Computation of diffusive shock acceleration using stochastic differential equations

The present work considers diffusive shock acceleration at non-relativistic shocks using a system of stochastic differential equations (SDE) equivalen t to the Fokker-Planck equation. We compute approximate solutions of the tr ansport of cosmic particles at shock fronts with a SDE numerical scheme. On ly the first order Fermi process is considered. The momentum gain is given by implicit calculations of the fluid velocity gradients using a linear int erpolation between two consecutive time steps. We first validate our proced ure in the case of single shock acceleration and retrieve previous analytic al derivations of the spectral index for different values of the Peclet num ber. The spectral steepening effect by synchrotron losses is also reproduce d. A comparative discussion of implicit and explicit schemes for different shock thickness shows that implicit calculations extend the range of applic ability of SDE schemes to infinitely thin 1D shocks. The method is then app lied to multiple shock acceleration that can be relevant for Blazar jets an d accretion disks and for galactic centre sources. We only consider a syste m of identical shocks which free parameters are the distance between two co nsecutive shocks, the synchrotron losses time and the escape time of the pa rticles. The stationary distribution reproduces quite well the flat differe ntial logarithm energy distribution produced by multiple shock effect, and also the piling-up effect due synchrotron losses at a momentum where they e quilibrate the acceleration rate. At higher momenta particle losses dominat e and the spectrum drops. The competition between acceleration and loss eff ects leads to a pile-up shaped distribution which appears to be effective o nly in a restrict range of inter-shock distances of similar to 10-100 diffu sion lengths. We finally compute the optically thin synchrotron spectrum pr oduced such periodic pattern which can explain flat and/or inverted spectra observed in Flat Radio spectrum Quasars and in the galactic centre.