differential equations that model the evolution of a collisionless plasma. It describes the time evolution of the distribution function f_s(t, x, v) for each charged species s (for example electrons and ions) in phase space, together with the self-consistent electromagnetic fields E(t, x) and B(t, x). The framework captures how particles move under electromagnetic forces and how the particle distribution generates electric and magnetic fields.

For each species s, the Vlasov equation governs the distribution function:

∂f_s/∂t + v · ∇_x f_s + (q_s/m_s)(E + v × B) · ∇_v f_s = 0.

The electromagnetic fields satisfy Maxwell’s equations:

∇ · E = ρ/ε0, ∇ · B = 0, ∇ × E = -∂B/∂t, ∇ × B = μ0 J + μ0 ε0 ∂E/∂t.

Charge and current densities are obtained from the distribution functions:

ρ(t, x) = ∑_s ∫ q_s f_s(t, x, v) d^3v, J(t, x) = ∑_s ∫ q_s v f_s(t, x, v) d^3v.

The system is self-consistent: the fields accelerate particles, while the particle distributions shape the fields.

Variants include the electrostatic Vlasov-Poisson limit, the relativistic Vlasov-Maxwell equations, and reduced models such as gyrokinetic descriptions. Numerically, the system is commonly solved by particle-in-cell methods or high-dimensional Eulerian solvers. Applications span space and laboratory plasmas, including fusion devices, laser-plasma interactions, and magnetospheric physics.