= 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 $\langle\Delta p\rangle/p=4(u_1-u_2)/(3v)$ 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.
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