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In this paper, we investigate the motion of a set of charged particles actedupon by a growing electrostatic wave, in the limit when the initial waveamplitude is vanishingly small and when all the particles have the same initialaction, $I_0$. We show, both theoretically and numerically that, when all theparticles have been trapped in the wave potential, the distribution in actionexhibits a very sharp peak about the smallest action. Moreover, as the wavekeeps growing, the most probable action tends towards a constant, $I_f$, whichwe estimate theoretically. In particular, we show that $I_f$ may be calculatedvery accurately, when the particles' motion before trapping is far fromadiabatic, by making use of a perturbation analysis in the wave amplitude. Thisfills a gap regarding the computation of the action change which, in the past,has only been addressed for slowly varying dynamics. Moreover, when thevariations of the dynamics are fast enough, we show that the Fourier componentsof the particles' distribution function can be calculated by connectingestimates from our perturbation analysis with those obtained by assuming thatall the particles have the same constant action, $I=I_f$. This result is usedto compute theoretically the imaginary part of the electron susceptibility ofan electrostatic wave in a plasma. Moreover, using our formula for the electronsusceptibility, we can extend the range in $\epsilon_a$ (the parameter thatquantifies 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 becalculated theoretically, whatever the rate of variation of the dynamics, byconnecting perturbative results with neo-adiabatic ones.

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