Phase transition in the collisionless damping regime for wave-particle interaction.

Gibbs statistical mechanics is derived for the Hamiltonian system coupling a wave to N particles self-consistently. This identifies Landau damping with a regime where a second order phase transition occurs. For nonequilibrium initial data with warm particles, a critical initial wave intensity is found: above it, thermodynamics predicts a finite wave amplitude in the limit N-->infinity; below it, the equilibrium amplitude vanishes. Simulations support these predictions providing new insight into the long-time nonlinear fate of the wave due to Landau damping in plasmas.