Energetic particles are ubiquitous in space and astrophysical plasmas, and interplanetary shocks are widely regarded as one of the main particle accelerators in the heliosphere. Indeed, in situ measurements typically show that energetic particle fluxes peak at the shock, indicating a local acceleration process. Furthermore, the time profile of energetic particle fluxes is highly influenced by particle transport properties upstream and downstream of the shock. By advancing previous numerical test-particle models that simulate the transport of monoenergetic particles around an infinite planar shock, in this work we add the acceleration of such particles via energy gains at each shock crossing, in a first-order Fermi-type mechanism. Moreover, the acceleration of a 70 keV particle population, namely the seed population, is reproduced by integrating a Langevin-type equation upstream and downstream of an infinite planar shock. Particles can diffuse in the simulation box via random `kicks’, which belong either to a Gaussian distribution (normal diffusion) or to a Lévy distribution (superdiffusion). We performed several simulations by varying the parameters of the model. The particle energy spectra in both diffusive and superdiffusive simulations are in remarkable agreement with the theoretical predictions. The output energetic particle densities have been compared with those observed by the ACE spacecraft during an interplanetary shock crossing on December 14 2006. We show not only that particle fluxes in different energy bins reproduce very well the observed ones upstream and downstream when superdiffusion is at work, but also that anomalous, superdiffusive transport speeds up the acceleration process and leads to particle energy values that are consistent with observations.
Prete et al. (2026) studied this question.