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The Wigner–Poisson System

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Quantum Plasmas

Part of the book series: Springer Series on Atomic, Optical, and Plasma Physics ((SSAOPP,volume 65))

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Abstract

In electrostatic quantum plasmas, the Wigner–Poisson system plays the same rôle as the Vlasov–Poisson system in classical plasmas. This chapter considers the basic properties of the Wigner–Poisson system, including the essentials on the Wigner function method and the derivation of the Wigner–Poisson system in the context of a mean field theory. This chapter also contains a discussion on the Schrödinger–Poisson system as well as extensions to include correlation and collisional effects. The Wigner–Poisson system is shown to imply, in the high-frequency limit, the Bohm–Pines dispersion relation for linear waves, which is the quantum analog of the Bohm–Gross dispersion relation for classical plasmas.

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References

  1. Arnold, A., López, J. L., Markowich, P., Soler, J.: An analysis of quantum Fokker-Planck models: a Wigner function approach. Rev. Mat. Iberoamericana 20, 771–814 (2004)

    MathSciNet  MATH  Google Scholar 

  2. Ashcroft, N. W. and Mermin, N. D.: Solid state physics. Saunders College Publishing, Orlando (1976)

    Google Scholar 

  3. Bogoliubov, N. N.: Kinetic equations. J. Exp. Theor. Phys. 16, 691–702 (1946)

    Google Scholar 

  4. Bohm, D. and Gross, E.: Theory of plasma oscillations. A. Origin of medium-like behavior. Phys. Rev. 75, 1851–1864 (1949)

    MATH  Google Scholar 

  5. Bohm, D., Pines, D.: A collective description of electron interactions: III. Coulomb interactions in a degenerate electron gas. Phys. Rev. 92, 609–625 (1953)

    MathSciNet  MATH  Google Scholar 

  6. Born, M., Green, H. S.: A general kinetic theory of liquids I. The molecular distribution functions. Proc. Roy. Soc. A 188, 10–18 (1946)

    Article  MathSciNet  MATH  Google Scholar 

  7. Caldeira, A. O., Leggett, A. J.: Path integral approach to quantum Brownian motion. Physica A 121, 587–616 (1983)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  8. Carruthers, P., Zachariasen, F.: Quantum collision theory with phase-space distributions. Rev. Mod. Phys. 55, 245–285 (1983)

    Article  MathSciNet  ADS  Google Scholar 

  9. Castella, F., Erdõs, L., Frommlet, F., Markowich, P.: Fokker-Planck equations as scaling limits of reversible quantum systems. J. Stat. Phys. 100, 543–601 (2000)

    Article  MATH  Google Scholar 

  10. DiVentra, M.: Electrical Transport in Nanoscale Systems. Cambridge, New York (2008)

    Book  Google Scholar 

  11. Glauber, R. J.: Coherent and incoherent states of the radiation field. Phys. Rev. 131, 2766–2788 (1963)

    Article  MathSciNet  ADS  Google Scholar 

  12. Glauber, R.J.: Optical coherence and photon statistics. In: Dewitt, C., Blandin, A., Cohen-Tannoudji, C. (eds.) Quantum Optics and Electronics, pp. 63–185, Gordon and Breach, New York (1965)

    Google Scholar 

  13. Gusev, G. M., Quivy, A. A., Laman, T. E., Leite, J. R., Bakarov, A. K., Topov, A. I., Estibals, O., Portal, J. C.: Magnetotransport of a quasi-three-dimensional electron gas in the lowest Landau level. Phys. Rev. B 65, 205316–205325 (2002)

    Article  ADS  Google Scholar 

  14. Haas, F.: On quantum plasma kinetic equations with a Bohmian force. J. Plasma Phys. 76, 389–393 (2010)

    Article  ADS  Google Scholar 

  15. Haas, F. and Shukla, P. K.: Nonlinear stationary solutions of the Wigner and Wigner–Poisson equations. Phys. Plasmas 15, 112302-112302-6 (2008)

    Google Scholar 

  16. Haas, F., Eliasson, B., Shukla, P. K. and Manfredi, G.: Phase-space structures in quantum-plasma wave turbulence. Phys. Rev. E 78, 056407–056414 (2008)

    Article  ADS  Google Scholar 

  17. Haug, H. J. W., Jauho, A. P.: Quantum Kinetics in Transport and Optics of Semiconductors. Springer, Berlin-Heidelberg (2008)

