A new method and a new scaling for deriving fermionic mean-field dynamics Journal Article

Author(s): Petrat, Sören P.; Pickl, Peter
Article Title: A new method and a new scaling for deriving fermionic mean-field dynamics
Affiliation IST Austria
Abstract: We introduce a new method for deriving the time-dependent Hartree or Hartree-Fock equations as an effective mean-field dynamics from the microscopic Schrödinger equation for fermionic many-particle systems in quantum mechanics. The method is an adaption of the method used in Pickl (Lett. Math. Phys. 97 (2) 151–164 2011) for bosonic systems to fermionic systems. It is based on a Gronwall type estimate for a suitable measure of distance between the microscopic solution and an antisymmetrized product state. We use this method to treat a new mean-field limit for fermions with long-range interactions in a large volume. Some of our results hold for singular attractive or repulsive interactions. We can also treat Coulomb interaction assuming either a mild singularity cutoff or certain regularity conditions on the solutions to the Hartree(-Fock) equations. In the considered limit, the kinetic and interaction energy are of the same order, while the average force is subleading. For some interactions, we prove that the Hartree(-Fock) dynamics is a more accurate approximation than a simpler dynamics that one would expect from the subleading force. With our method we also treat the mean-field limit coupled to a semiclassical limit, which was discussed in the literature before, and we recover some of the previous results. All results hold for initial data close (but not necessarily equal) to antisymmetrized product states and we always provide explicit rates of convergence.
Keywords: Fermionic mean-field dynamics; Many-body quantum mechanics; Mean-field limit; Reduced Hartree-Fock
Journal Title: Mathematical Physics, Analysis and Geometry
Volume: 19
Issue 1
ISSN: 1572-9656
Publisher: Springer  
Date Published: 2016-03-01
Start Page: Article number: 3
Copyright Statement: CC BY 4.0
DOI: 10.1007/s11040-016-9204-2
Notes: Open access funding provided by Institute of Science and Technology (IST Austria). We thank Detlef Dürr, Laszlo Erdös, Maximilian Jeblick, David Mitrouskas and Robert Seiringer for many helpful discussions. We thank the referee for many comments that helped to improve the presentation of the article. S.P. gratefully acknowledges support from Cusanuswerk. S. P.’s research has received funding from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement n°291734. The article is based on one of the author’s (S.P.) PhD thesis [48].
Open access: yes (OA journal)
IST Austria Authors
  1. Sören Petrat
    2 Petrat
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