Mean-Field- and Classical Limit of Many-Body Schrödinger Dynamics for Bosons

Froehlich, Juerg; Graffi, Sandro; Schwarz, Simon

doi:10.1007/s00220-007-0207-5

Cited by 85 publications

(128 citation statements)

References 15 publications

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“…We prove 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 [4,6,11]. …”

supporting

confidence: 90%

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

Liard

Pawilowski

2014

Journal of Mathematical Physics

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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 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 [4,6,11].

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“…We prove 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 [4,6,11]. …”

supporting

confidence: 90%

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

Liard

Pawilowski

2014

Journal of Mathematical Physics

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“…The derivation of a hierarchy similar to (1) coming from the limit of N -body Schrödinger dynamics was obtained in [1,2,11,[15][16][17]45,48,[51][52][53][54][55][56][57]74,[84][85][86][87]124,162] in various different contexts. The first rate of convergence result was obtained by Rodnianski and Schlein [145] and subsequent rate of convergence results have been obtained in [6,17,38,49,50,75,85,[93][94][95]110,115,121,124,137,138].…”

Section: Previously Known Resultsmentioning

confidence: 99%

Randomization and the Gross–Pitaevskii Hierarchy

Sohinger

Staffilani

2015

Arch Rational Mech Anal

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We study the Gross-Pitaevskii hierarchy on the spatial domain T 3 . By using an appropriate randomization of the Fourier coefficients in the collision operator, we prove an averaged form of the main estimate which is used in order to contract the Duhamel terms that occur in the study of the hierarchy. In the averaged estimate, we do not need to integrate in the time variable. An averaged spacetime estimate for this range of regularity exponents then follows as a direct corollary. The range of regularity exponents that we obtain is α > 3 4 . It was shown in our previous joint work with Gressman (J Funct Anal 266(7): 2014) that the range α > 1 is sharp in the corresponding deterministic spacetime estimate. This is in contrast to the non-periodic setting, which was studied by Klainerman and Machedon (Commun Math Phys 279(1): [169][170][171][172][173][174][175][176][177][178][179][180][181][182][183][184][185] 2008), where the spacetime estimate is known to hold whenever α ≥ 1. The goal of our paper is to extend the range of α in this class of estimates in a probabilistic sense. We use the new estimate and the ideas from its proof in order to study randomized forms of the Gross-Pitaevskii hierarchy. More precisely, we consider hierarchies similar to the Gross-Pitaevskii hierarchy, but in which the collision operator has been randomized. For these hierarchies, we show convergence to zero in low regularity Sobolev spaces of Duhamel expansions of fixed deterministic density matrices. We believe that the study of the randomized collision operators could be the first step in the understanding of a nonlinear form of randomization.

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“…In series expansion(25) the operator N * int (j 1 , j 2 ) f n = −N int (j 1 , j 2 ) f n is an adjoint operator to operator (12) and the group G * 1 (t, i) = G 1 (−t, i) is dual to group (10) in the sense of functional (24). For bounded interaction potentials series (25) is norm convergent on the space (25) is a solution of the Cauchy problem of the quantum Vlasov-type kinetic equation with initial correlations:…”

Section: The Quantum Vlasov-type Kinetic Equation With Initial Correlmentioning

confidence: 99%

“…The conventional approach to the rigorous derivation of the quantum kinetic equations is based on the consideration of an asymptotic behavior of a solution of the quantum BBGKY hierarchy for marginal density operators constructed within the framework of the theory of perturbations in case of initial states specified by a one-particle (marginal) density operator without correlations [11][12][13][14], i.e. such that satisfy a chaos condition.…”

Section: Introductionmentioning

confidence: 99%

New approach to derivation of quantum kinetic equations with initial correlations

Gerasimenko

2015

Carpathian Math. Publ.

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We propose a new approach to the derivation of kinetic equations from dynamics of large particle quantum systems, involving correlations of particle states at initial time. The developed approach is based on the description of the evolution within the framework of marginal observables in scaling limits. As a result the quantum Vlasov-type kinetic equation with initial correlations is constructed and the statement relating to the property of a propagation of initial correlations is proved in a mean field limit.

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Mean-Field- and Classical Limit of Many-Body Schrödinger Dynamics for Bosons

Cited by 85 publications

References 15 publications

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

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

Randomization and the Gross–Pitaevskii Hierarchy

New approach to derivation of quantum kinetic equations with initial correlations

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