Quantum Hydrodynamic and Diffusion Models Derived from the Entropy Principle

  • Pierre Degond
  • Samy Gallego
  • Florian Méhats
  • Christian Ringhofer
Part of the Lecture Notes in Mathematics book series (LNM, volume 1946)


In these notes, we review the recent theory of quantum hydrodynamic and diffusion models derived from the entropy minimization principle. These models are obtained by taking the moments of a collisional Wigner equation and closing the resulting system of equations by a quantum equilibrium. Such an equilibrium is defined as a minimizer of the quantum entropy subject to local constraints of given moments. We provide a framework to develop this minimization approach and successively apply it to quantum hydrodynamic models and quantum diffusion models. The results of numerical simulations show that these models capture well the various features of quantum transport.


Density Operator Collision Operator Entropy Minimization Pressure Tensor Weyl Quantization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Ancona, M. G., Diffusion-Drift modeling of strong inversion layers, COMPEL 6, 11–18 (1987)Google Scholar
  2. 2.
    Ancona, M. G., Iafrate, G. J., Quantum correction of the equation of state of an electron gas in a semiconductor, Phys. review B, 39, 9536–9540 (1989)CrossRefGoogle Scholar
  3. 3.
    Ancona, M. G., Tiersten, H. F., Macroscopic physics of the silicon inversion layer, Phys. review B, 35, 7959–7965 (1987)CrossRefGoogle Scholar
  4. 4.
    Argyres, P. N., Quantum kinetic equations for electrons in high electric and phonon fields, Physics Lett. A, 171, 373–379 (1992)CrossRefGoogle Scholar
  5. 5.
    Arnold, A., Lopez, J. L., Markowich, P., Soler, J., An analysis of quantum Fokker - Planck models: A Wigner function approach, Rev. Mat. Iberoamericana, 20 771–814 (2004)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Bardos, C., Golse, F., Mauser, N. J., Weak coupling limit of the N-particle Schrodinger equation, Methods Appl. Anal., 7, 275–293 (2000)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Bardos, C., Golse, F., Gottlieb, A. D., Mauser, N. J., Mean field dynamics of fermions and the time-dependent Hartree-Fock equation, J. Math. Pures Appl., 82, 665–683 (2003)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Bardos, C., Golse, F., Gottlieb, A. D., Mauser, N. J., Accuracy of the time-dependent Hartree-Fock approximation for uncorrelated initial states, J. Statist. Phys., 115, 1037–1055 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Ben Abdallah, N., Degond, P., On a hierarchy of macroscopic models for semiconductors, J. Math. Phys., 37, 3306–3333 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Ben Abdallah, N., Degond, P., Gnieys, S., An energy-transport model for semiconductors derived from the Boltzmann equation, J. Stat. Phys., 84, 205-231 (1996)zbMATHCrossRefGoogle Scholar
  11. 11.
    Ben Abdallah, N., Unterreiter, A., On the stationary quantum drift-diffusion model, Z. Angew. Math. Phys., 49, 251–275 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Brenier, Y., Grenier, E., Sticky particles and scalar conservation laws, SIAM J. Numer. Anal., 35, 2317–2328 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Bouchut, F., On zero pressure gas dynamics, in Advances in kinetic theory and computing, B. Perthame (ed), World Scientific (1994)Google Scholar
  14. 14.
    Burghardt, I., Cederbaum, L. S., Hydrodynamic equations for mixed quantum states I. General formulation, Journal of Chemical Physics, 115, 10303–10311 (2001)CrossRefGoogle Scholar
  15. 15.
    Burghardt, I., Cederbaum, L. S., Hydrodynamic equations for mixed quantum states II. Coupled electronic states, Journal of Chemical Physics, 115, 10312–10322 (2001)CrossRefGoogle Scholar
  16. 16.
    Burghardt, I., Moller, K. B., Quantum dynamics for dissipative systems: a hydrodynamic perspective, Journal of Chemical Physics, 117, 7409–7425 (2002)CrossRefGoogle Scholar
  17. 17.
    Burghardt, I., Parlant, G., On the dynamics of coupled Bohmian and phase-space variables, a new hybrid quantum-classical approach, Journal of Chemical Physics, 120, 3055–3058 (2004)CrossRefGoogle Scholar
  18. 18.
    Car, R., Parrinello, M., Unified approach for molecular dynamics and density-functional theory, Phys. Rev. Lett., 55, 2471–2474 (1985)CrossRefGoogle Scholar
  19. 19.
