The Vlasov equation is the standard description of many-body systems with mean-field coupling [1,4,5]. However, chaotic dynamics imply a departure from the Vlasov behaviour on relatively short time scales, so that the many-body evolution often looks more stochastic than predicted by the Vlasov equation. We consider two examples. First, for a plasma beam with N particles interacting with a wave, finite-N effects (granularity of the empirical, microscopic distribution function) have been shown to drive the system over long times towards a thermal equilibrium, which is not the (metastable) state corresponding to the saturation of the beam-wave instability in the Vlasovian picture. [1,3] Second,

quasilinear 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 quasilinear equations in the chaotic saturation regime within the Vlasovian description of the problem, and is not yet settled. It is worth noting that, in contrast to the Vlasov equation, the quasilinear equations imply irreversible behaviour, with an H-theorem. We derive the quasilinear equations in the strongly nonlinear chaotic regime without resorting to the previous description. Instead, the Langmuir wave-beam system is described as a Hamiltonian system with a finite number of degrees of freedom. A new technique enables one to derive the quasilinear evolution in the limit of a continuous wave spectrum (i.e. of infinite Chirikov resonance overlap parameter s). The close to rigorous argument takes advantage of the fact that the motion of any particle and the evolution of any wave depend only weakly on each other. [1,2]

[1] Y. Elskens and D. Escande, Microscopic dynamics of plasmas and chaos (IoP Publishing, Bristol, 2002).

[2] D. Escande and Y. Elskens, Phys. Lett. A 302 (2002) 110-118 ; Phys. Plasmas 10 (2003) 1588-1594 ; Plasma Phys. Contr. Fusion 45 (2003) A115-124.

[3] M-C. Firpo, F. Doveil, Y. Elskens, P. Bertrand, M. Poleni and D. Guyomarc’h, Phys. Rev. E 64 (2001) 016411.

[4] M-C. Firpo and Y. Elskens, J. Stat. Phys. 93 (1998) 193-209