Landau Damping — Collisionless Plasma

An electrostatic wave in a plasma damps without collisions via resonance: particles moving near the wave phase velocity v_ph = ω/k get trapped or accelerated. Particles slightly slower than v_ph gain energy (damp the wave), faster ones lose energy (anti-damp). Net effect depends on the slope of the distribution f'(v_ph).

Parameters

Damping rate γ/ω_p:
Regime:
Time: 0
Landau 1946: γ/ω ∝ (π/8)^½ · (ω/k)³/v_th³ · exp(−ω²/2k²v_th²)

f'(v_ph) < 0 → damping
f'(v_ph) > 0 → growth (bump-on-tail instability)

No collisions needed — purely kinetic, phase-space resonance