Application of random walk theory to the first-order Fermi acceleration in shock waves

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We formulate the first-order Fermi acceleration in shock waves in terms of the random walk theory. The formulation is applicable to any value of the shock speed and the particle speed – in particular, to the acceleration in relativistic shocks and to the injection problem, where the particle speed is comparable to the fluid speed – as long as large-angle scattering is suitable for the scattering process of particles. We first show that the trajectory of a particle suffering from large-angle scattering can be treated as a random walk in a moving medium with an absorbing boundary (e.g., the shock front). We derive an integral equation to determine the density of scattering points of the random walk, and by solving it approximately we obtain approximate solutions of the probability density of pitch angle at, and the return probability after, the shock crossing in analytic form. These approximate solutions include corrections of several non-diffusive effects to the conventional diffusion approximation, and we show that they agree well with Monte Carlo results for the isotropic scattering model for any shock speed and particle speed. When we neglect effects of ‘a few-step return’, we obtain ‘the multistep approximation’ which includes only the effect of finite mean free path and which is equivalent to ‘the relativistic diffusion approximation’ used by Peacock if the correct diffusion length is used in his expression. We find that the multistep approximation is not appropriate to describe the probability densities of individual particles for relativistic shocks, but that the pitch angle distribution at the shock front in steady state is in practice quite well approximated by that given by the multistep approximation, because the effects of a finite mean free path and a few-step return compensate each other when averaged over the pitch angle distribution. Finally, we give an analytical expression for the spectral index of accelerated particles in parallel shocks valid for arbitrary shock speed using this approximation.