Skip to main content
SpringerLink
Log in

Quantum Fluctuations and Rate of Convergence Towards Mean Field Dynamics

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

The nonlinear Hartree equation describes the macroscopic dynamics of initially factorized N-boson states, in the limit of large N. In this paper we provide estimates on the rate of convergence of the microscopic quantum mechanical evolution towards the limiting Hartree dynamics. More precisely, we prove bounds on the difference between the one-particle density associated with the solution of the N-body Schrödinger equation and the orthogonal projection onto the solution of the Hartree equation.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /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.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adami, R., Golse, F., Teta, A.: Rigorous derivation of the cubic NLS in dimension one. Preprint: Univ. Texas Math. Physics Archive, http://www.ma.utexas.edu, No. 05-211, 2005

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

    MATH  MathSciNet  Google Scholar 

  3. Elgart A., Erdős L., Schlein B., Yau H.-T.: Gross–Pitaevskii equation as the mean field limit of weakly coupled bosons. Arch. Rat. Mech. Anal. 179(2), 265–283 (2006)

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  5. Erdős L., Schlein B., Yau H.-T.: Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems. Invent. Math. 167(3), 515–614 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  6. Erdős, L., Schlein, B., Yau, H.-T.: Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate. To appear in Ann. of Math. http://arxiv.org/abs/math-ph/0606017v3, 2006

  7. 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 

  8. Ginibre, J., Velo, G.: The classical field limit of scattering theory for non-relativistic many-boson systems. I and II. Commun. Math. Phys. 66, 37–76 (1979), and 68, 45–68 (1979)

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

    Article  ADS  MathSciNet  Google Scholar 

  10. Krasikov I.: Inequalities for Laguerre polynomials. East J. Approx. 11, 257–268 (2005)

    MATH  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  12. Szegö, G.: Orthogonal Polynomials. Colloq. pub. AMS. V. 23, New York: Amer. Math. Soc., 1959

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Princeton University, Princeton, NJ, USA

    Igor Rodnianski

  2. Institute of Mathematics, University of Munich, Theresienstr. 39, D-80333, Munich, Germany

    Benjamin Schlein

Authors
  1. Igor Rodnianski
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Benjamin Schlein
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Benjamin Schlein.

Additional information

Communicated by H.-T. Yau

Partially supported by NSF grant DMS-0702270.

On leave from the University of Cambridge. Supported by a Sofja Kovalevskaja Award of the Humboldt Foundation. Current address: University of Cambridge, Centre for Mathematical Sciences, DPMMS, Wilberforce Rd, Cambridge CB3 0WB, UK.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rodnianski, I., Schlein, B. Quantum Fluctuations and Rate of Convergence Towards Mean Field Dynamics. Commun. Math. Phys. 291, 31–61 (2009). https://doi.org/10.1007/s00220-009-0867-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-009-0867-4

Keywords

Search

Navigation