TY - JOUR

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

AU - Arous, Gérard Ben

AU - Kirkpatrick, Kay

AU - Schlein, Benjamin

N1 - Funding Information:
Benjamin Schlein, Partially supported by an ERC Starting Grant.
Funding Information:
Kirkpatrick, Partially supported by NSF grants OISE-0730136 and DMS-1106770.

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.

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U2 - 10.1007/s00220-013-1722-1

DO - 10.1007/s00220-013-1722-1

M3 - Article

AN - SCOPUS:84878772792

SN - 0010-3616

VL - 321

SP - 371

EP - 417

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 2

ER -