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Quantum Moment Hydrodynamics and the Entropy Principle

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Abstract

This paper presents how a non-commutative version of the entropy extremalization principle allows to construct new quantum hydrodynamic models. Our starting point is the moment method, which consists in integrating the quantum Liouville equation with respect to momentum p against a given vector of monomials of p. Like in the classical case, the so-obtained moment system is not closed. Inspired from Levermore's procedure in the classical case,(26) we propose to close the moment system by a quantum (Wigner) distribution function which minimizes the entropy subject to the constraint that its moments are given. In contrast to the classical case, the quantum entropy is defined globally (and not locally) as the trace of an operator. Therefore, the relation between the moments and the Lagrange multipliers of the constrained entropy minimization problem becomes nonlocal and the resulting moment system involves nonlocal operators (instead of purely local ones in the classical case). In the present paper, we discuss some practical aspects and consequences of this nonlocal feature.

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REFERENCES

  1. P. Andriès, P. Le Tallec, J. P. Perlat, and B. Perthame, The Gaussian-BGK model of Boltzmann equation with small Prandtl number, preprint.

  2. P. Argyres, Quantum kinetic equations for electrons in high electric and phonon fields, Phys. Lett. A 171:373-379 (1992).

    Google Scholar 

  3. A. Arnold, J. Lopèz, P. Markowich, and J. Soler, An analysis of quantum Fokker-Planck models: A Wigner function approach, preprint (2002).

  4. R. Balian, From Microphysics to Macrophysics (Springer, 1982).

  5. J. Barker and D. Ferry, Phys. Rev. Lett. 42(1997).

  6. A. Caldeira and A Leggett, A path integral approach to quantum Brownian motion, Phys. A 121:587-616 (1983).

    Google Scholar 

  7. P. Degond, Mathematical Modelling of Microelectronics Semiconductor Devices, Proceedings of the Morningside Mathematical Center, Beijing (AMS/IP Studies in Advanced Mathematics, AMS Society and International Press, 2000), pp. 77-109.

  8. P. Degond, F. Méhats, and C. Ringhofer, On a quantum energy-transport model, in preparation.

  9. P. Degond and C. Ringhofer, A note on quantum moment hydrodynamics and the entropy principle, C. R. Acad. Sci. Paris Sér. I Math. 335:967-972 (2002).

    Google Scholar 

  10. P. Degond and C. Ringhofer, A note on binary quantum collision operators conserving mass momentum and energy, submitted.

  11. D. Ferry and H. Grubin, Modelling of quantum transport in semiconductor devices, Solid State Phys. 49:283-448 (1995).

    Google Scholar 

  12. F. Fromlet, P. Markowich, and C. Ringhofer, A Wigner function approach to phonon scattering, VLSI Design 9:339-350 (1999).

    Google Scholar 

  13. C. Gardner, The quantum hydrodynamic model for semiconductor devices, SIAM J. Appl. Math. 54:409-427 (1994).

    Google Scholar 

  14. C. Gardner and C. Ringhofer, The smooth quantum potential for the hydrodynamic model, Phys. Rev. E 53:157-167 (1996).

    Google Scholar 

  15. C. Gardner and C. Ringhofer, The Chapman-Enskog expansion and the quantum hydrodynamic model for semiconductor devices, VLSI Design 10:415-435 (2000).

    Google Scholar 

  16. I. Gasser and A. Jüngel, The quantum hydrodynamic model for semiconductors in thermal equilibrium, Z. Angew. Math. Phys. 48:45-59 (1997).

    Google Scholar 

  17. I. Gasser and P. A. Markowich, Quantum hydrodynamics, Wigner transforms, and the classical limit, Asymptot. Anal. 14:97-116 (1997).

    Google Scholar 

  18. I. Gasser, P. Markowich, and C. Ringhofer, Closure conditions for classical and quantum moment hierarchies in the small temperature limit, Transport Theory Statist. Phys. 25:409-423 (1996).

    Google Scholar 

  19. H. Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math. 2:331-407 (1949).

    Google Scholar 

  20. H. Grubin and J. Krekovski, Quantum moment balance equations and resonant tunneling structures, Solid State Electron. 32:1071-1075 (1989).

    Google Scholar 

  21. B. Hellfer and J. Sjöstrand, Equation de Harper, Lecture Notes in Physics, Vol. 345 (Springer, 1989), pp. 118-197.

    Google Scholar 

  22. S. Jin and X. T. Li, Multi-phase computations of the semiclassical limit of the Schrödinger equation and related problems: Whitham vs. Wigner, preprint, University of Wisconsin (http://www.math.wisc.edu/∼jin/publications.html).

  23. A. Jüngel and P. Pietra, A discretization scheme for a quasi-hydrodynamic semiconductor model, Math. Models Methods Appl. Sci. 7:935-955 (1997).

    Google Scholar 

  24. M. Junk, Domain of definition of Levermore's five-moment system, J. Stat. Phys. 93:1143-1167 (1998).

    Google Scholar 

  25. M. Junk, Maximum entropy for reduced moment problems, Math. Models Methods Appl. Sci. 10:1001-1025 (2000).

    Google Scholar 

  26. C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Stat. Phys. 83:1021-1065 (1996).

    Google Scholar 

  27. R. Luzzi, Untitled, electronic preprint archive, reference arXiv:cond-mat/9909160 v2 11, September 1999.

  28. V. G. Morozov and G. Röpke, Zubarev's method of a nonequilibrium statistical operator and some challenges in the theory of irreversible processes, Condensed Matter Physics 1:673-686 (1998).

    Google Scholar 

  29. I. Muller and T. Ruggeri, Rational Extended Thermodynamics, Springer Tracts in Natural Philosophy, Vol. 37, 2nd ed. (Springer, New York, 1998).

    Google Scholar 

  30. F. Nier, A variational formulation of Schrödinger-Poisson systems in dimension d≤3, Comm. Partial Differential Equations 18:1125-1147 (1993).

    Google Scholar 

  31. M. Reed and B. Simon, Methods of Modern Mathematical Physics. Vol. 1: Functional Analysis (Academic Press, 1980).

  32. J. Schneider, Entropic Approximation in Kinetic Theory, preprint, Analyse non linéaire appliquée et modélisation, Université de Toulon et du Var, BP 132, 83957 La Garde cedex, France (2001).

  33. D. Serre, Systèmes de lois de conservation, Tome I et II (Diderot, Paris, 1996).

    Google Scholar 

  34. M. A. Shubin, Pseudodifferential Operators and Spectral Theory (Springer, 1980).

  35. J. Smoller, Shock Waves and Reaction-Diffusion Equations, Grundlehren der mathematischen Wissenschaften, Vol. 258, 2nd ed. (Springer, New York, 1994).

    Google Scholar 

  36. D. N. Zubarev, V. G. Morozov, and G. Röpke, Statistical Mechanics of Nonequilibrium Processes, Vol. 1, Basic Concepts, Kinetic Theory (Akademie Verlag, Berlin, 1996).

    Google Scholar 

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Authors and Affiliations

  1. MIP, UMR 5640 (CNRS-UPS-INSA) Université Paul Sabatier, 118 route de Narbonne, 31062, Toulouse cedex, France

    P. Degond

  2. Department of Mathematics, Arizona State University, Tempe, Arizona, 85287-1804

    C. Ringhofer

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  1. P. Degond
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  2. C. Ringhofer
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Degond, P., Ringhofer, C. Quantum Moment Hydrodynamics and the Entropy Principle. Journal of Statistical Physics 112, 587–628 (2003). https://doi.org/10.1023/A:1023824008525

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