A random cloud model for the Wigner equation. (English) Zbl 1329.60352
Summary: 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.
MSC:
| 60K35 | Interacting random processes; statistical mechanics type models; percolation theory |
| 60J25 | Continuous-time Markov processes on general state spaces |
| 81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
Keywords:
stochastic particle model; Wigner equation; piecewise deterministic Markov process; probabilistic representationReferences:
| [1] | L. Breiman, <em>Probability</em>,, Addison-Wesley Publishing Company (1968) · Zbl 0174.48801 |
| [2] | M. H. A. Davis, <em>Markov Models and Optimization</em>,, Chapman & Hall (1993) · Zbl 0780.60002 · doi:10.1007/978-1-4899-4483-2 |
| [3] | I. M. Gamba, An adaptable discontinuous Galerkin scheme for the Wigner-Fokker-Planck equation,, Commun. Math. Sci., 7, 635 (2009) · Zbl 1185.65180 · doi:10.4310/CMS.2009.v7.n3.a7 |
| [4] | I. M. Gamba, Direct simulation of the uniformly heated granular Boltzmann equation,, Math. Comput. Modelling, 42, 683 (2005) · Zbl 1088.35049 · doi:10.1016/j.mcm.2004.02.047 |
| [5] | T. M. M. Homolle, A low-variance deviational simulation Monte Carlo for the Boltzmann equation,, J. Comput. Phys., 226, 2341 (2007) · Zbl 1214.82089 · 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, 459 (1989) · Zbl 0696.47042 · doi:10.1002/mma.1670110404 |
| [7] | P. A. Markowich, An analysis of the quantum Liouville equation,, Z. Angew. Math. Mech., 69, 121 (1989) · Zbl 0682.46047 · doi:10.1002/zamm.19890690303 |
| [8] | M. Nedjalkov, Unified particle approach to Wigner-Boltzmann transport in small semiconductor devices,, Phys. Rev. B, 70 (2004) · doi:10.1103/PhysRevB.70.115319 |
| [9] | R. I. A. Patterson, A stochastic weighted particle method for coagulation-advection problems,, SIAM J. Sci. Comput., 34 (2012) · Zbl 1250.65014 · doi:10.1137/110843319 |
| [10] | R. I. A. Patterson, Stochastic weighted particle methods for population balance equations,, J. Comput. Phys., 230, 7456 (2011) · Zbl 1252.65021 · doi:10.1016/j.jcp.2011.06.011 |
| [11] | D. Querlioz, <em>The Wigner Monte Carlo Method for Nanoelectronic Devices</em>,, Wiley (2010) · Zbl 1195.81128 · doi:10.1002/9781118618479 |
| [12] | G. A. Radtke, Low-noise Monte Carlo simulation of the variable hard sphere gas,, Phys. Fluids, 23, 1 (2011) · doi:10.1063/1.3558887 |
| [13] | S. Rjasanow, A stochastic weighted particle method for the Boltzmann equation,, J. Comput. Phys., 124, 243 (1996) · Zbl 0883.65118 · doi:10.1006/jcph.1996.0057 |
| [14] | S. Rjasanow, Simulation of rare events by the stochastic weighted particle method for the Boltzmann equation,, Math. Comput. Modelling, 33, 907 (2001) · Zbl 1050.82038 · doi:10.1016/S0895-7177(00)00289-2 |
| [15] | J. M. Sellier, A benchmark study of the Wigner Monte Carlo method,, Monte Carlo Methods Appl., 20, 43 (2014) · Zbl 1285.81044 · doi:10.1515/mcma-2013-0018 |
| [16] | W. Wagner, Deviational particle Monte Carlo for the Boltzmann equation,, Monte Carlo Methods Appl., 14, 191 (2008) · Zbl 1158.82002 · doi:10.1515/MCMA.2008.010 |
| [17] | W. Wagner, A random cloud model for the Schrödinger equation,, Kinetic and Related Models, 7, 361 (2014) · Zbl 1304.35596 · doi:10.3934/krm.2014.7.361 |
| [18] | E. Wigner, On the quantum correction for thermodynamic equilibrium,, Phys. Rev., 40, 749 (1932) · JFM 58.0948.07 |
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