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Anomalous Transport and Particle Acceleration at Shocks

The theory of first order Fermi acceleration at shocks assumes that particles diffuse due to scattering off slow-moving magnetic irregularities. However, cosmic rays are closely tied to magnetic field lines, and the transport process, particularly across the direction of the field, is likely to be more complicated. To describe cross field transport we employ recent extensions of the Rechester-Rosenbluth theory in localised stochastic regions of magnetic field. During acceleration at a shock and when the motion along the field is diffusive, there is a transition at a critical energy from \lq\lq sub-diffusive\rq\rq\ motion, where the mean square displacement of a particle increases with time as $t^{1/2}$, to compound diffusion, a combined process involving diffusion along a magnetic field which is itself wandering. Requiring this critical energy to be less than the system cut-off in a SNR of radius $R$ places an upper limit on the coherence length, $λ$, of the magnetic field for which diffusive shock acceleration can occur at quasi-perpendicular shocks when field line wandering dominates the effective transport of particles: $λ<R(U/v)^{1/2}$ where $U$ is the shock speed and $v$ the particle speed. The acceleration rate for particles in the sub-diffusive regime is derived and its implication for the origin of high energy cosmic rays is discussed.