Diffusive shock acceleration

Diffusive shock acceleration repeatedly returns particles to a shock wave by scattering them in the upstream and downstream plasma. For fluid speeds $u_1>u_2$, particle speed $v\gg u_1$, and shock compression ratio $r=u_1/u_2$, the cycle gain is $⟨\Delta p⟩/p=4(u_1-u_2)/(3v)$ and the cycle escape probability is $4u_2/v$. Their ratio gives $f(p)\propto p^{-3r/(r-1)}$ for the isotropic phase-space one-particle distribution function. For $r=4$ this is $p^{-4}$, corresponding to $dN/dE\propto E^{-2}$ in the ultrarelativistic regime.