    Chen, R-C., Liu, J-L., A quantum corrected energy-transport model for nanoscale semiconductor devices, J. Comput. Phys., 204, 131–156 (2005)zbMATHCrossRefGoogle Scholar
  20. 20.
    Degond, P., Mathematical modelling of microelectronics semiconductor devices, AMS/IP Studies in Advanced Mathematics, AMS Society and International Press, 77–109, (2000)Google Scholar
  21. 21.
    Degond, P., Gallego, S., Mhats, F., An entropic quantum drift-diffusion model for electron transport in resonant tunneling diodes, J. Comp. Phys., to appearGoogle Scholar
  22. 22.
    Degond, P., Gallego, S., Mhats, F., Isothermal quantum hydrodynamics: derivation, asymptotic analysis and simulation, manuscript, submittedGoogle Scholar
  23. 23.
    Degond, P., Mhats, F., Ringhofer, C., Quantum energy-transport and drift-diffusion models, J. Stat. Phys., 118, 625–667 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Degond, P., Mhats, F., Ringhofer, C., Quantum hydrodynamic models derived from the entropy principle, Contemp. Math., 371, 107–131 (2005)Google Scholar
  25. 25.
    Degond, P., Ringhofer, C., Quantum moment hydrodynamics and the entropy principle, J. Stat. Phys. 112, 587–628 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Degond, P., Ringhofer, C., A note on quantum moment hydrodynamics and the entropy principle, C. R. Acad. Sci. Paris Ser 1, 335, 967–972 (2002)MathSciNetGoogle Scholar
  27. 27.
    de Falco, C., Gatti, E., Lacaita, A. L., Sacco, R., Quantum-Corrected Drift-Diffusion Models for Transport in Semiconductor Devices, J. Comput. Phys., 204 533–561 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Dreizler, R. M., Gross, E. K. U., Density Functional Theory, Springer, Berlin (1990)zbMATHGoogle Scholar
  29. 29.
    E, W., Rykov, Y. G., Sinai, Y. G., Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics, Comm. Math. Phys. 177, 349–380 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Fischetti, M. V., Theory of electron transport in small semiconductor devices using the Pauli Master equation, J. Appl. Phys., 83, 270–291 (1998)CrossRefGoogle Scholar
  31. 31.
    Fromlet, F., Markowich, P., Ringhofer, C., A Wignerfunction Approach to Phonon Scattering, VLSI Design, 9, 339–350 (1999)CrossRefGoogle Scholar
  32. 32.
    Gallego, S., Mhats, F., Entropic discretization of a quantum drift-diffusion model, SIAM J. Numer. Anal., 43, 1828–1849 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Gallego, S., Mhats, F., Numerical approximation of a quantum drift-diffusion model, C. R. Math. Acad. Sci. Paris, 339, 519–524 (2004)zbMATHMathSciNetGoogle Scholar
  34. 34.
    Gardner, C., The quantum hydrodynamic model for semiconductor devices, SIAM J. Appl. Math. 54, 409–427 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Gardner, C., Ringhofer, C., The smooth quantum potential for the hydrodynamic model, Phys. Rev. E 53, 157–167 (1996)CrossRefGoogle Scholar
  36. 36.
    Gardner, C., Ringhofer, C., The Chapman-Enskog Expansion and the Quantum Hydrodynamic Model for Semiconductor Devices, VLSI Design 10, 415–435 (2000)CrossRefGoogle Scholar
  37. 37.
    Gasser, I., Markowich, P. A., Quantum Hydrodynamics, Wigner Transforms and the Classical Limit, Asympt. Analysis, 14, 97–116 (1997)zbMATHMathSciNetGoogle Scholar
  38. 38.
    Gasser, I., Markowich, P. A., Ringhofer, C., Closure conditions for classical and quantum moment hierarchies in the small temperature limit, Transp. Th. Stat. Phys. 25 409–423 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    Hohenberg, P., Kohn, W. Inhomogeneous electron gas, Phys. Rev. B, 136, 864–871 (1964)CrossRefMathSciNetGoogle Scholar
  40. 40.
    Jngel, A., Matthes, D., A derivation of the isothermal quantum hydrodynamic equations using entropy minimization, ZAMM Z. Angew. Math. Mech., 85, 806–814 (2005)CrossRefMathSciNetGoogle Scholar
  41. 41.
    Jngel, A., Matthes, D., Milisic, P., Derivation of new quantum hydrodynamic equations using entropy minimization, submittedGoogle Scholar
  42. 42.
