Non-markovian pitch-angle scattering as the origin of particle superdiffusion parallel to the magnetic field

We develop a theoretical model for particle superdiffusive transport parallel to the average magnetic field, due to the pitch-angle scattering times having a non-Markovian, power-law probability distribution. We show that a non- Markovian Fokker-Planck equation can be derived, where the traditional time derivative is changed for a fractional time derivative. By solving the fractional Fokker-Planck equation, with the time-dependent part having solutions that are expressed by the Mittag-Leffler functions, it is found that an initial pitch-angle distribution slowly decays toward isotropy. This leads to a parallel velocity autocorrelation function that also has a slow power-law decay in time, thus implying superdiffusive transport in the direction parallel to the background magnetic field. In this framework, we derive for the first time the anomalous diffusion coefficient as a function of physical parameters like the background magnetic field, the resonant turbulence level, and the particle speed.