  We set up a mapping procedure able to translate the evolution of the radial
profile of fast ions, interacting with Toroidal Alfv\'en Eigenmodes, into the
dynamics of an equivalent one dimensional bump-on-tail system. We apply this
mapping technique to reproduce ITER relevant simulations, which clearly
outlined deviations from the diffusive quasi-linear (QL) model. Our analysis
demonstrates the capability of the one-dimensional beam-plasma dynamics to
predict the relevant features of the non-linear hybrid LIGKA/HAGIS simulations.
In particular, we clearly identify how the deviation from the QL evolutive
profiles is due to the presence of avalanche processes. A detailed analysis
regarding the reduced dimensionality is also addressed, by means of phase-space
slicing based on constants of motion. In the conclusions, we outline the main
criticalities and outcomes of the procedure, which must be satisfactorily
addressed to make quantitative prediction on the observed outgoing fluxes in a
Tokamak device.
