Landau damping is one of plasma physics’ most counterintuitive results: collisionless plasma oscillations decay spontaneously, not because energy is lost, but because it’s redistributed among particles traveling at different speeds. The mathematical proof took half a century to make rigorous. But even the celebrated Mouhot-Villani result requires initial data close to equilibrium.

This paper proves Landau damping for a two-species plasma — electrons and ions — with initial data far from equilibrium. The decay rate is optimal: t^(-d) in d dimensions. And the proof works in all of physical space, not just on a torus.

The two-species case is harder than single-species for a structural reason: the different masses create different timescales. Electrons oscillate fast; ions oscillate slow. The screening between species — each species partially shields the other’s electric field — introduces a coupling that doesn’t exist in single-species theory. The quasi-neutrality condition (electrons and ions must approximately cancel each other’s charge at large scales) acts as a constraint that shapes the long-time behavior.

The result matters because real plasmas are never single-species and never start near equilibrium. Solar wind, fusion devices, and interstellar media all involve multiple particle populations initialized by violent events. Proving that damping persists under these conditions extends Landau’s prediction from a beautiful theoretical statement to one that applies to the plasmas that actually exist.