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arXiv:0807.3108 · Journal of Mathematical Physics

Mean field limit for bosons and propagation of Wigner measures

Zied Ammari, Francis Nier

Published Apr 1, 2009• 36 citations
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Abstract

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.

DOI
10.1063/1.3115046

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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 a quantum system constituted by N identical particles interacting by means of a mean-field Hamiltonian. It is well known that, in the limit N → ∞, the one-particle state obeys to the Hartree equation. Moreover, propagation of chaos holds. In this paper, we take care of the � dependence by considering the semiclassical expansion of the N-particle system. We prove that each term of the expansion agrees, in the limit N → ∞, with the corresponding one associated with the Hartree equation. We work in the classical phase space by using the Wigner formalism, which seems to be the most appropriate for the present problem.
Mean-field limits for bosonic systems explain how an effective one-particle nonlinear equationemerges from an N-body Schrödinger dynamics. In the standard mean-field setting, themicroscopic Hamiltonian contains a scaled pair interaction, and the limiting dynamics is aHartree equation. This paper considers a closely related but structurally different situation: inaddition to the usual pair interaction, the microscopic system carries a phase drive which isgenerated by the evolving N-body state itself.The distinction is important. The phase field considered here is not a prescribed timedependent potential and is not imported from the limiting Hartree trajectory. At finite N, it iscomputed from the normalized one-particle marginal density
The mean-field approximations of many-boson dynamics are known to be effective in many physical relevant situations. The mathematical justifications of such approximations rely generally on specific considerations which depend too much on the model and on the initial states of the system which are required to be well-prepared. In this article, using the method of Wigner measures, we prove in a fairly complete generality the accuracy of the mean-field approximation. Roughly speaking, we show that the dynamics of a many-boson system are well approximated, in the limit of a large number of particles, by a one particle mean-field equation if the following general principles are satisfied: ∙\bullet∙ The Hamiltonian is in a mean-field regime (i.e.: The interaction and the free energy parts are of the same order with respect to the number of particles). ∙\bullet∙ The interaction is relatively compact with respect to a one particle and it is dominated by the free energy part. ∙\bullet∙ There exists at most one weak solution for the mean-field equation for each initial condition.The convergence towards the mean field limit is described in terms of Wigner (probability) measures and it holds for any initial quantum states with a finite free energy average. The main novelty of this article lies in the use of fine properties of uniqueness for the Liouville equations in infinite dimensional spaces.