Diffusive shock acceleration: non-classical model of cosmic ray transport

Abstract

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 ${\hat{I}}^3=[\hat{I}\pm (\mathrm{r}+5)-6{\hat{I}}^2]/[\hat{I}\pm (\mathrm{r}-1)]$, where $0<\hat{I}\pm <2$ and $0<{\hat{I}}^2<1$ are the exponents of power-law behavior of Lévy flights and Lévy traps, respectively. We note that this result coincides with standard slope at $\hat{I}\pm =2,{\hat{I}}^2=1$ (normal diffusion), and also includes those obtained earlier for the subdiffusion ($\hat{I}\pm =2,{\hat{I}}^2<1$) and superdiffusion ($\hat{I}\pm <2,{\hat{I}}^2=1$) regimes.

17 pages, the 5th International Symposium on Cosmic Rays and Astrophysics (ISCRA-2025)