American Institute of Mathematical Sciences

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March  2016, 9(1): 217-235. doi: 10.3934/krm.2016.9.217

A random cloud model for the Wigner equation

Wolfgang Wagner 1,

1. 

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

Received  December 2014 Revised  July 2015 Published  October 2015

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.
Mathematics Subject Classification: Primary: 60J25; Secondary: 81Q0.
Citation: Wolfgang Wagner. A random cloud model for the Wigner equation. Kinetic and Related Models, 2016, 9 (1) : 217-235. doi: 10.3934/krm.2016.9.217
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.

show all references

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.

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