Global change in action due to trapping, how to derive it whatever the rate of variation of the dynamics

In this paper, we investigate the motion of a set of charged particles acted upon by a growing electrostatic wave, in the limit when the initial wave amplitude is vanishingly small and when all the particles have the same initial action, $I_0$. We show, both theoretically and numerically that, when all the particles have been trapped in the wave potential, the distribution in action exhibits a very sharp peak about the smallest action. Moreover, as the wave keeps growing, the most probable action tends towards a constant, $I_f$, which we estimate theoretically. In particular, we show that $I_f$ may be calculated very accurately, when the particles' motion before trapping is far from adiabatic, by making use of a perturbation analysis in the wave amplitude. This fills a gap regarding the computation of the action change which, in the past, has only been addressed for slowly varying dynamics. Moreover, when the variations of the dynamics are fast enough, we show that the Fourier components of the particles' distribution function can be calculated by connecting estimates from our perturbation analysis with those obtained by assuming that all the particles have the same constant action, $I=I_f$. This result is used to compute theoretically the imaginary part of the electron susceptibility of an electrostatic wave in a plasma. Moreover, using our formula for the electron susceptibility, we can extend the range in $\epsilon_a$ (the parameter that quantifies the slowness of the dynamics) for our perturbative estimate of $I_f-I_0$. This range can actually be pushed down to values of $\epsilon_a$ allowing the use of neo-adiabatic techniques to compute the jump in action. Hence, this paper shows that the action change due to trapping can be calculated theoretically, whatever the rate of variation of the dynamics, by connecting perturbative results with neo-adiabatic ones.