The numerical analysis of random particle methods applied to Vlasov-Poisson-Fokker-Planck kinetic equations

We devise and study a random particle method for approximating Vlasov-Poisson-Fokker-Planck systems. Such a proposed scheme takes into account the fact that the trajectories of a particle, undergoing Brownian motion due to collisions with the medium or background particles, can be obtained as the solutions of stochastic differential equations, i.e., the Langevin equations. These equations are the precise analogs of the deterministic Hamiltonian system in the collisionless model. The particle approximation, in particular, simulates the action of viscosity by the use of independent Wiener processes (Brownian motions). The error analysis generalizes that of Ganguly and Victory [SIAM J. Numer. Anal., 26 (1989), pp. 249-288] by incorporating into the consistency estimates the sampling errors generated by the random motion of the particles. These particular errors are the dominant component of the consistency errors and are estimated by applying Bennett's inequality for estimating tail probabilities for standardized sums of independent random variables with zero means. The stability estimates for the particle approximations to the collisionless model are extended to the Vlasov-Poisson-Fokker-Planck setting by means of this inequality.