Skip to main content

Rate of Convergence Towards Hartree Dynamics

Abstract

We consider a system of N bosons interacting through a two-body potential with, possibly, Coulomb-type singularities. We show that the difference between the many-body Schrödinger evolution in the mean-field regime and the effective nonlinear Hartree dynamics is at most of the order 1/N, for any fixed time. The N-dependence of the bound is optimal.

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

References

  1. Bardos, C., Golse, F., Mauser, N.: Weak coupling limit of the N-particle Schrödinger equation. Methods Appl. Anal. 7, 275–293 (2000)

    MathSciNet  MATH  Google Scholar 

  2. Chen, L., Lee, J.O.: Rate of convergence in nonlinear Hartree dynamics with factorized initial data. J. Math. Phys. 52, 052108 (2011)

    ADS  Article  MathSciNet  Google Scholar 

  3. Elgart, A., Schlein, B.: Mean field dynamics of boson stars. Commun. Pure Appl. Math. 60(4), 500–545 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  4. Erdős, L., Schlein, B.: Quantum dynamics with mean field interactions: a new approach. J. Stat. Phys. 134(5), 859–870 (2009)

    MathSciNet  ADS  Article  Google Scholar 

  5. Erdős, L., Yau, H.-T.: Derivation of the nonlinear Schrödinger equation from a many body Coulomb system. Adv. Theor. Math. Phys. 5(6), 1169–1205 (2001)

    MathSciNet  Google Scholar 

  6. Erdős, L., Schlein, B., Yau, H.-T.: Derivation of the cubic nonlinear Schrödinger equation from quantum dynamics of many-body systems. Invent. Math. 167, 515–614 (2007)

    MathSciNet  ADS  Article  Google Scholar 

  7. Erdős, L., Schlein, B., Yau, H.-T.: Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate. Preprint arXiv:math-ph/0606017. To appear in Ann. Math.

  8. Erdős, L., Schlein, B., Yau, H.-T.: Rigorous derivation of the Gross-Pitaevskii equation with a large interaction potential. Preprint arXiv:0802.3877. To appear in J. Am. Math. Soc.

  9. Grillakis, M., Machedon, M., Margetis, D.: Second-order corrections to mean field evolution of weakly interacting bosons. I. Commun. Math. Phys. 294(1), 273–301 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  10. Grillakis, M., Machedon, M., Margetis, D.: Second-order corrections to mean field evolution of weakly interacting bosons. II. Preprint arXiv:1003.4713

  11. Ginibre, J., Velo, G.: The classical field limit of scattering theory for non-relativistic many-boson systems. I. Commun. Math. Phys. 66, 37–76 (1979)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  12. Ginibre, J., Velo, G.: The classical field limit of scattering theory for non-relativistic many-boson systems. II. Commun. Math. Phys. 68, 45–68 (1979)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  13. Hepp, K.: The classical limit for quantum mechanical correlation functions. Commun. Math. Phys. 35, 265–277 (1974)

    MathSciNet  ADS  Article  Google Scholar 

  14. Knowles, A., Pickl, P.: Mean-field dynamics: singular potentials and rate of convergence. Preprint arXiv:0907.4313

  15. Michelangeli, A., Schlein, B.: Dynamical collapse of boson stars. Preprint arXiv:1005.3135

  16. Pickl, P.: Derivation of the time dependent Gross Pitaevskii equation with external fields. Preprint arXiv:1001.4894

  17. Rodnianski, I., Schlein, B.: Quantum fluctuations and rate of convergence towards mean field dynamics. Commun. Math. Phys. 291(1), 31–61 (2009)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  18. Spohn, H.: Kinetic equations from Hamiltonian dynamics. Rev. Mod. Phys. 52(3), 569–615 (1980)

    MathSciNet  ADS  Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Benjamin Schlein.

Additional information

Li Chen is partially supported by National Natural Science Foundation of China (NSFC), grant number 10871112. Benjamin Schlein is partially supported by an ERC Starting Grant.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Chen, L., Lee, J.O. & Schlein, B. Rate of Convergence Towards Hartree Dynamics. J Stat Phys 144, 872 (2011). https://doi.org/10.1007/s10955-011-0283-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10955-011-0283-y

Keywords

  • Many body quantum dynamics
  • Hartree equation
  • Mean field limit