Probabilistic description of the first-order Fermi acceleration in shock waves: time-dependent solution by the single-particle approach

We give a new coherent description of the first-order Fermi acceleration of particles in shock waves from the point of view of the stochastic process of the individual particles, under the test particle approximation. Thetime development of the particle distribution function can be dealt with by this description, especially for relativistic shocks. We formulate the acceleration process of a particle as a two-dimensional Markov process in a logarithmic momentum-time space, and relate the solution of the Markov process to the particle distribution function at the shock front, for both steady and time-dependent cases. For the case where the probability density function of the energy gain and cycle-time at each shock crossing of the particles obeys a scaling law in momentum, which is usually assumed in the literature, it is confirmed in more general form that the energy distribution of particles has a power-law feature in steady state. The equation to determine the exact power-law index that is applicable for any shock speed is derived and it is shown that the power-law index, in general, depends on the shape of the probability density function of the energy gain at each shock crossing; in particular for relativistic shocks, the dispersion of the energy gain can influence the power-law index. It is also shown that the time-dependent solution has a self-similarity for the same case.