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Numerical analysis of the deterministic particle method applied to the Wigner equation

Authors: Anton Arnold and Francis Nier
Journal: Math. Comp. 58 (1992), 645-669
MSC: Primary 65M12; Secondary 35Q40, 81Q05
MathSciNet review: 1122055
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Abstract: The Wigner equation of quantum mechanics has the form of a kinetic equation with a pseudodifferential operator in a Fourier integral form which requires great care in the numerical approximation. This paper is concerned with the numerical analysis of the weighted particle method, introduced by S. Mas-Gallic and P. A. Raviart, applied to this equation. In particular, we will prove convergence of the method in a physically relevant case, where the Wigner equation models the quantum tunneling of electrons through a potential barrier.

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  • [1] S. Mas-Gallic and P.-A. Raviart, A particle method for first-order symmetric systems, Numer. Math. 51 (1987), no. 3, 323–352. MR 895090, 10.1007/BF01400118
  • [2] E. P. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40 (1932), 749-759.
  • [3] W. R. Frensley, Wigner function model of a resonant-tunneling semiconductor device, Phys. Rev. B 36 (1987), 1570-1580.
  • [4] U. Ravaioli, M. A. Osman, W. Pötz, N. Kluksdahl, and D. K. Ferry, Investigation of ballistic transport through resonant-tunneling quantum wells using Wigner function approach, Physica B 134 (1985), 36-40.
  • [5] Christian Ringhofer, A spectral method for the numerical simulation of quantum tunneling phenomena, SIAM J. Numer. Anal. 27 (1990), no. 1, 32–50. MR 1034919, 10.1137/0727003
  • [6] S. Mas-Gallic and F. Poupaud, Approximation of the transport equation by a weighted particle method, Transport Theory Statist. Phys. 17 (1988), no. 4, 311–345. MR 968659, 10.1080/00411458808230870
  • [7] Sylvie Mas-Gallic, A deterministic particle method for the linearized Boltzmann equation, Proceedings of the conference on mathematical methods applied to kinetic equations (Paris, 1985), 1987, pp. 855–887. MR 906929, 10.1080/00411458708204318
  • [8] P. Degond and B. Niclot, Numerical analysis of the weighted particle method applied to the semiconductor Boltzmann equation, Numer. Math. 55 (1989), no. 5, 599–618. MR 998912, 10.1007/BF01398918
  • [9] Peter A. Markowich, On the equivalence of the Schrödinger and the quantum Liouville equations, Math. Methods Appl. Sci. 11 (1989), no. 4, 459–469. MR 1001097, 10.1002/mma.1670110404
  • [10] P.-A. Raviart, An analysis of particle methods, Numerical methods in fluid dynamics (Como, 1983) Lecture Notes in Math., vol. 1127, Springer, Berlin, 1985, pp. 243–324. MR 802214, 10.1007/BFb0074532
  • [11] Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR 0450957
  • [12] P. Degond and F. Guyot-Delaurens, Particle simulations of the semiconductor Boltzmann equation for one-dimensional inhomogeneous structures, J. Comput. Phys. 90 (1990), no. 1, 65–97. MR 1070472, 10.1016/0021-9991(90)90197-9
  • [13] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486
  • [14] G.-H. Cottet and P.-A. Raviart, Particle methods for the one-dimensional Vlasov-Poisson equations, SIAM J. Numer. Anal. 21 (1984), no. 1, 52–76. MR 731212, 10.1137/0721003
  • [15] P. Degond and P. A. Markowich, A mathematical analysis of quantum transport in three-dimensional crystals, Ann. Mat. Pura Appl. (to appear).
  • [16] -, A quantum transport model for semiconductors: the Wigner-Poisson problem on a bounded Brillouin zone, Math. Mod. Numer. Anal. (to appear).
  • [17] R. Dautray and J. L. Lions, Analyse mathématique et calcul numérique, Masson, Paris, 1985.
  • [18] P. A. Markowich, C. A. Ringhofer, and C. Schmeiser, Semiconductor equations, Springer-Verlag, Vienna, 1990. MR 1063852
  • [19] F. Nier, Application de la méthode particulaire à l'équation de Wigner-mise en oeuvre numérique, Thèse de l'Ecole Polytechnique, Palaiseau, France, 1991.
  • [20] Michael Reed and Barry Simon, Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0493420

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Article copyright: © Copyright 1992 American Mathematical Society