Fractional Fick's Law and Anomalous Transport of Energetic Particles at Interplanetary Shocks.

This study examines the impact of non-local phenomena on the transport of energetic particles, which are frequently observed in interplanetary space near collisionless shocks. The prevailing theory of particle acceleration at shocks, known as diffusive shock acceleration (DSA), is based on the standard diffusion equation and predicts a specific density profile for energetic particles: upstream of the shock, the density is expected to exhibit exponential growth, while downstream, it should form a constant, flat profile. However, these predictions often conflict with observations around shocks in interplanetary space. In practice, data analysis and numerical simulations  indicate a long power-law-like upstream density profile, while the downstream region exhibits a decreasing, non-flat density profile. This discrepancy leads to consider of a modified version of the ordinary Fick's law relating the macroscopic diffusive flux of energetic particles with the number density. As described by Calvo et. al (2007), in this formulation the spatial derivative of the number density is replaced by an extended spatial integration of a function depending on the density profile multiplied by a statistical weight that decreases with increasing distance from the point where the flux is being evaluated, which yields a non-local diffusive flux. Such expression is called the Fractional Fick's Law as it involves fractional order derivatives. It can be shown that substituting this fractional flux into the continuity equation recovers the Fractional Diffusion Equation describing superdiffusion, that is, an anomalous diffusion regime in which the mean square displacement of particles grows super-linearly with time. We use a numerical Fortran 90 code for evaluating the fractional flux using the trapezoidal rule for integration. After verifying that the numerical method used is consistent and correct, this code is tested using the density profiles of accelerated particles at a numerically simulated shock in two scenarios: one involving normal diffusion and the other involving superdiffusion. Notably, in both cases, a downstream negative flux is observed, indicating the presence of uphill transport, i.e., transport in the same direction as the density gradient. Finally, the fractional flux is numerically evaluated using data from the ACE and Wind spacecrafts for two different shock crossings. In both events, uphill transport is observed, which is an intriguing and counterintuitive result that allows for the correct interpretation of satellite observations, as well as shedding light on the physical processes underlying the acceleration of energetic particles at space and astrophysical shocks.