The Vlasov Equation

The collisionless kinetic equation that evolves a particle distribution function in six-dimensional phase space.

The equation

The Vlasov equation describes how the distribution function f(x, v, t) of a species evolves under smooth, collective electromagnetic forces with collisions neglected. It states that f is conserved along particle trajectories in phase space:

df/dt + v . grad_x f + (q/m)(E + v x B) . grad_v f = 0

Here grad_x is the gradient in space and grad_v the gradient in velocity. The three terms are the explicit time change, spatial streaming, and acceleration by the self-consistent fields.

Vlasov-Maxwell and Vlasov-Poisson

The fields E and B are not external: they are computed from the charge and current moments of f through Maxwell's equations. The electrostatic reduction, Vlasov-Poisson, closes the loop with div E = rho_c/epsilon0 only. This coupled nonlinearity produces Landau damping, plasma waves, and kinetic instabilities that no fluid model captures.

Why it is hard

f lives in six dimensions plus time. Direct grids are expensive, and fine velocity structure (filamentation) develops without bound, so the distribution becomes ever more oscillatory in velocity space.

How it is solved numerically

• Particle-in-cell: represent f by macro-particles sampled from it, push them in the fields, and deposit moments on a grid
• Eulerian (continuum) Vlasov: discretize phase space directly with high-order or semi-Lagrangian schemes
• Spectral velocity methods (Hermite, Fourier) that manage filamentation with closure or hyper-collision terms

Landau damping, a purely kinetic collisionless energy transfer from wave to particles, is the canonical test problem. Accurate collisionless kinetics underpins fast-ion and alpha-particle modeling, relevant to any D-T device.