{"abstract":[{"text":"We prove that nonlinear Gibbs measures can be obtained from the corresponding many-body, grand-canonical, quantum Gibbs states, in a mean-field limit where the temperature T diverges and the interaction strength behaves as 1/T. We proceed by characterizing the interacting Gibbs state as minimizing a functional counting the free-energy relatively to the non-interacting case. We then perform an infinite-dimensional analogue of phase-space semiclassical analysis, using fine properties of the quantum relative entropy, the link between quantum de Finetti measures and upper/lower symbols in a coherent state basis, as well as Berezin-Lieb type inequalities. Our results cover the measure built on the defocusing nonlinear Schrödinger functional on a finite interval, as well as smoother interactions in dimensions d 2.","lang":"eng"}],"department":[{"_id":"RoSe"}],"publication_status":"published","date_published":"2015-01-01T00:00:00Z","date_updated":"2021-01-12T08:00:52Z","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by-nd/4.0/legalcode","name":"Creative Commons Attribution-NoDerivatives 4.0 International (CC BY-ND 4.0)","image":"/image/cc_by_nd.png","short":"CC BY-ND (4.0)"},"month":"01","day":"01","license":"https://creativecommons.org/licenses/by-nd/4.0/","type":"journal_article","file":[{"access_level":"open_access","creator":"system","file_name":"IST-2018-951-v1+1_2015_Thanh-Nam_Derivation_of.pdf","content_type":"application/pdf","checksum":"a40eb4016717ddc9927154798a4c164a","relation":"main_file","date_created":"2018-12-12T10:12:53Z","file_size":1084254,"file_id":"4974","date_updated":"2020-07-14T12:46:35Z"}],"status":"public","quality_controlled":"1","title":"Derivation of nonlinear gibbs measures from many-body quantum mechanics","scopus_import":1,"publisher":"Ecole Polytechnique","date_created":"2018-12-11T11:46:40Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa_version":"Published Version","oa":1,"_id":"473","language":[{"iso":"eng"}],"pubrep_id":"951","citation":{"mla":"Lewin, Mathieu, et al. “Derivation of Nonlinear Gibbs Measures from Many-Body Quantum Mechanics.” <i>Journal de l’Ecole Polytechnique - Mathematiques</i>, vol. 2, Ecole Polytechnique, 2015, pp. 65–115, doi:<a href=\"https://doi.org/10.5802/jep.18\">10.5802/jep.18</a>.","ieee":"M. Lewin, P. Nam, and N. Rougerie, “Derivation of nonlinear gibbs measures from many-body quantum mechanics,” <i>Journal de l’Ecole Polytechnique - Mathematiques</i>, vol. 2. Ecole Polytechnique, pp. 65–115, 2015.","ama":"Lewin M, Nam P, Rougerie N. Derivation of nonlinear gibbs measures from many-body quantum mechanics. <i>Journal de l’Ecole Polytechnique - Mathematiques</i>. 2015;2:65-115. doi:<a href=\"https://doi.org/10.5802/jep.18\">10.5802/jep.18</a>","chicago":"Lewin, Mathieu, Phan Nam, and Nicolas Rougerie. “Derivation of Nonlinear Gibbs Measures from Many-Body Quantum Mechanics.” <i>Journal de l’Ecole Polytechnique - Mathematiques</i>. Ecole Polytechnique, 2015. <a href=\"https://doi.org/10.5802/jep.18\">https://doi.org/10.5802/jep.18</a>.","ista":"Lewin M, Nam P, Rougerie N. 2015. Derivation of nonlinear gibbs measures from many-body quantum mechanics. Journal de l’Ecole Polytechnique - Mathematiques. 2, 65–115.","apa":"Lewin, M., Nam, P., &#38; Rougerie, N. (2015). Derivation of nonlinear gibbs measures from many-body quantum mechanics. <i>Journal de l’Ecole Polytechnique - Mathematiques</i>. Ecole Polytechnique. <a href=\"https://doi.org/10.5802/jep.18\">https://doi.org/10.5802/jep.18</a>","short":"M. Lewin, P. Nam, N. Rougerie, Journal de l’Ecole Polytechnique - Mathematiques 2 (2015) 65–115."},"ec_funded":1,"publication":"Journal de l'Ecole Polytechnique - Mathematiques","author":[{"first_name":"Mathieu","last_name":"Lewin","full_name":"Lewin, Mathieu"},{"id":"404092F4-F248-11E8-B48F-1D18A9856A87","full_name":"Phan Thanh, Nam","first_name":"Nam","last_name":"Phan Thanh"},{"first_name":"Nicolas","last_name":"Rougerie","full_name":"Rougerie, Nicolas"}],"has_accepted_license":"1","page":"65 - 115","doi":"10.5802/jep.18","project":[{"call_identifier":"FP7","grant_number":"291734","_id":"25681D80-B435-11E9-9278-68D0E5697425","name":"International IST Postdoc Fellowship Programme"}],"ddc":["539"],"file_date_updated":"2020-07-14T12:46:35Z","year":"2015","publist_id":"7344","volume":2,"intvolume":"         2"}