Abstract
We discuss the recent developments in the theory of diffusive shock acceleration (DSA) by using both first-principle kinetic plasma simulations and analytical theory based on the solution of the convection/diffusion equation. In particular, we show how simulations reveal that the spectra of accelerated particles are significantly steeper than the $E^{-2}$ predicted by the standard theory of DSA for strong shocks, in agreement with several observational pieces of evidence. We single out which standard assumptions of test-particle and non-linear DSA are violated in the presence of strong (self-generated) magnetic turbulence and put forward a novel theory in better agreement with the particle spectra inferred with multi-wavelength observations of young SN remnants, radio-SNe, and Galactic cosmic rays.
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