Quantum Fluctuations and Rate of Convergence Towards Mean Field Dynamics

By Igor Rodnianski, B. Schlein. 2007

270 citations recorded by the source corpus.

The nonlinear Hartree equation describes the macroscopic dynamics of initially factorized N-boson states, in the limit of large N. In this paper we provide estimates on the rate of convergence of the microscopic quantum mechanical evolution towards the limiting Hartree dynamics. More precisely, we prove bounds on the difference between the one-particle density associated with the solution of the N-body Schrödinger equation and the orthogonal projection onto the solution of the Hartree equation.

Subfields: Atomic and Molecular Physics, and Optics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/31e1993478bb87e8b57983d6cf22b7dd82c85608

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Citations in the corpus

Effective Evolution Equations from Quantum Dynamics (2015, methodology): “We will omit the details which can be found in [56, 21], and just write the final estimate ∣∣Tr J (1) ( γ (1) N,t −N |φt〉〈φt| ∣∣ ≤ D‖J ((1))‖e.”
Mean–Field Evolution of Fermionic Systems (2013, methodology): “For a certain class of initial data, and for interaction potentials allowing Coulomb singularities, it was shown in [36, 7] that”
Bogoliubov Spectrum of Interacting Bose Gases (2012, methodology): “19) is well understood in the time-dependent setting [58, 20, 1, 2, 50, 14], and in this case there has been recent interest in corrections to the Hartree equation [25, 26, 6], where our method might also apply.”
Quantitative Derivation of the Gross‐Pitaevskii Equation (2012, background): “It seems natural to ask whether the coherent state approach introduced in [22] can be used to obtain a rigorous derivation of the Gross-Pitaevskii equation (11), providing at the same time bounds on the rate of the convergence of the form (21).”
On the Mean-Field Limit of Bosons with Coulomb Two-Body Interaction (2008, methodology): “Rodnianski, I., Schlein, B.: Quantum fluctuations and rate of convergence towards mean field dynamics. http://arXiv.org/abs/0711.3087v1[math-ph], 2007 22.”
The quintic NLS as the mean field limit of a boson gas with three-body interactions (2008, methodology): “[7] A. Elgart, L. Erdös, B. Schlein, H.-T. Yau, Gross-Pitaevskii equation as the mean field limit of weakly coupled bosons, Arch.”
A Simple Derivation of Mean Field Limits for Quantum Systems (2009, background): “Nevertheless our result here is already better than previous results [2, 6, 7, 15] in the sense that we can handle stronger singularities than Coulomb in the interaction.”
Second-Order Corrections to Mean Field Evolution of Weakly Interacting Bosons. I. (2009, methodology): “Our work was inspired by: (i) Fock-space estimates provided by Rodnianski and Schlein [23], with regard to the rate of convergence for Hartree dynamics; and (ii) a second-order correction formulated by Wu [25,26], who introduced a kernel for the scattering of atoms in pairs from the condensate to ot”
Derivation of Hartree's theory for generic mean-field Bose systems (2013, background): “There are many works on the related (but still different) derivation of the time-dependent Hartree theory from the time-dependent Schrödinger equation associated with the Hamiltonian HN , see for instance [29, 25, 63, 7, 20, 21, 3, 22, 24, 54, 34, 51].”
Mean-Field Dynamics: Singular Potentials and Rate of Convergence (2009, unknown): “This was first observed by Seiringer; see [12].”