A random cloud model for the Wigner equation
Wolfgang Wagner
Kinetic & Related Models 2016, 9(1): 217-235 doi: 10.3934/krm.2016.9.217
A probabilistic model for the Wigner equation is studied. The model is based on a particle system with the time evolution of a piecewise deterministic Markov process. Each particle is characterized by a real-valued weight, a position and a wave-vector. The particle position changes continuously, according to the velocity determined by the wave-vector. New particles are created randomly and added to the system. The main result is that appropriate functionals of the process satisfy a weak form of the Wigner equation.
keywords: piecewise deterministic Markov process. probabilistic representation Wigner equation stochastic particle model
Some properties of the kinetic equation for electron transport in semiconductors
Wolfgang Wagner
Kinetic & Related Models 2013, 6(4): 955-967 doi: 10.3934/krm.2013.6.955
The paper studies the kinetic equation for electron transport in semiconductors. New formulas for the heat generation rate are derived by analyzing the basic scattering mechanisms. In addition, properties of the steady state distribution are discussed and possible extensions of the deviational particle Monte Carlo method to the area of electron transport are proposed.
keywords: Electron transport equation Monte Carlo algorithm. semiconductors steady state distribution heat generation
A stochastic algorithm without time discretization error for the Wigner equation
Orazio Muscato Wolfgang Wagner
Kinetic & Related Models 2019, 12(1): 59-77 doi: 10.3934/krm.2019003

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.

keywords: Monte Carlo methods Wigner distribution semiconductors
A random cloud model for the Schrödinger equation
Wolfgang Wagner
Kinetic & Related Models 2014, 7(2): 361-379 doi: 10.3934/krm.2014.7.361
The paper is concerned with the construction of a stochastic model for the spatially discretized time-dependent Schrödinger equation. The model is based on a particle system with a Markov jump evolution. The particles are characterized by a sign (plus or minus), a position (discrete grid) and a type (real or imaginary). The jumps are determined by the creation of offspring. The main result is the construction of a family of complex-valued random variables such that their expected values coincide with the solution of the Schrödinger equation.
keywords: Markov jump process. stochastic particle model probabilistic representation Schrödinger equation
Properties of the steady state distribution of electrons in semiconductors
Orazio Muscato Wolfgang Wagner Vincenza Di Stefano
Kinetic & Related Models 2011, 4(3): 809-829 doi: 10.3934/krm.2011.4.809
This paper studies a Boltzmann transport equation with several electron-phonon scattering mechanisms, which describes the charge transport in semiconductors. The electric field is coupled to the electron distribution function via Poisson's equation. Both the parabolic and the quasi-parabolic band approximations are considered. The steady state behaviour of the electron distribution function is investigated by a Monte Carlo algorithm. More precisely, several nonlinear functionals of the solution are calculated that quantify the deviation of the steady state from a Maxwellian distribution with respect to the wave-vector. On the one hand, the numerical results illustrate known theoretical statements about the steady state and indicate directions for further studies. On the other hand, the nonlinear functionals provide tools that can be used in the framework of Monte Carlo algorithms for detecting regions in which the steady state distribution has a relatively simple structure, thus providing a basis for domain decomposition methods.
keywords: electron transport Monte Carlo algorithm. Boltzmann-Poisson equation steady state distribution semiconductors

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