---
_id: '1493'
abstract:
- lang: eng
text: 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.
acknowledgement: 'Open access funding provided by Institute of Science and Technology
(IST Austria). '
article_number: '3'
article_processing_charge: Yes (via OA deal)
author:
- first_name: Sören P
full_name: Petrat, Sören P
id: 40AC02DC-F248-11E8-B48F-1D18A9856A87
last_name: Petrat
orcid: 0000-0002-9166-5889
- first_name: Peter
full_name: Pickl, Peter
last_name: Pickl
citation:
ama: Petrat SP, Pickl P. A new method and a new scaling for deriving fermionic mean-field
dynamics. Mathematical Physics, Analysis and Geometry. 2016;19(1). doi:10.1007/s11040-016-9204-2
apa: Petrat, S. P., & Pickl, P. (2016). A new method and a new scaling for deriving
fermionic mean-field dynamics. Mathematical Physics, Analysis and Geometry.
Springer. https://doi.org/10.1007/s11040-016-9204-2
chicago: Petrat, Sören P, and Peter Pickl. “A New Method and a New Scaling for Deriving
Fermionic Mean-Field Dynamics.” Mathematical Physics, Analysis and Geometry.
Springer, 2016. https://doi.org/10.1007/s11040-016-9204-2.
ieee: S. P. Petrat and P. Pickl, “A new method and a new scaling for deriving fermionic
mean-field dynamics,” Mathematical Physics, Analysis and Geometry, vol.
19, no. 1. Springer, 2016.
ista: Petrat SP, Pickl P. 2016. A new method and a new scaling for deriving fermionic
mean-field dynamics. Mathematical Physics, Analysis and Geometry. 19(1), 3.
mla: Petrat, Sören P., and Peter Pickl. “A New Method and a New Scaling for Deriving
Fermionic Mean-Field Dynamics.” Mathematical Physics, Analysis and Geometry,
vol. 19, no. 1, 3, Springer, 2016, doi:10.1007/s11040-016-9204-2.
short: S.P. Petrat, P. Pickl, Mathematical Physics, Analysis and Geometry 19 (2016).
date_created: 2018-12-11T11:52:20Z
date_published: 2016-03-01T00:00:00Z
date_updated: 2021-01-12T06:51:08Z
day: '01'
ddc:
- '510'
- '530'
department:
- _id: RoSe
doi: 10.1007/s11040-016-9204-2
ec_funded: 1
file:
- access_level: open_access
checksum: eb5d2145ef0d377c4f78bf06e18f4529
content_type: application/pdf
creator: system
date_created: 2018-12-12T10:16:55Z
date_updated: 2020-07-14T12:44:58Z
file_id: '5246'
file_name: IST-2016-514-v1+1_s11040-016-9204-2.pdf
file_size: 911310
relation: main_file
file_date_updated: 2020-07-14T12:44:58Z
has_accepted_license: '1'
intvolume: ' 19'
issue: '1'
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
month: '03'
oa: 1
oa_version: Published Version
project:
- _id: 25681D80-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '291734'
name: International IST Postdoc Fellowship Programme
- _id: B67AFEDC-15C9-11EA-A837-991A96BB2854
name: IST Austria Open Access Fund
publication: Mathematical Physics, Analysis and Geometry
publication_status: published
publisher: Springer
publist_id: '5690'
pubrep_id: '514'
quality_controlled: '1'
scopus_import: 1
status: public
title: A new method and a new scaling for deriving fermionic mean-field dynamics
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 19
year: '2016'
...