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 $N\to \infty$; 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.

• Received 6 August 1999

DOI:https://doi.org/10.1103/PhysRevLett.84.3318