Linear and nonlinear regimes of bump-on-tail instability through Vlasov and toy model simulations

Resonant wave-particle interaction in collisionless unmagnetized plasmas is numer- ically investigated by means of a Fermi-like model, focusing on the linear and nonlinear regimes of the well-known bump-on-tail instability. Within this toy model, particle trapping effects are described through elastic collisions of particles with two barriers separated by a fixed length and whose amplitude (proportional to the wave energy) can increase or decrease in time, due to the sequence of stochastic collisions. The systematic comparison of the toy model numerical results with those obtained from Vlasov-Poisson simulations as well as with the predictions of kinetic theory, shows that the nonlinear map, on which the Fermi-like model is based, captures the basic physics of the linear growth of the bump-on-tail instability and of the particle trapping effects which produce the saturation of the instability and drive the nonlinear phase of wave-particle interaction.