Weak Turbulence Theory of Velocity Space Diffusion and the Nonlinear Landau Damping of Waves

The particle-wave interaction to arbitrary order in perturbation theory is investigated. One finds that to every order the ensemble average distribution function obeys a diffusion equation. The diffusion constant specifies how the energy and momentum of particles change in time. By conservation of energy and momentum between waves and particles, the growth or damping rates of waves to any order can be determined. The growth rates for nonlinear instabilities and the damping rates for nonlinear Landau damping are explicitly calculated.