  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.
