### 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.

Original language | English (US) |
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Pages (from-to) | 371-417 |

Number of pages | 47 |

Journal | Communications in Mathematical Physics |

Volume | 321 |

Issue number | 2 |

DOIs | |

State | Published - Jul 2013 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Communications in Mathematical Physics*,

*321*(2), 371-417. https://doi.org/10.1007/s00220-013-1722-1

**A Central Limit Theorem in Many-Body Quantum Dynamics.** / Ben Arous, Gerard; Kirkpatrick, Kay; Schlein, Benjamin.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 321, no. 2, pp. 371-417. https://doi.org/10.1007/s00220-013-1722-1

}

TY - JOUR

T1 - A Central Limit Theorem in Many-Body Quantum Dynamics

AU - Ben Arous, Gerard

AU - Kirkpatrick, Kay

AU - Schlein, Benjamin

PY - 2013/7

Y1 - 2013/7

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84878772792&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84878772792&partnerID=8YFLogxK

U2 - 10.1007/s00220-013-1722-1

DO - 10.1007/s00220-013-1722-1

M3 - Article

AN - SCOPUS:84878772792

VL - 321

SP - 371

EP - 417

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 2

ER -