    Google Scholar 

  18. Hillery, M., O’Connell, R. F., Scully, M. O., Wigner, E. P.: Distribution functions in physics - fundamentals. Phys. Rep. 106, 121–330 (1990)

    Article  MathSciNet  ADS  Google Scholar 

  19. Husimi, K.: Some formal properties of the density matrix. Prog. Phys. Math. Soc. Japan 22, 264–314 (1940)

    MATH  Google Scholar 

  20. Jungel, A.: Transport Equations for Semiconductors. Springer, Berlin-Heidelberg (2009)

    Book  Google Scholar 

  21. Kadanoff, L.P., Baym, G.: Quantum Statistical Mechanics: Green’s Function Methods in Equilibrium and Non-Equilibrium Problems. Benjamin, New York (1962)

    Google Scholar 

  22. Kirkwood, J. G.: Quantum statistics of almost classical assemblies. Phys. Rev. 44, 31–37 (1933)

    Article  ADS  MATH  Google Scholar 

  23. Kirkwood, J. G.: The statistical mechanical theory of transport processes I. General theory. J. Chem. Phys. 14, 180–201 (1946)

    Article  ADS  Google Scholar 

  24. Klimontovich Y. and Silin, V. P.: The spectra of systems of interacting particles. In: Drummond, J. E. (ed.) Plasma Physics, pp. 35–87, McGraw-Hill, New York (1961)

    Google Scholar 

  25. Kluksdahl, N. C., Kriman, A. M., Ferry, D. K., Ringhofer, C.: Self-consistent study of the resonant tunneling diode. Phys. Rev. B. 39, 7720–7735 (1989)

    Article  ADS  Google Scholar 

  26. Lee, H.: Theory and application of the quantum phase-space distribution functions. Phys. Rep. 259, 147–211 (1995)

    Article  MathSciNet  Google Scholar 

  27. Lindblad, G.: On the generators of quantum dynamical semigroups. Commun. Math. Phys. 48, 119–130 (1976)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  28. López, J. L.: Nonlinear Ginzburg-Landau-type approach to quantum dissipation. Phys. Rev. E 69, 026110–026125 (2004)

    Article  ADS  Google Scholar 

  29. Markowich, P. A.: On the equivalence of the Schrödinger and the quantum Liouville equations. Math. Meth. in the Appl. Sci. 11, 459–469 (1989)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  30. Markowich, P. A., Ringhofer, C. A., Schmeiser, C.: Semiconductor Equations. Springer, Wien (1990)

    MATH  Google Scholar 

  31. Manfredi, G., Haas, F.: Self-consistent fluid model for a quantum electron gas. Phys. Rev. B 64, 075316–075323 (2001)

    Article  ADS  Google Scholar 

  32. Mehta, C. L.: Phase-space formulation of the dynamics of canonical variables. J. Math. Phys. 5, 677–686 (1964)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  33. Santer, M., Mehlig, B., Moseler, M.: Optical response of two-dimensional electron fluids beyond the Kohn regime: strong nonparabolic confinement and intense laser light. Phys. Rev. Lett. 89, 266801–266804 (2002)

    Article  ADS  Google Scholar 

  34. Sudarshan, E. C. G.: Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams. Phys. Rev. Lett. 10, 277–279 (1963)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  35. Suh, N., Feix, M. R. and Bertrand, P.: Numerical simulation of the quantum Liouville-Poisson system. J. Comput. Phys. 94, 403–418 (1991)

    Article  ADS  MATH  Google Scholar 

  36. Tatarskii, V. I.: The Wigner representation of quantum mechanics. Sov. Phys. Usp. 26, 311–327 (1983)

    Article  MathSciNet  ADS  Google Scholar 

  37. Weyl, H.: Quantenmechanik und gruppentheorie. Z. Phys. 46, 1–46 (1927)

    Article  Google Scholar 

  38. Wigner, E.: On the quantum correction for thermodynamic equilibrium. Phys. Rev. 40, 749–759 (1932)

    Article  ADS  MATH  Google Scholar 

  39. Yvon, J.: La Théorie Statistique des Fluides. Hermann, Paris (1935)

    Google Scholar 

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Authors and Affiliations

  1. Universidade Federal do Paraná, Curitiba, Brazil

    Fernando Haas

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  1. Fernando Haas
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Haas, F. (2011). The Wigner–Poisson System. In: Quantum Plasmas. Springer Series on Atomic, Optical, and Plasma Physics, vol 65. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-8201-8_2

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