Beam–plasma instability and fast particles: the Lynden-Bell approach

Beam–plasma instability, i.e., the response of plasma bulk to the injection of supra-thermal charged-particle beams, can be appropriately characterized by a long-range interaction system. This physical system hosts a number of very interesting phenomena and, in particular, the emergence of long-lived quasi-stationary states. We characterize the self-consistent distribution functions of such out-of-equilibrium states by means of the Lynden-Bell theory. The prediction of this theory, based on the statistical mechanics of the Vlasov equation, are checked against the outcomes of numerical simulations of the discrete system. Moreover, a phenomenological study of the effective resonance band for the system response is also addressed. A threshold value is found in the initial spread of beam-particle momenta. This threshold allows discrimination between the resonant and non-resonant regimes. The analysis of the thermalization of a few percent of the beam population characterized by large initial momenta (with respect to the main part of the beam itself) is also performed and it confirms and deepens our understanding of the physical meaning of the mentioned threshold.