are akin to ping pong balls trapped between converging paddles, here represented by the large scale plasma turbulence. It is simple to show that on average each time they cycle from upstream to downstream and back again they gain momentum given by $$\frac{\delta p}{p} \approx \frac{4}{3} \frac{u_1 - u_2}{v} = \alpha <<1,$$ where $u_1$ represents the upstream bulk plasma speed into the shock. Once the particles are relativistic the fractional momentum gain, $\alpha$ is a constant, as is the escape probability, ${\sf P}_e$.

Under such circumstances Fermi showed that the steady state momentum distribution is a powerlaw (since there is no associated momentum scale) given by $f(p) \propto p^{-q}$, with $q = 3 + \frac{{\sf P}_e}{\alpha}$. Fermi was not thinking of shocks, but in the late 1970s several people independently recognized how this Fermi-like process would work in shocks (Axford, 1977; Krymskii 1977; Bell 1978; Blandford & Ostriker 1978). In that case the powerlaw index of the momentum distribution becomes $q = \frac{3u_1}{u_1 - u_2} = \frac{3r}{r-1},$ where $r = u_1/u_2 = \rho_2/\rho_1$, is the compression through the shock. For a strong, nonrelativistic adiabatic shock $r\rightarrow 4$, so $q\rightarrow 4^+$. For even moderately strong shocks the powerlaw slope is close 