A stochastic algorithm without time discretization error for the Wigner equation

Abstract

Stochastic particle methods for the numerical treatment of the Wigner equation are considered. The approximation properties of these methods depend on several numerical parameters. Such parameters are the number of particles, a time step (if transport and other processes are treated separately) and the grid size (used for the discretization of the position and the wave-vector). A stochastic algorithm without time discretization error is introduced. Its derivation is based on the theory of piecewise deterministic Markov processes. Numerical experiments are performed in a one-dimensional test case. Approximation properties with respect to the grid size and the number of particles are studied. Convergence of a time-splitting scheme to the no-splitting algorithm is demonstrated. The no-splitting algorithm is shown to be more efficient in terms of computational effort.

Wigner Ensemble Monte Carlo Simulation Without Discretization Error of a GaAs Resonant Tunneling Diode

“…The deltas in the formula (46) mean that the k y , k z of the particle do not change during the creation process. The majorant (13) in this case iŝ…”

“…Here, the Wigner potential is treated as a scattering source which determines the electron-potential interaction, and consequently new particles with different signs are stochastically added to the system. Recently this method has also been be understood in terms of the Markov jump process theory [12,9,13], producing a class of new stochastic algorithms. In this paper a thorough validation of a stochastic algorithm without time discretization error has been introduced in the case of a GaAs Resonant Tunneling Diode.…”

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A Monte Carlo technique for the solution of the Wigner transport equation has been developed during these years, based on the generation and annihilation of signed particles [16]. A stochastic algorithm without time discretization error has been recently introduced [13]. Its derivation is based on the theory of piecewise deterministic Markov processes.

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Wigner Monte Carlo simulation without discretization error of the tunneling rectangular barrier

“…where r ∈ U (0, 1) is a uniform random number, and the random waiting time τ is completely determined. In this case, a new no-splitting generation algorithm has been introduced in [18] without time discretization error, which will be validated with a well-known benchmark model in the next section.…”

“…Here, the Wigner potential is treated as a scattering source which determines the electron-potential interaction, and consequently new particles with different signs are stochastically added to the system. Recently this method has also been be understood in terms of the Markov jump process theory [15][16][17][18], producing a class of new stochastic algorithms.…”