Diffusive shock acceleration: non-classical model of cosmic ray transport
In this work the theory of diffusive shock acceleration is extended to the case of non-classical particle transport with Lévy flights and Lévy traps, when the mean square displacement grows nonlinearly with time. In this approach the Green function is not a Gaussian but it exhibits power-law tails. By using the propagator appropriate for non-classical diffusion, it is found for the first time that energy spectral index of particles accelerated at shock front is $γ= [α(\mathrm{r} + 5) - 6 β]/[α(\mathrm{r}-1)]$, where $0 < α< 2$ and $0 <β< 1$ are the exponents of power-law behavior of Lévy flights and Lévy traps, respectively. We note that this result coincides with standard slope at $α=2, β=1$ (normal diffusion), and also includes those obtained earlier for the subdiffusion ($α=2, β<1$) and superdiffusion ($α<2, β=1$) regimes.