We consider the dynamics of a large number [Formula: see text] of nonrelativistic bosons in the mean field limit for a class of interaction potentials that includes Coulomb interaction. In order to describe the fluctuations around the mean field Hartree state, we introduce an auxiliary Hamiltonian on the [Formula: see text]-particle space that is similar to the one obtained from Bogoliubov theory. We show convergence of the auxiliary time evolution to the fully interacting dynamics in the norm of the [Formula: see text]-particle space. This result allows us to prove several other results: convergence of reduced density matrices in trace norm with optimal rate, convergence in energy trace norm, and convergence to a time evolution obtained from the Bogoliubov Hamiltonian on Fock space with expected optimal rate. We thus extend and quantify several previous results, e.g., by providing the physically important convergence rates, including time-dependent external fields and singular interactions, and allowing for more general initial states, e.g., those that are expected to be ground states of interacting systems.We consider the dynamics of a large number [Formula: see text] of nonrelativistic bosons in the mean field limit for a class of interaction potentials that includes Coulomb interaction. In order to describe the fluctuations around the mean field Hartree state, we introduce an auxiliary Hamiltonian on the [Formula: see text]-particle space that is similar to the one obtained from Bogoliubov theory. We show convergence of the auxiliary time evolution to the fully interacting dynamics in the norm of the [Formula: see text]-particle space. This result allows us to prove several other results: convergence of reduced density matrices in trace norm with optimal rate, convergence in energy trace norm, and convergence to a time evolution obtained from the Bogoliubov Hamiltonian on Fock space with expected optimal rate. We thus extend and quantify several previous results, e.g., by providing the physically important convergence rates, including time-dependent external fields and singular interactions, and allowing for more general initial states, e.g., those that are expected to be ground states of interacting systems.

arXiv:1609.06264Reviews in Mathematical PhysicsFeb 8, 201941 citations

Stochastic quantum dynamics beyond mean field

Denis Lacroix A, Sakir Ayik B

arXiv:1402.2393The European Physical Journal AJun 1, 2014110 citations

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.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.

Aug 16, 201628 citations

Ground state fluctuations of quantum mean field systems

A.K. Callebaut, M. Fannes

QP-PQ, quantum probability and white noise analysisJul 4, 19941 citation

Mean-Field Dynamics: Singular Potentials and Rate of Convergence

Antti Knowles, Peter Pickl

arXiv:0907.4313Communications in Mathematical PhysicsFeb 18, 2010166 citations

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.