Empirical Measures and Quantum Mechanics: Applications to the Mean-Field Limit
In this paper, we define a quantum analogue of the notion of empirical measure in the classical mechanics of N -particle systems. We establish an equation governing the evolution of our quantum analogue of the N -particle empirical measure, and we prove that this equation contains the Hartree equati...
Gespeichert in:
| Veröffentlicht in: | Communications in mathematical physics 2019-08, Vol.369 (3), p.1021-1053 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | eng |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| container_end_page | 1053 |
|---|---|
| container_issue | 3 |
| container_start_page | 1021 |
| container_title | Communications in mathematical physics |
| container_volume | 369 |
| creator | Golse, François Paul, Thierry |
| description | In this paper, we define a quantum analogue of the notion of empirical measure in the classical mechanics of
N
-particle systems. We establish an equation governing the evolution of our quantum analogue of the
N
-particle empirical measure, and we prove that this equation contains the Hartree equation as a special case. Applications to the mean-field limit of the
N
-particle Schrödinger equation include an
O
(
1
/
N
)
convergence rate in some appropriate dual Sobolev norm for the Wigner transform of the single-particle marginal of the
N
-particle density operator, uniform in
ħ
∈
(
0
,
1
]
provided that
V
and
(
-
Δ
)
3
/
2
+
d
/
4
V
have integrable Fourier transforms. |
| doi_str_mv | 10.1007/s00220-019-03357-z |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_2249088027</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2249088027</sourcerecordid><originalsourceid>FETCH-LOGICAL-c363t-c31cd0f8c7b1e73be5f246d76d88090a085b36b351e772c2245c689a01f0a9463</originalsourceid><addsrcrecordid>eNp9kEFLwzAYhoMoOKd_wFPBc_RL0iattzG2KUxF0HNI09RltGlN2oP79WZW8OblC3x53veDB6FrArcEQNwFAEoBAykwMJYJfDhBM5IyiqEg_BTNAAhgxgk_Rxch7AGgoJzP0POq7a23WjXJk1Fh9CYkylXJ66jcMLZxqXfKWR3uk0XfNxEcbOdCMnTJsDPHjMNra5oq2drWDpforFZNMFe_7xy9r1dvywe8fdk8LhdbrBlnQ5xEV1DnWpTECFaarKYprwSv8hwKUJBnJeMly-KvoJrSNNM8LxSQGlSRcjZHN1Nv77vP0YRB7rvRu3hSRriAWENFpOhEad-F4E0te29b5b8kAXn0JidvMnqTP97kIYbYFAoRdh_G_1X_k_oGK4xvew</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2249088027</pqid></control><display><type>article</type><title>Empirical Measures and Quantum Mechanics: Applications to the Mean-Field Limit</title><source>SpringerLink Journals</source><creator>Golse, François ; Paul, Thierry</creator><creatorcontrib>Golse, François ; Paul, Thierry</creatorcontrib><description>In this paper, we define a quantum analogue of the notion of empirical measure in the classical mechanics of
N
-particle systems. We establish an equation governing the evolution of our quantum analogue of the
N
-particle empirical measure, and we prove that this equation contains the Hartree equation as a special case. Applications to the mean-field limit of the
N
-particle Schrödinger equation include an
O
(
1
/
N
)
convergence rate in some appropriate dual Sobolev norm for the Wigner transform of the single-particle marginal of the
N
-particle density operator, uniform in
ħ
∈
(
0
,
1
]
provided that
V
and
(
-
Δ
)
3
/
2
+
d
/
4
V
