Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1111.5918 · ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE

Mean field propagation of infinite dimensional Wigner measures with a singular two-body interaction potential

Zied Ammari, Francis Nier

Published Mar 31, 2015• 38 citations
arXivHTMLPDF

Abstract

We consider the quantum dynamics of many bosons systems in the mean field limit with a singular pair-interaction potential, including the attractive or repulsive Coulombic case in three dimensions.By using a measure transportation technique developed in [3], we show that Wigner measures propagate along the nonlinear Hartree flow.Such property was previously proved only for bounded potentials in our works [5,6] with a slightly different strategy.

DOI
10.2422/2036-2145.201112_004

Related papers

Mean field limit for bosons and propagation of Wigner measures

Zied Ammari, Francis Nier

We consider the N-body Schrödinger dynamics of bosons in the mean field limit with a bounded pair-interaction potential. According to the previous work [Ammari, Z. and Nier, F., “Mean field limit for bosons and infinite dimensional phase-space analysis,” Ann. Henri Poincare 9, 1503 (2008)], the mean field limit is translated into a semiclassical problem with a small parameter ε→0, after introducing an ε-dependent bosonic quantization. The limits of quantum correlation functions are expressed as a push forward by a nonlinear flow (e.g., Hartree) of the associated Wigner measures. These object and their basic properties were introduced by Ammari and Nier in the infinite dimensional setting. The additional result presented here states that the transport by the nonlinear flow holds for a rather general class of quantum states in their mean field limit.

arXiv:0807.3108•Journal of Mathematical Physics•Apr 1, 2009•36 citations

Mean field limit for bosons and propagation of Wigner measures

Zied Ammari, Francis Nier

We consider the N-body Schrödinger dynamics of bosons in the mean field limit with a bounded pair-interaction potential. According to the previous work [AmNi], the mean field limit is translated into a semiclassical problem with a small parameter ε → 0, after introducing an ε-dependent bosonic quantization. The limit is expressed as a push-forward by a nonlinear flow (e.g. Hartree) of the associated Wigner measures. These object and their basic properties were introduced in [AmNi] in the infinite dimensional setting. The additional result presented here states that the transport by the nonlinear flow holds for rather general class of quantum states in their mean field limit.

•Jan 1, 2009•43 citations

Mean field limit for bosons with compact kernels interactions by Wigner measures transportation

Quentin Liard, Boris Pawilowski

We consider a class of many-body Hamiltonians composed of a free (kinetic) part and a multi-particle (potential) interaction with a compactness assumption on the latter part. We investigate the mean field limit of such quantum systems following the Wigner measures approach. We prove in particular the propagation of these measures along the flow of a nonlinear (Hartree) field equation. This enhances and complements some previous results of the same type shown in Z. Ammari and F. Nier and Fröhlich et al. [“Mean field limit for bosons and propagation of Wigner measures,” J. Math. Phys. 50(4), 042107 (2009); Z. Ammari and F. Nier and Fröhlich et al., “Mean field propagation of Wigner measures and BBGKY hierarchies for general bosonic states,” J. Math. Pures Appl. 95(6), 585–626 (2011); Z. Ammari and F. Nier and Fröhlich et al., “Mean-field- and classical limit of many-body Schrödinger dynamics for bosons,” Commun. Math. Phys. 271(3), 681–697 (2007)].

arXiv:1402.4261•Journal of Mathematical Physics•Sep 1, 2014•11 citations

Mean Field Limits in Strongly Confined Systems

Johannes von Keler

We consider the dynamics of N interacting Bosons in three dimensions which are strongly confined in one or two directions. We analyze the mean field scaling and the nonlinear Schrödinger scaling of the interaction potential w and choose the initial wavefunction to be close to a product wavefunction. For both scalings we prove that in the mean field limit the dynamics of the N-particle system is described by a nonlinear equation in two or one dimensions. In the case of the mean field scaling this equation is the Hartree equation and for the second scaling the nonlinear Schrödinger equation. In both cases we obtain explicit bounds for the rate of convergence of the N-particle dynamics to the one-particle dynamics.

•Feb 1, 2015•2 citations

Rate of convergence toward Hartree dynamics with singular interaction potential

Li Chen, Ji Oon Lee, Jinyeop Lee

We consider a system of N-bosons with a two-body interaction potential V∈L2(R3)+L∞(R3), possibly more singular than the Coulomb interaction. We show that, with H1(R3) initial data, the difference between the many-body Schrödinger evolution in the mean-field regime and the corresponding Hartree dynamics is of order 1/N, for any fixed time. The N-dependence of the bound is optimal.

arXiv:1708.07278•Journal of Mathematical Physics•Mar 1, 2018•12 citations

Rate of convergence towards mean-field evolution for weakly interacting bosons with singular three-body interactions

