The fundamental assumptions of the adiabatic theory do not apply in presence of sharp field gradients as well as in presence of well developed magnetohydrodynamic turbulence. For this reason in such conditions the magnetic moment $mu$ is no longer expected to be constant. This can influence particle acceleration and have considerable implications in many astrophysical problems. Starting with the resonant interaction between ions and a single parallel propagating electromagnetic wave, we derive expressions for the magnetic moment trapping width $Delta mu$ (defined as the half peak-to-peak difference in the particle magnetic moment) and the bounce frequency $omega_b$. We perform test-particle simulations to investigate magnetic moment behavior when resonances overlapping occurs and during the interaction of a ring-beam particle distribution with a broad-band slab spectrum. We find that magnetic moment dynamics is strictly related to pitch angle $alpha$ for a low level of magnetic fluctuation, $delta B/B_0 = (10^{-3}, , 10^{-2})$, where $B_0$ is the constant and uniform background magnetic field. Stochasticity arises for intermediate fluctuation values and its effect on pitch angle is the isotropization of the distribution function $f(alpha)$. This is a transient regime during which magnetic moment distribution $f(mu)$ exhibits a characteristic one-sided long tail and starts to be influenced by the onset of spatial parallel diffusion, i.e., the variance $<(Delta z)^2 >$ grows linearly in time as in normal diffusion. With strong fluctuations $f(alpha)$ isotropizes completely, spatial diffusion sets in and $f(mu)$ behavior is closely related to the sampling of the varying magnetic field associated with that spatial diffusion.
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