2012ApJ...749..178B

We present a simple continuous-time random-walk model for the transport of energetic particles accelerated by a collection of supernova explosions in a galactic superbubble, developed to simulate and highlight signatures of anomalous transport on the particles' evolution and their spectra in a multi-shock context. We assume standard diffusive shock acceleration (DSA) theory for each shock encounter. The superbubble (an OB stars association) is idealized as a heterogeneous region of particle sources and sinks bounded by a random surface. The model is based on two coupled stochastic differential equations and is applied for protons and alpha particles. Using characteristic values for a typical bubble, our simulations suggest that acceleration and transport in the bubble may be sub-diffusive. In addition, a spectral break in the particles' evolution and spectra is evident located at ~10¹⁵ eV for protons and ~3x10¹⁵ eV for alphas. Our simulations are consistent with a bubble's mean magnetic field strength of ~1 µG and a shock separation distance ∼0.1xthe characteristic radius of the bubble. The simulations imply that the diffusion coefficient (for the elementary shock acceleration process) is ≲ 10²⁷ cm2/s at 1 GeV/c. While the sub-diffusive transport is readily attributed to the stochastic nature of the acceleration time according to DSA theory, the spectral break appears to be an artifact of transport in a finite medium. These simulations point to a new and intriguing phenomenon associated with the statistical nature of collective acceleration of high-energy cosmic rays in galactic superbubbles.