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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.
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.
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.
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.
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