Simultaneously regular and statistical acceleration

The propagation of a shock wave is accompanied by the excitation of space plasma, which leads to the development of different types of instabilities and the generation of magneto-hydrodynamic turbulence. Therefore, as was considered above in Sections 4.2-4.9, the particle scattering by moving magnetic inhomogeneities will give some additional statistical acceleration of particles simultaneously with the shock wave regular acceleration. This problem was considered approximately in the one-dimensional approximation in Berezhko et al. (1988). As it was shown in Section 4.9, the statistical acceleration by turbulent plasma can be considered as diffusion in the momentum space:

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