have integrable Fourier transforms.</description><identifier>ISSN: 0010-3616</identifier><identifier>EISSN: 1432-0916</identifier><identifier>DOI: 10.1007/s00220-019-03357-z</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer Berlin Heidelberg</publisher><subject>Classical and Quantum Gravitation ; Classical mechanics ; Complex Systems ; Empirical equations ; Fourier transforms ; Mathematical and Computational Physics ; Mathematical Physics ; Particle density (concentration) ; Physics ; Physics and Astronomy ; Quantum mechanics ; Quantum Physics ; Relativity Theory ; Schrodinger equation ; Theoretical</subject><ispartof>Communications in mathematical physics, 2019-08, Vol.369 (3), p.1021-1053</ispartof><rights>Springer-Verlag GmbH Germany, part of Springer Nature 2019</rights><rights>Copyright Springer Nature B.V. 2019</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c363t-c31cd0f8c7b1e73be5f246d76d88090a085b36b351e772c2245c689a01f0a9463</citedby><cites>FETCH-LOGICAL-c363t-c31cd0f8c7b1e73be5f246d76d88090a085b36b351e772c2245c689a01f0a9463</cites><orcidid>0000-0002-7715-6682</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s00220-019-03357-z$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00220-019-03357-z$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,777,781,27905,27906,41469,42538,51300</link.rule.ids></links><search><creatorcontrib>Golse, François</creatorcontrib><creatorcontrib>Paul, Thierry</creatorcontrib><title>Empirical Measures and Quantum Mechanics: Applications to the Mean-Field Limit</title><title>Communications in mathematical physics</title><addtitle>Commun. Math. Phys</addtitle><description>In this paper, we define a quantum analogue of the notion of empirical measure in the classical mechanics of
N
-particle systems. We establish an equation governing the evolution of our quantum analogue of the
N
-particle empirical measure, and we prove that this equation contains the Hartree equation as a special case. Applications to the mean-field limit of the
N
-particle Schrödinger equation include an
O
(
1
/
N
)
convergence rate in some appropriate dual Sobolev norm for the Wigner transform of the single-particle marginal of the
N
-particle density operator, uniform in
ħ
∈
(
0
,
1
]
provided that
V
and
(
-
Δ
)
3
/
2
+
d
/
4
V
have integrable Fourier transforms.</description><subject>Classical and Quantum Gravitation</subject><subject>Classical mechanics</subject><subject>Complex Systems</subject><subject>Empirical equations</subject><subject>Fourier transforms</subject><subject>Mathematical and Computational Physics</subject><subject>Mathematical Physics</subject><subject>Particle density (concentration)</subject><subject>Physics</subject><subject>Physics and Astronomy</subject><subject>Quantum mechanics</subject><subject>Quantum Physics</subject><subject>Relativity Theory</subject><subject>Schrodinger equation</subject><subject>Theoretical</subject><issn>0010-3616</issn><issn>1432-0916</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2019</creationdate><recordtype>article</recordtype><recordid>eNp9kEFLwzAYhoMoOKd_wFPBc_RL0iattzG2KUxF0HNI09RltGlN2oP79WZW8OblC3x53veDB6FrArcEQNwFAEoBAykwMJYJfDhBM5IyiqEg_BTNAAhgxgk_Rxch7AGgoJzP0POq7a23WjXJk1Fh9CYkylXJ66jcMLZxqXfKWR3uk0XfNxEcbOdCMnTJsDPHjMNra5oq2drWDpforFZNMFe_7xy9r1dvywe8fdk8LhdbrBlnQ5xEV1DnWpTECFaarKYprwSv8hwKUJBnJeMly-KvoJrSNNM8LxSQGlSRcjZHN1Nv77vP0YRB7rvRu3hSRriAWENFpOhEad-F4E0te29b5b8kAXn0JidvMnqTP97kIYbYFAoRdh_G_1X_k_oGK4xvew</recordid><startdate>20190801</startdate><enddate>20190801</enddate><creator>Golse, François</creator><creator>Paul, Thierry</creator><general>Springer Berlin Heidelberg</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0002-7715-6682</orcidid></search><sort><creationdate>20190801</creationdate><title>Empirical Measures and Quantum Mechanics: Applications to the Mean-Field Limit</title><author>Golse, François ; Paul, Thierry</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c363t-c31cd0f8c7b1e73be5f246d76d88090a085b36b351e772c2245c689a01f0a9463</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2019</creationdate><topic>Classical