Skip to main content
SpringerLink
Log in

A Central Limit Theorem in Many-Body Quantum Dynamics

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

Abstract

We study the many body quantum evolution of bosonic systems in the mean field limit. The dynamics is known to be well approximated by the Hartree equation. So far, the available results have the form of a law of large numbers. In this paper we go one step further and we show that the fluctuations around the Hartree evolution satisfy a central limit theorem. Interestingly, the variance of the limiting Gaussian distribution is determined by a time-dependent Bogoliubov transformation describing the dynamics of initial coherent states in a Fock space representation of the system.

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

Access this article

Log in via an institution

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., Schlein B.: Rate of convergence towards Hartree dynamics. J. Stat. Phys. 144(4), 872–903 (2011)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  3. Chen T., Pavlović N.: On the Cauchy problem for focusing and defocusing Gross-Pitaevskii hierarchies. Discrete Contin. Dyn. Syst. 27(2), 715–739 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen T., Pavlović N.: The quintic NLS as the mean field limit of a boson gas with three-body interactions. J. Funct. Anal. 260(4), 959–997 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cramer M., Eisert J.: A quantum central limit theorem for non-equilibrium systems: exact local relaxation of correlated states. New J. Phys. 12, 055020 (2009)

    Article  Google Scholar 

  6. Cushen C.D., Hudson R.L.: A quantum-mechanical central limit theorem. J. Appl. Prob. 8, 454 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen X.: Second order corrections to mean field evolution for weakly interacting bosons in the case of 3-body interactions. Arch. Rati. Mech. Anal. 203(2), 455–497 (2012)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. 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 

  10. 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 

  11. Erdős L., Schlein B., Yau H.-T.: Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate. Ann. Math. (2) 172, 291–370 (2010)

    Article  Google Scholar 

  12. Erdős L., Schlein B., Yau H.-T.: Rigorous derivation of the Gross-Pitaevskii equation with a large interaction potential. J. Amer. Math. Soc. 22(4), 1099–1156 (2009)

    Article  MathSciNet  Google Scholar 

  13. 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 

  14. 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 

  15. Grillakis M., Machedon M., Margetis D.: Second-order corrections to mean field evolution of weakly interacting bosons. II. Adv. Math. 228(3), 1788–1815 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. 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)

  17. Goderis, D., Verbeure, A., Vets, P.: About the mathematical theory of quantum fluctuations. In: Mathematical Methods in Statistical Mechanics. Leuven Notes in Mathematical and Theoretical Physics. Series A: Mathematical Physics, 1. Leuven: Leuven University Press, 1989

  18. Hayashi M.: Quantum estimation and the quantum central limit theorem. Sci. and Tech. 227, 95 (2006)

    Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

  20. Hepp K., Lieb E.H.: Phase transitions in reservoir-driven open systems with applications to lasers and superconductors. Helv. Phys. Acta 46, 573 (1973)

    Google Scholar 

  21. Jakšić V., Pautrat Y., Pillet C.-A.: A quantum central limit theorem for sums of iid random variables. J. Math. Phys. 51, 015208 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  22. Kuperberg G.: A tracial quantum central limit theorem. Trans. Amer. Math. Soc. 357, 549 (2005)

    Article  MathSciNet  Google Scholar 

  23. Lee J.O.: Rate of convergence towards semi-relativistic Hartree dynamics. Ann. Henri Poincaré 114, 313–346 (2013)

    Article  ADS  Google Scholar 

  24. Kirkpatrick K., Schlein B., Staffilani G.: Derivation of the two dimensional nonlinear Schrödinger equation from many body quantum dynamics. Amer. J. Math. 133(1), 91–130 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. Klainerman S., Machedon M.: On the uniqueness of solutions to the Gross-Pitaevskii hierarchy. Commun. Math. Phys. 279(1), 169–185 (2008)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  26. Michelangeli A., Schlein B.: Dynamical Collapse of Boson Stars. Commun. Math. Phys. 311(3), 645–687 (2012)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  27. Knowles A., Pickl P.: Mean-field dynamics: singular potentials and rate of convergence. Commun. Math. Phys. 298(1), 101–138 (2010)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  28. Pickl, P.: Derivation of the time dependent Gross Pitaevskii equation with external fields. Preprint, http://arxiv.org/abs/1001.4894v2 [math-ph], 2010

  29. 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 

  30. Speicher R.: A noncommutative central limit theorem. Math. Z. 209(1), 55–66 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  31. 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

  1. Courant Institute of Mathematics, New York University, 251 Mercer Street, New York, NY, 10012, USA

    Gérard Ben Arous

  2. Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, IL, 61801, USA

    Kay Kirkpatrick

  3. Institute of Applied Mathematics, Endenicher Allee 60, 53115, Bonn, Germany

    Benjamin Schlein

Authors
  1. Gérard Ben Arous
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kay Kirkpatrick
    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.

Additional information

Communicated by H.-T. Yau

Kirkpatrick, Partially supported by NSF grants OISE-0730136 and DMS-1106770.

Benjamin Schlein, Partially supported by an ERC Starting Grant.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Arous, G.B., Kirkpatrick, K. & Schlein, B. A Central Limit Theorem in Many-Body Quantum Dynamics. Commun. Math. Phys. 321, 371–417 (2013). https://doi.org/10.1007/s00220-013-1722-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-013-1722-1

Keywords

Search

Navigation