Proof of quasilinear equations in the strongly nonlinear regime of the weak warm beam instability

Quasilinear (QL) theory was developed in 1962 to describe the saturation of the weak warm beam-plasma instability, which involves the development of a Langmuir turbulence and the formation of a plateau in the electron velocity distribution function. The original derivations assume that particle orbits are weakly perturbed (quasi linear description), though the plateau formation is the result of a strong chaotic diffusion of the beam particles. Over two decades a controversy has developed about the validity of QL equations in the chaotic saturation regime within the Vlasovian description of the problem, and is not yet settled. Here a proof of these equations is proposed that does not resort to this description. Instead the Langmuir wave-beam system is described as a finite-number-of-degrees-of-freedom Hamiltonian system. The calculation of the chaotic drag on a beam particle is performed using Picard fixed point equation derived from Hamilton equations, and by making approximations justified by the spatial speading of chaotic orbits and by the weak mutual influence between any particle and any wave. The same techniques enable the computation of the particle diffusion coefficient and of the wave growth rate.