On the Mean-Field Limit of Bosons with Coulomb Two-Body Interaction

By Juerg Froehlich, Antti Knowles, Simon Schwarz. 2008

161 citations recorded by the source corpus.

In the mean-field limit the dynamics of a quantum Bose gas is described by a Hartree equation. We present a simple method for proving the convergence of the microscopic quantum dynamics to the Hartree dynamics when the number of particles becomes large and the strength of the two-body potential tends to 0 like the inverse of the particle number. Our method is applicable for a class of singular interaction potentials including the Coulomb potential. We prove and state our main result for the Heisenberg- picture dynamics of “observables”, thus avoiding the use of coherent states. Our formulation shows that the mean-field limit is a “semi-classical” limit.

Subfields: Atomic and Molecular Physics, and Optics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/48910691ffc88cf12e59de07f4add283468251a2

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In Atomic and Molecular Physics, and Optics

In Mathematical Physics

Citations in the corpus

Quantitative Derivation of the Gross‐Pitaevskii Equation (2012, methodology): “A different point of view on the mean field limit was given in [20], where the convergence towards the Hartree dynamics was interpreted as a Egorov-type theorem.”
The Excitation Spectrum for Weakly Interacting Bosons in a Trap (2012, methodology): “In this paper, we shall prove the accuracy of the Bogoliubov approximation for the excitation spectrum of a trapped Bose gas, in the mean-field or Hartree limit [21,22], where the interaction is weak and long-range.”
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].”
The Excitation Spectrum for Weakly Interacting Bosons (2010, background): “We refer to [5, 6] for a review and further references.”
The quintic NLS as the mean field limit of a boson gas with three-body interactions (2008, background): “In dimensions d ≥ 3 it is currently not known whether the limit points obtained from the BBGKY hierarchy in the weak, subsequential limit N → ∞ indeed satisfy this space-time a priori bound.”
Mean-Field Dynamics: Singular Potentials and Rate of Convergence (2009, unknown): “Methods based on expansions, as used in [13] and [4], give very weak bounds on the error R (1) N (t), while weak compactness arguments, as used in [3] and [1], yield no information on the rate of convergence.”
Mean–Field Evolution of Fermionic Systems (2013, methodology): “Among many other important contributions to the mathematical understanding of the many body dynamics for bosonic systems in the mean field regime, let us also recall [17], [2], [35, 32], and the series of papers [21, 22, 23].”