Superdiffusion of energetic particles at shocks: A Lévy flight model for acceleration

Context. In the heliosphere, power-law particle distributions are observed, for example, upstream of interplanetary shocks, which can result from superdiffusive transport. This non-Gaussian transport regime may be due to intermittent magnetic field structures. Recently, we have shown that a Lévy flight model reproduces the observed features at shocks: power-law distributions upstream of the shock and enhanced intensities at the shock. Aims. In this work, we extend the Lévy flight model to study the impact of superdiffusive transport on particle acceleration at shocks. We compared the acceleration timescale and spectral slope to Gaussian diffusion and a Lévy walk model. Methods. We solved the fractional transport equation by sampling the number density with the corresponding stochastic differential equation that is driven by an alpha-stable Lévy distribution. For both Gaussian and superdiffusive transport, we used a modified version of the cosmic ray propagation framework CRPropa 3.2. Results. We obtained the number density and energy spectra for constant and energy-dependent anomalous diffusion, and we find, compared to the case of Gaussian diffusion, harder energy spectra at the shock as well as faster acceleration. The spectral slope is even harder than predicted for Lévy walks. Conclusions Lévy flight models of superdiffusive transport lead to observed features in the heliosphere. We further show that superdiffusive transport impacts the acceleration process by changing the probability of escaping the shock. The flexibility of the Lévy flight model allows for further studies in the future that can take the shock geometry and magnetic field structure into account.