Skip to main content
Log in

Rate of Convergence Towards Hartree Dynamics

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

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, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Similar content being viewed by others

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)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  ADS  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)

    Article  MathSciNet  ADS  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)

    Article  MathSciNet  MATH  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)

    Article  MathSciNet  ADS  MATH  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)

    Article  MathSciNet  ADS  MATH  Google Scholar 

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

    Article  MathSciNet  ADS  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)

    Article  MathSciNet  ADS  MATH  Google Scholar 

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

    Article  MathSciNet  ADS  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, 872–903 (2011). https://doi.org/10.1007/s10955-011-0283-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

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

Keywords

Navigation