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

TL;DR

The paper addresses diffusive shock acceleration (DSA) under non-classical transport, where cosmic rays experience Lévy flights and Lévy traps leading to nonlinear mean-square displacement. It develops a 1D fractional diffusion framework using a fractional Laplacian and a fractional derivative, derives the Green's function via a fractional-stable propagator, and applies Bell's DSA framework to non-classical transport. A central result is the explicit spectral index $\\gamma = \\\frac{\\alpha( r+5 ) - 6\\beta}{\\alpha(r-1)}$, which reduces to the classical $\\gamma = \\\frac{\\r+2}{\\r-1}$ in the normal-diffusion limit and encompasses subdiffusive and superdiffusive regimes. This work broadens the DSA paradigm to fractal and intermittently turbulent media, offering a unified description of CR spectra under non-homogeneous diffusion.

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 $γ= [α(\mathrm{r} + 5) - 6 β]/[α(\mathrm{r}-1)]$, 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.