Fractional Fick’s law and uphill transport of energetic particles at collisionless shocks

In plasmas and in astrophysical systems, particle diffusion faster than normal, namely superdiffusion, has been detected, calling for a generalisation of Fick’s law and of the transport equation. Formally, superdiffusive transport is often described by fractional diffusion equations, where the second-order spatial derivative is changed into a spatial derivative of fractional order less than two, usually in the form of the so-called Riesz derivative. Fractional operators are non-local, so that this involves the contribution of very distant points (far from the particle source) to the particle flux at a given position in the system. To consider the property of non-locality in the case of anomalous transport, we give a simple analytical derivation of the fractional Fick’s law, where the contribution to the flux of distant points is weighted by an inverse power law, and show that this is consistent with use of the Riesz derivative in the transport equation. A numerical procedure for the computation of the non-local flux is presented and applied to both a simple Gaussian density profile and also to density profiles coming from test particle simulations of one-dimensional collisionless shocks. In these simulations, energetic particles can move diffusively or superdiffusively. The latter case naturally gives rise to the emergence of uphill transport in the downstream region, which means a flux of particles in the same direction of the density gradient. This analysis contributes to the interpretation of energetic particle fluxes accelerated at collisionless shock waves in the interplanetary medium.