Operator-Split Method for Variance Reduction in Stochastic Solutions of the Wigner Equation

The standard stochastic approach for simulation of carrier transport described by the Wigner equation introduces hard computational requirements. Averaged values of physical quantities are evaluated by means of numerical trajectories which accumulate statistical weight. The weight can take large positive and negative values which introduce large variance of the calculations. Aiming at variance reduction, we utilize the idea to split the weight so that a part is assigned to the trajectory and a part is left on a phase space grid for future processing. Formally this corresponds to a splitting of the kernel of the Wigner equation into two components. An operator equation is derived which couples the two kernel components and gives an answer how to further process the stored weight. The obtained Monte Carlo algorithm resembles a physical process of generation or annihilation of positive and negative particles. Variance reduction is achieved due to the partial annihilation of particles with opposite sign. Simulation results for a resonant-tunneling diode are presented.