Rodnianski, Igor; Schlein, Benjamin Quantum fluctuations and rate of convergence towards mean field dynamics. (English) Zbl 1186.82051 Commun. Math. Phys. 291, No. 1, 31-61 (2009). The authors consider a system of \(N\)-boson described by a mean field Hamiltonian, with a coupling constant \(1/N\) in front of the two-body interaction potential terms. In the limit of large \(N\) an initially factorized state can be approximated by a factorized state, whose one-particle wave function obeys the nonlinear Hartree equation. This approximation is to be understood in terms of convergence of marginal densities. The authors address this problem using a different technique with respect to the study of the BBGKY hierarchy, and obtain improved bounds on the difference between the one-particle density associated with the solution of the \(N\)-body Schrödinger equation and the one-particle wavefunction solution of the Hartree equation. Reviewer: Bassano Vacchini (Milano) Cited in 3 ReviewsCited in 122 Documents MSC: 82C10 Quantum dynamics and nonequilibrium statistical mechanics (general) 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics Keywords:Hartree equation; mean field dynamics; BBGKY hierarchy PDFBibTeX XMLCite \textit{I. Rodnianski} and \textit{B. Schlein}, Commun. Math. Phys. 291, No. 1, 31--61 (2009; Zbl 1186.82051) Full Text: DOI arXiv References: [1] Adami, R., Golse, F., Teta, A.: Rigorous derivation of the cubic NLS in dimension one. Preprint: Univ. Texas Math. Physics Archive, http://www.ma.utexas.edu , No. 05-211, 2005 · Zbl 1118.81021 [2] Bardos C., Golse F., Mauser N.: Weak coupling limit of the N-particle Schrödinger equation. Meth. Appl. Anal. 7, 275–293 (2000) · Zbl 1003.81027 [3] Elgart A., Erdos L., Schlein B., Yau H.-T.: Gross–Pitaevskii equation as the mean field limit of weakly coupled bosons. Arch. Rat. Mech. Anal. 179(2), 265–283 (2006) · Zbl 1086.81035 · doi:10.1007/s00205-005-0388-z [4] Elgart A., Schlein B.: Mean field dynamics of Boson stars. Commun. Pure Appl. Math. 60(4), 500–545 (2007) · Zbl 1113.81032 · doi:10.1002/cpa.20134 [5] Erdos L., Schlein B., Yau H.-T.: Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems. Invent. Math. 167(3), 515–614 (2007) · Zbl 1123.35066 · doi:10.1007/s00222-006-0022-1 [6] Erdos, L., Schlein, B., Yau, H.-T.: Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate. To appear in Ann. of Math. http://arxiv.org/abs/math-ph/0606017v3 , 2006 [7] Erdos L., Yau H.-T.: Derivation of the nonlinear Schrödinger equation from a many body Coulomb system. Adv. Theor. Math. Phys. 5(6), 1169–1205 (2001) · Zbl 1014.81063 [8] Ginibre, J., Velo, G.: The classical field limit of scattering theory for non-relativistic many-boson systems. I and II. Commun. Math. Phys. 66, 37–76 (1979), and 68, 45–68 (1979) · Zbl 0443.35067 [9] Hepp K.: The classical limit for quantum mechanical correlation functions. Commun. Math. Phys. 35, 265–277 (1974) · doi:10.1007/BF01646348 [10] Krasikov I.: Inequalities for Laguerre polynomials. East J. Approx. 11, 257–268 (2005) · Zbl 1137.33306 [11] Spohn H.: Kinetic equations from Hamiltonian dynamics. Rev. Mod. Phys. 52(3), 569–615 (1980) · Zbl 0465.76069 · doi:10.1103/RevModPhys.52.569 [12] Szegö, G.: Orthogonal Polynomials. Colloq. pub. AMS. V. 23, New York: Amer. Math. Soc., 1959 · Zbl 0089.27501 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.