Journal of Statistical Physics

, Volume 154, Issue 1–2, pp 113–152 | Cite as

Multivariate Central Limit Theorem in Quantum Dynamics

  • Simon Buchholz
  • Chiara Saffirio
  • Benjamin Schlein


We consider the time evolution of N bosons in the mean field regime for factorized initial data. In the limit of large N, the many body evolution can be approximated by the non-linear Hartree equation. In this paper we are interested in the fluctuations around the Hartree dynamics. We choose k self-adjoint one-particle operators O 1,…,O k on \(L^{2} ({\mathbb{R}}^{3})\), and we average their action over the N-particles. We show that, for every fixed \(t \in{\mathbb{R}}\), expectations of products of functions of the averaged observables approach, as N→∞, expectations with respect to a complex Gaussian measure, whose covariance matrix can be expressed in terms of a Bogoliubov transformation describing the dynamics of quantum fluctuations around the mean field Hartree evolution. If the operators O 1,…,O k commute, the Gaussian measure is real and positive, and we recover a “classical” multivariate central limit theorem. All our results give explicit bounds on the rate of the convergence.


Many body quantum dynamics Hartree equation Mean field limit Central limit theorem Bogoliubov transformations 


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Simon Buchholz
    • 1
  • Chiara Saffirio
    • 1
  • Benjamin Schlein
    • 1
  1. 1.Institute of Applied MathematicsUniversity of BonnBonnGermany

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