and Quantum Gravitation</topic><topic>Classical mechanics</topic><topic>Complex Systems</topic><topic>Empirical equations</topic><topic>Fourier transforms</topic><topic>Mathematical and Computational Physics</topic><topic>Mathematical Physics</topic><topic>Particle density (concentration)</topic><topic>Physics</topic><topic>Physics and Astronomy</topic><topic>Quantum mechanics</topic><topic>Quantum Physics</topic><topic>Relativity Theory</topic><topic>Schrodinger equation</topic><topic>Theoretical</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Golse, François</creatorcontrib><creatorcontrib>Paul, Thierry</creatorcontrib><collection>CrossRef</collection><jtitle>Communications in mathematical physics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Golse, François</au><au>Paul, Thierry</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Empirical Measures and Quantum Mechanics: Applications to the Mean-Field Limit</atitle><jtitle>Communications in mathematical physics</jtitle><stitle>Commun. Math. Phys</stitle><date>2019-08-01</date><risdate>2019</risdate><volume>369</volume><issue>3</issue><spage>1021</spage><epage>1053</epage><pages>1021-1053</pages><issn>0010-3616</issn><eissn>1432-0916</eissn><abstract>In this paper, we define a quantum analogue of the notion of empirical measure in the classical mechanics of
N
-particle systems. We establish an equation governing the evolution of our quantum analogue of the
N
-particle empirical measure, and we prove that this equation contains the Hartree equation as a special case. Applications to the mean-field limit of the
N
-particle Schrödinger equation include an
O
(
1
/
N
)
convergence rate in some appropriate dual Sobolev norm for the Wigner transform of the single-particle marginal of the
N
-particle density operator, uniform in
ħ
∈
(
0
,
1
]
provided that
V
and
(
-
Δ
)
3
/
2
+
d
/
4
V
have integrable Fourier transforms.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/s00220-019-03357-z</doi><tpages>33</tpages><orcidid>https://orcid.org/0000-0002-7715-6682</orcidid><oa>free_for_read</oa></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0010-3616 |
| ispartof | Communications in mathematical physics, 2019-08, Vol.369 (3), p.1021-1053 |
| issn | 0010-3616 1432-0916 |
| language | eng |
| recordid | cdi_proquest_journals_2249088027 |
| source | SpringerLink Journals |
| subjects | Classical and Quantum Gravitation Classical mechanics Complex Systems Empirical equations Fourier transforms Mathematical and Computational Physics Mathematical Physics Particle density (concentration) Physics Physics and Astronomy Quantum mechanics Quantum Physics Relativity Theory Schrodinger equation Theoretical |
| title | Empirical Measures and Quantum Mechanics: Applications to the Mean-Field Limit |
| url | https://sfx.bib-bvb.de/sfx_tum?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2025-01-21T08%3A56%3A59IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-proquest_cross&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=Empirical%20Measures%20and%20Quantum%20Mechanics:%20Applications%20to%20the%20Mean-Field%20Limit&rft.jtitle=Communications%20in%20mathematical%20physics&rft.au=Golse,%20Fran%C3%A7ois&rft.date=2019-08-01&rft.volume=369&rft.issue=3&rft.spage=1021&rft.epage=1053&rft.pages=1021-1053&rft.issn=0010-3616&rft.eissn=1432-0916&rft_id=info:doi/10.1007/s00220-019-03357-z&rft_dat=%3Cproquest_cross%3E2249088027%3C/proquest_cross%3E%3Curl%3E%3C/url%3E&disable_directlink=true&sfx.directlink=off&sfx.report_link=0&rft_id=info:oai/&rft_pqid=2249088027&rft_id=info:pmid/&rfr_iscdi=true |