Constrained minimizers of the von Neumann entropy and their characterization

Abstract

We consider in this work the problem of minimizing the von Neumann entropy under the constraints that the density of particles, the current, and the kinetic energy of the system is fixed at each point of space. The unique minimizer is a self-adjoint positive trace class operator, and our objective is to characterize its form. We will show that this minimizer is solution to a self-consistent nonlinear eigenvalue problem. One of the main difficulties in the proof is to parametrize the feasible set in order to derive the Euler–Lagrange equation, and we will proceed by constructing an appropriate form of perturbations of the minimizer. The question of deriving quantum statistical equilibria is at the heart of the quantum hydrodynamical models introduced by Degond and Ringhofer (J Statist Phys 112:(3–4), 587-628, 2003). An original feature of the problem is the local nature of constraints, i.e. they depend on position, while more classical models consider the total number of particles, the total current and the total energy in the system to be fixed.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Degond, P., Gallego, S., Méhats, F.: An entropic quantum drift-diffusion model for electron transport in resonant tunneling diodes. J. Comput. Phys. 221(1), 226–249 (2007)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Degond, P., Gallego, S., Méhats, F.: Isothermal quantum hydrodynamics: derivation, asymptotic analysis, and simulation. Multiscale Model. Simul. 6(1), 246–272 (2007)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Degond, P., Gallego, S., Méhats, F.: On quantum hydrodynamic and quantum energy transport models. Commun. Math. Sci. 5(4), 887–908 (2007)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Degond, P., Gallego, S., Méhats, F., Ringhofer, C.: Quantum hydrodynamic and diffusion models derived from the entropy principle. In: Quantum Transport, Volume 1946 of Lecture Notes in Mathematics, pp. 111–168. Springer, Berlin (2008)

  5. 5.

    Degond, P., Ringhofer, C.: Quantum moment hydrodynamics and the entropy principle. J. Statist. Phys. 112(3–4), 587–628 (2003)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Dolbeault, J., Felmer, P., Loss, M., Paturel, E.: Lieb–Thirring type inequalities and Gagliardo–Nirenberg inequalities for systems. J. Funct. Anal. 238(1), 193–220 (2006)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Duboscq, R., Olivier P.: A constrained optimization problem in quantum statistical physics. arXiv preprint arXiv:1904.00600 (2019)

  8. 8.

    Duboscq, R., Pinaud, O.: On the minimization of quantum entropies under local constraints. J. Math. Pure Appl. 128, 87–118 (2019)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Jüngel, A., Matthes, D.: A derivation of the isothermal quantum hydrodynamic equations using entropy minimization. ZAMM Z. Angew. Math. Mech. 85(11), 806–814 (2005)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Jüngel, A., Matthes, D., Milišić, J.P.: Derivation of new quantum hydrodynamic equations using entropy minimization. SIAM J. Appl. Math. 67(1), 46–68 (2006)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Junk, M.: Domain of definition of Levermore’s five-moment system. J. Statist. Phys. 93(5–6), 1143–1167 (1998)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Levermore, C.D.: Moment closure hierarchies for kinetic theories. J. Statist. Phys. 83(5–6), 1021–1065 (1996)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Lions, P.-L., Paul, T.: Sur les mesures de Wigner. Rev. Mat. Iberoamericana 9, 553–618 (1993)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Méhats, F., Pinaud, O.: An inverse problem in quantum statistical physics. J. Stat. Phys. 140(3), 565–602 (2010)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Méhats, F., Pinaud, O.: A problem of moment realizability in quantum statistical physics. Kinet. Relat. Models 4(4), 1143–1158 (2011)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Méhats, F., Pinaud, O.: The quantum Liouville-BGK equation and the moment problem. J. Differ. Eq. 263(7), 3737–3787 (2017)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Nachtergaele, B., Yau, H.-T.: Derivation of the Euler equations from quantum dynamics. Commun. Math. Phys. 243(3), 485–540 (2003)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Pinaud, O.: The quantum drift-diffusion model: existence and exponential convergence to the equilibrium. Annales de l’Institut Henri Poincaré C, Analyse non linéaire 36(3), 811–836 (2019)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Reed, M., Simon, B.: Methods of Modern Mathematical Physics. I. Functional Analysis, 2nd edn. Academic Press Inc, New York (1980)

    Google Scholar 

  20. 20.

    von Neumann, J.: Mathematical Foundations of Quantum Mechanics. Princeton University Press, Princeton (1955). Translated by Robert T. Beyer

    Google Scholar 

Download references

Acknowledgements

OP is supported by NSF CAREER Grant DMS-1452349.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Romain Duboscq.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by J. Ball.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Duboscq, R., Pinaud, O. Constrained minimizers of the von Neumann entropy and their characterization. Calc. Var. 59, 105 (2020). https://doi.org/10.1007/s00526-020-01753-1

Download citation

Mathematics Subject Classification

  • Primary 35Q40
  • Secondary 82B10