    Jngel, A., Pinnau, R., A positivity preserving numerical scheme for a fourth order parabolic equation, SIAM J. Num. Anal., 39, 385–406 (2001)CrossRefGoogle Scholar
  43. 43.
    Kaiser, H-C., Rehberg, J., On stationary Schrdinger-Poisson equations modelling an electron gas with reduced dimension, Math. Methods Appl. Sci., 20, 1283–1312 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    C. Le Bris (ed), Handbook of numerical analysis. Vol. X. Special volume: computational chemistry, North-Holland, Amsterdam (2003)Google Scholar
  45. 45.
    Levermore, C. D., Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83, 1021–1065 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  46. 46.
    Luzzi, R., untitled, electronic preprint archive, reference arXiv:cond-mat/9909160 v2 11 Sep 1999Google Scholar
  47. 47.
    Lions, P-L., Paul, T., Sur les mesures de Wigner, Rev. Mat. Iberoamericana, 9, 553–618 (1993)zbMATHMathSciNetGoogle Scholar
  48. 48.
    Lopreore, C. L., Wyatt, R. E., Quantum Wave Packet Dynamics with Trajectories, Phys. Rev. Lett., 82, 5190–5193 (1999)CrossRefGoogle Scholar
  49. 49.
    Maddox, J. B., Bittner, E. R., Quantum dissipation in the hydrodynamic moment hierarchy, a semiclassical truncation stategy, J. Phys. Chem. B, 106, 7981–7990 (2002)CrossRefGoogle Scholar
  50. 50.
    Markowich, P. A., Mauser, N. J., The classical limit of a self-consistent quantum-Vlasov equation in 3D, Math. Models Methods Appl. Sci., 3, 109–124 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  51. 51.
    Micheletti, S., Sacco, R., Simioni, P., Numerical Simulation of Resonant Tunnelling Diodes with a Quantum-Drift-Diffusion Model, Scientific Computing in Electrical Engineering, Lecture Notes in Computer Science, Springer-Verlag, pp. 313–321 (2004)Google Scholar
  52. 52.
    Morozov, V. G., Röpke, G., 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
  53. 53.
    Nier, F., A stationary Schrdinger-Poisson system arising from the modelling of electronic devices, Forum Math., 2, 489–510 (1990)zbMATHMathSciNetCrossRefGoogle Scholar
  54. 54.
    Nier, F., A variational formulation of Schrdinger-Poisson systems in dimension d ≤ 3, Comm. Partial Differential Equations, 18, 1125–1147 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  55. 55.
    Nier, F., Schrdinger-Poisson systems in dimension d ≤ 3: the whole-space case, Proc. Roy. Soc. Edinburgh Sect. A, 123, 1179–1201 (1993)zbMATHMathSciNetGoogle Scholar
  56. 56.
    Pinnau, R., The Linearized Transient Quantum Drift Diffusion Model - Stability of Stationary States, Z. Angew. Math. Mech., 80 327–344 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  57. 57.
    Pinnau, R., Unterreiter, A., The Stationary Current-Voltage Characteristics of the Quantum Drift Diffusion Model, SIAM J. Numer. Anal., 37, 211–245 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  58. 58.
    Pirovano, A., Lacaita, A., Spinelli, A., Two-Dimensional Quantum effects in Nanoscale MOSFETs, IEEE Trans. Electron Devices, 47, 25–31 (2002)CrossRefGoogle Scholar
  59. 59.
    Spohn, H., Large scale dynamics of interacting particles, Springer, Berlin (1991)zbMATHGoogle Scholar
  60. 60.
    Wyatt, R. E., Bittner, E. R., Quantum wave packet dynamics with trajectories: Implementation with adaptive Lagrangian grids, The Journal of Chemical Physics, 113, 8898–8907 (2000)CrossRefGoogle Scholar
  61. 61.
    Zubarev, D. N., Morozov, V. G., Röpke, G., Statistical mechanics of nonequilibrium processes. Vol 1, basic concepts, kinetic theory, Akademie Verlag, Berlin (1996)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Pierre Degond
    • 1
  • Samy Gallego
    • 1
  • Florian Méhats
    • 2
  • Christian Ringhofer
    • 3
  1. 1.Institut de MathématiquesUniversité de ToulouseToulouse Cedex 9France
  2. 2.IRMAR (UMR CNRS 6625)Université de RennesRennes CedexFrance
  3. 3.Department of MathematicsArizona State UniversityTempeUSA

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