Jinyeop Lee

In this paper, we investigate the dynamics of a system of $N$ weakly interacting bosons with singular three-body interactions in three dimensions. By assuming factorized initial data $Ψ_{N,0}=φ_{0}^{\otimes N}$ and triple collisions, we prove that in the many-particle limit, its mean-field approximation converges to quintic Hartree dynamics. Moreover, we prove that the rate of convergence towards the mean-field quintic Hartree evolution is of $O(N^{-(1+4a)/(3+2a)})$ for $φ_{0}\in H^{(3/2)+a}(\mathbb{R}^{3})$ where $0\leq a<1/2$ and $O(N^{-1})$ for $a>1$. Our proof is based on and extends the Fock space approach.

arXiv:2006.13040•Jun 22, 2020•2 citations
We consider the quantum dynamics of many bosons systems in the mean field limit with a singular pair-interaction potential, including the attractive or repulsive Coulombic case in three dimensions.By using a measure transportation technique developed in [3], we show that Wigner measures propagate along the nonlinear Hartree flow.Such property was previously proved only for bounded potentials in our works [5,6] with a slightly different strategy.
We consider the N-body Schrödinger dynamics of bosons in the mean field limit with a bounded pair-interaction potential. According to the previous work [Ammari, Z. and Nier, F., “Mean field limit for bosons and infinite dimensional phase-space analysis,” Ann. Henri Poincare 9, 1503 (2008)], the mean field limit is translated into a semiclassical problem with a small parameter ε→0, after introducing an ε-dependent bosonic quantization. The limits of quantum correlation functions are expressed as a push forward by a nonlinear flow (e.g., Hartree) of the associated Wigner measures. These object and their basic properties were introduced by Ammari and Nier in the infinite dimensional setting. The additional result presented here states that the transport by the nonlinear flow holds for a rather general class of quantum states in their mean field limit.
We consider the N-body Schrödinger dynamics of bosons in the mean field limit with a bounded pair-interaction potential. According to the previous work [AmNi], the mean field limit is translated into a semiclassical problem with a small parameter ε → 0, after introducing an ε-dependent bosonic quantization. The limit is expressed as a push-forward by a nonlinear flow (e.g. Hartree) of the associated Wigner measures. These object and their basic properties were introduced in [AmNi] in the infinite dimensional setting. The additional result presented here states that the transport by the nonlinear flow holds for rather general class of quantum states in their mean field limit.
We consider a class of many-body Hamiltonians composed of a free (kinetic) part and a multi-particle (potential) interaction with a compactness assumption on the latter part. We investigate the mean field limit of such quantum systems following the Wigner measures approach. We prove in particular the propagation of these measures along the flow of a nonlinear (Hartree) field equation. This enhances and complements some previous results of the same type shown in Z. Ammari and F. Nier and Fröhlich et al. [“Mean field limit for bosons and propagation of Wigner measures,” J. Math. Phys. 50(4), 042107 (2009); Z. Ammari and F. Nier and Fröhlich et al., “Mean field propagation of Wigner measures and BBGKY hierarchies for general bosonic states,” J. Math. Pures Appl. 95(6), 585–626 (2011); Z. Ammari and F. Nier and Fröhlich et al., “Mean-field- and classical limit of many-body Schrödinger dynamics for bosons,” Commun. Math. Phys. 271(3), 681–697 (2007)].
We consider the dynamics of N interacting Bosons in three dimensions which are strongly confined in one or two directions. We analyze the mean field scaling and the nonlinear Schrödinger scaling of the interaction potential w and choose the initial wavefunction to be close to a product wavefunction. For both scalings we prove that in the mean field limit the dynamics of the N-particle system is described by a nonlinear equation in two or one dimensions. In the case of the mean field scaling this equation is the Hartree equation and for the second scaling the nonlinear Schrödinger equation. In both cases we obtain explicit bounds for the rate of convergence of the N-particle dynamics to the one-particle dynamics.
In this paper, we investigate the dynamics of a system of NNN weakly interacting bosons with singular three-body interactions in three dimensions. By assuming factorized initial data ΨN,0=φ0⊗NΨ_{N,0}=φ_{0}^{\otimes N}ΨN,0​=φ0⊗N​ and triple collisions, we prove that in the many-particle limit, its mean-field approximation converges to quintic Hartree dynamics. Moreover, we prove that the rate of convergence towards the mean-field quintic Hartree evolution is of O(N−(1+4a)/(3+2a))O(N^{-(1+4a)/(3+2a)})O(N−(1+4a)/(3+2a)) for φ0∈H(3/2)+a(R3)φ_{0}\in H^{(3/2)+a}(\mathbb{R}^{3})φ0​∈H(3/2)+a(R3) where 0≤a<1/20\leq a<1/20≤a<1/2 and O(N−1)O(N^{-1})O(N−1) for a>1a>1a>1. Our proof is based on and extends the Fock space approach.