Calculation of diffusive shock acceleration of charged particles by skew Brownian motion

In the study of diffusive shock acceleration of charged particles, Fokker-Planck diffusion equations can be replaced by stochastic differential equations that describe the trajectory of the guiding center and the momentum of randomly walking individual particles. Numerical solution of stochastic differential equations is much easier to achieve, and very complicated shock acceleration cases can be simulated. However, the divergence of plasma velocity is a delta-function at the shock, resulting in a singularity for the momentum gain rate. The straightforward way of calculating shock acceleration is very slow because it requires that the shock be treated with finite thickness and the particles diffuse many steps inside the velocity gradient region. To overcome this difficulty, we suggest the use of skew Brownian motion, a diffusion process that has asymmetric reflection probability on both sides of the shock. The skew Brownian motion can be solved by a scaling method that eliminates the delta-function in the stochastic differential equation. The particle momentum gain is proportional to the local time spent by the diffusion process at the shock. In this way, the shock can be treated as infinitely thin, and thus the speed of numerical simulation is greatly improved. This method has been applied to a few cases of shack acceleration models, and results from the stochastic process simulation completely agree with analytical calculation. In addition, we have outlined a method using time backward stochastic processes to solve general diffusive shock acceleration problems with an extended source of particle injection.