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Superdiffusive Stochastic Fermi Acceleration in Space and Energy

We analyze the transport properties of charged particles (ions and electrons) interacting with randomly formed magnetic scatterers (e.g.\ large scale local ``magnetic fluctuations&#39;&#39; or ``coherent magnetic irregularities&#39;&#39; usually present in strongly turbulent plasmas), using the energization processes proposed initially by Fermi in 1949. The scatterers are formed by large scale local fluctuations ($δB/B \approx 1$) and are randomly distributed inside the unstable magnetic topology. We construct a 3D grid on which a small fraction of randomly chosen grid points are acting as scatterers. In particular, we study how a large number of test particles are accelerated and transported inside a collection of scatterers in a finite volume. Our main results are: (1) The spatial mean-square displacement $<(Δr)^2>$ inside the stochastic Fermi accelerator is superdiffusive, $<(Δr)^2> \sim t^{a_{r}},$ with $a_r \sim 1.2-1.6$, for the high energy electrons with kinetic energy $(W)$ larger than $1 MeV$, and it is normal ($a_r=1$) for the heated low energy $(W< 10 keV)$ electrons. (2) The transport properties of the high energy particles are closely related with the mean-free path that the particles travel in-between the scatterers ($λ_{sc}$). The smaller $λ_{sc}$ is, the faster the electrons and ions escape from the acceleration volume. (3) The mean displacement in energy $<ΔW> \sim t^{a_{W}}$ is strongly enhanced inside the acceleration volume $(a_W=1.5- 2.5)$ for the high energy particles compared to the thermal low energy particles ($a_W=0.4$), i.e.\ high energy particles undergo an enhanced systematic gain in energy.(4) The mean-square displacement in energy $<W^2>$ is superdiffusive for the high energy particles and normal for the low energy, heated particles.