A random cloud model for the Wigner equation

Wolfgang Wagner

doi:10.3934/krm.2016.9.217

Article Contents

2016, Volume 9, Issue 1: 217-235. Doi: 10.3934/krm.2016.9.217

This issue Previous Article A realizability-preserving high-order kinetic scheme using WENO reconstruction for entropy-based moment closures of linear kinetic equations in slab geometry Next Article

A random cloud model for the Wigner equation

Wolfgang Wagner^1,

1.
Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39 - 10117 Berlin

Received: November 30, 2014

Revised: June 30, 2015

Published: February 2016

Abstract Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 60J25; Secondary: 81Q05.

Citation:

Related Papers

Cited by

References

[1]	L. Breiman, Probability, Addison-Wesley Publishing Company, Reading, Mass., 1968.
[2]	M. H. A. Davis, Markov Models and Optimization, Chapman & Hall, London, 1993.doi: 10.1007/978-1-4899-4483-2.
[3]	I. M. Gamba, M. P. Gualdani and R. W. Sharp, An adaptable discontinuous Galerkin scheme for the Wigner-Fokker-Planck equation, Commun. Math. Sci., 7 (2009), 635-664, URL http://projecteuclid.org/euclid.cms/1256562817.doi: 10.4310/CMS.2009.v7.n3.a7.
[4]	I. M. Gamba, S. Rjasanow and W. Wagner, Direct simulation of the uniformly heated granular Boltzmann equation, Math. Comput. Modelling, 42 (2005), 683-700.doi: 10.1016/j.mcm.2004.02.047.
[5]	T. M. M. Homolle and N. G. Hadjiconstantinou, A low-variance deviational simulation Monte Carlo for the Boltzmann equation, J. Comput. Phys., 226 (2007), 2341-2358.doi: 10.1016/j.jcp.2007.07.006.
[6]	P. A. Markowich, On the equivalence of the Schrödinger and the quantum Liouville equations, Math. Methods Appl. Sci., 11 (1989), 459-469.doi: 10.1002/mma.1670110404.
[7]	P. A. Markowich and C. A. Ringhofer, An analysis of the quantum Liouville equation, Z. Angew. Math. Mech., 69 (1989), 121-127.doi: 10.1002/zamm.19890690303.
[8]	M. Nedjalkov, H. Kosina, S. Selberherr, C. Ringhofer and D. K. Ferry, Unified particle approach to Wigner-Boltzmann transport in small semiconductor devices, Phys. Rev. B, 70 (2004), 115319.doi: 10.1103/PhysRevB.70.115319.
[9]	R. I. A. Patterson and W. Wagner, A stochastic weighted particle method for coagulation-advection problems, SIAM J. Sci. Comput., 34 (2012), B290-B311.doi: 10.1137/110843319.
[10]	R. I. A. Patterson, W. Wagner and M. Kraft, Stochastic weighted particle methods for population balance equations, J. Comput. Phys., 230 (2011), 7456-7472.doi: 10.1016/j.jcp.2011.06.011.
[11]	D. Querlioz and P. Dollfus, The Wigner Monte Carlo Method for Nanoelectronic Devices, Wiley, 2010.doi: 10.1002/9781118618479.
[12]	G. A. Radtke, N. G. Hadjiconstantinou and W. Wagner, Low-noise Monte Carlo simulation of the variable hard sphere gas, Phys. Fluids, 23 (2011), 030606(1-12).doi: 10.1063/1.3558887.
[13]	S. Rjasanow and W. Wagner, A stochastic weighted particle method for the Boltzmann equation, J. Comput. Phys., 124 (1996), 243-253.doi: 10.1006/jcph.1996.0057.
[14]	S. Rjasanow and W. Wagner, Simulation of rare events by the stochastic weighted particle method for the Boltzmann equation, Math. Comput. Modelling, 33 (2001), 907-926.doi: 10.1016/S0895-7177(00)00289-2.
[15]	J. M. Sellier, M. Nedjalkov, I. Dimov and S. Selberherr, A benchmark study of the Wigner Monte Carlo method, Monte Carlo Methods Appl., 20 (2014), 43-51.doi: 10.1515/mcma-2013-0018.
[16]	W. Wagner, Deviational particle Monte Carlo for the Boltzmann equation, Monte Carlo Methods Appl., 14 (2008), 191-268.doi: 10.1515/MCMA.2008.010.
[17]	W. Wagner, A random cloud model for the Schrödinger equation, Kinetic and Related Models, 7 (2014), 361-379.doi: 10.3934/krm.2014.7.361.
[18]	E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev., 40 (1932), 749-759.