Skip to main content

From Hartree Dynamics to the Relativistic Vlasov Equation

Journal of Statistical Physics volume 172, pages 398–433 (2018)Cite this article

Abstract

We derive the relativistic Vlasov equation from quantum Hartree dynamics for fermions with relativistic dispersion in the mean-field scaling, which is naturally linked with an effective semiclassic limit. Similar results in the non-relativistic setting have been recently obtained in Benedikter et al. (Arch Rat Mech Anal 221(1): 273–334, 2016). The new challenge that we have to face here, in the relativistic setting, consists in controlling the difference between the quantum kinetic energy and the relativistic transport term appearing in the Vlasov equation.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

References

  1. Amour, L., Khodja, M., Nourrigat, J.: The semiclassical limit of the time dependent Hartree-Fock equation: the Weyl symbol of the solution. Anal. PDE 6(7), 1649–1674 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Amour, L., Khodja, M., Nourrigat, J.: The classical limit of the Heisenberg and time dependent Hartree-Fock equations: the Wick symbol of the solution. Math. Res. Lett. 20(1), 119–139 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Athanassoulis, A., Paul, T., Pezzotti, F., Pulvirenti, M.: Strong semiclassical approximation of Wigner functions for the Hartree dynamics. Rend. Lincei Math. Appl. 22, 525–552 (2011)

    MathSciNet  MATH  Google Scholar 

  4. Bach, V., Breteaux, S., Petrat, S., Pickl, P., Tzaneteas, T.: Kinetic energy estimates for the accuracy of the time-dependent Hartree-Fock approximation with Coulomb interaction. J. Math. Pures Appl. 105(1), 1–30 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. Benedikter, N., Porta, M., Schlein, B.: Mean-field evolution of fermionic systems. Commun. Math. Phys. 331, 1087–1131 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Benedikter, N., Porta, M., Schlein, B.: Mean-field dynamics of fermions with relativistic dispersion. J. Math. Phys. 55, 021901 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Benedikter, N., Jaksic, V., Porta, M., Saffirio, C., Schlein, B.: Mean-field evolution of fermionic mixed states. Commun. Pure Appl. Math. 69(12), 2250–2303 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. Benedikter, N., Porta, M., Saffirio, C., Schlein, B.: From the Hartree dynamics to the Vlasov equation. Arch. Ration. Mech. Anal. 221(1), 273–334 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dobrushin, R.L.: Vlasov equations. Funct. Anal. Its Appl. 13(2), 115–123 (1979)

    Article  MATH  Google Scholar 

  10. Elgart, A., Erdős, L., Schlein, B., Yau, H.-T.: Nonlinear Hartree equation as the mean field limit of weakly coupled fermions. J. Math. Pures Appl. 83(10), 1241–1273 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gasser, I., Illner, R., Markowich, P.A., Schmeiser, C.: Semiclassical $t \rightarrow \infty $ asymptotics and dispersive effects for Hartree-Fock systems. Model. Math. Anal. Numer. 32, 699–713 (1998)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Markowich, P.A., Mauser, N.J.: The classical limit of a self-consistent Quantum-Vlasov equation in 3D. Math. Models Methods Appl. Sci. 03(1), 109 (1993)

    Article  MATH  Google Scholar 

  14. Narnhofer, H., Sewell, G.L.: Vlasov hydrodynamics of a quantum mechanical model. Commun. Math. Phys. 79(1), 9–24 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  15. Petrat, S., Pickl, P.: A new method and a new scaling for deriving fermionic mean-field dynamics. Math. Phys. Anal. Geom. 19, 1 (2016)

    Article  MathSciNet  Google Scholar 

  16. Pezzotti, F., Pulvirenti, M.: Mean-field limit and semiclassical expansion of a quantum particle system. Ann. Henri Poincaré 10(1), 145–187 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Spohn, H.: On the Vlasov hierarchy. Math. Methods Appl. Sci. 3(4), 445–455 (1981)

    Article  ADS  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We would like to thank the two anonymous referees for their useful comments that allowed us to improve the presentation of our paper. B.S. gratefully acknowledges support from the NCCR SwissMAP and from the Swiss National Foundation of Science through the SNF Grant “Effective equations from quantum dynamics” and the SNF Grant “Dynamical and energetic properties of Bose–Einstein condensates”.

Author information

Authors and Affiliations

  1. Institute for Mathematics, University of Zurich, Winterthurerstrasse 190, Zurich, 8057, Switzerland

    Elia Dietler, Simone Rademacher & Benjamin Schlein

Authors
  1. Elia Dietler
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Simone Rademacher
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Benjamin Schlein
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Benjamin Schlein.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Dietler, E., Rademacher, S. & Schlein, B. From Hartree Dynamics to the Relativistic Vlasov Equation. J Stat Phys 172, 398–433 (2018). https://doi.org/10.1007/s10955-018-1973-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-018-1973-5

Keywords

  • Hartree theory
  • Relativistic Vlasov equation
  • Many-body quantum mechanics