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Gibbs measures as unique KMS equilibrium states of nonlinear Hamiltonian PDEs
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Ammari, Zied and Sohinger, Vedran (2023) Gibbs measures as unique KMS equilibrium states of nonlinear Hamiltonian PDEs. Revista Matematica Iberoamericana, 39 (1). pp. 29-90. doi:10.4171/RMI/1366 ISSN 0213-2230.
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Official URL: https://doi.org/10.4171/RMI/1366
Abstract
The classical Kubo-Martin-Schwinger (KMS) condition is a fundamental property of statistical mechanics characterizing the equilibrium of infinite classical mechanical systems. It was introduced in the seventies by G. Gallavotti and E. Verboven as an alternative to the Dobrushin-Lanford-Ruelle (DLR) equation. In this article, we consider this concept in the framework of nonlinear Hamiltonian PDEs and discuss its relevance. In particular, we prove that Gibbs measures are the unique KMS equilibrium states for such systems. Our proof is based on Malliavin calculus and Gross-Sobolev spaces. The main feature of our work is the applicability of our results to the general context of white noise, abstract Wiener spaces and Gaussian probability spaces, as well as to fundamental examples of PDEs like the nonlinear Schrodinger, Hartree, and wave (Klein-Gordon) equations.
Item Type: | Journal Article | ||||||||
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Subjects: | Q Science > QA Mathematics | ||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
Library of Congress Subject Headings (LCSH): | Mathematical physics, Gaussian measures, Malliavin calculus, Differential equations, Nonlinear | ||||||||
Journal or Publication Title: | Revista Matematica Iberoamericana | ||||||||
Publisher: | European Mathematical Society Publishing House | ||||||||
ISSN: | 0213-2230 | ||||||||
Official Date: | 5 April 2023 | ||||||||
Dates: |
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Volume: | 39 | ||||||||
Number: | 1 | ||||||||
Page Range: | pp. 29-90 | ||||||||
DOI: | 10.4171/RMI/1366 | ||||||||
Status: | Peer Reviewed | ||||||||
Publication Status: | Published | ||||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||||
Copyright Holders: | © 2022 EMS Press EMS Press is an imprint of the European Mathematical Society - EMS - Publishing House GmbH, a subsidiary of the European Mathematical Society. | ||||||||
Date of first compliant deposit: | 22 July 2022 | ||||||||
Date of first compliant Open Access: | 15 September 2022 | ||||||||
RIOXX Funder/Project Grant: |
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Open Access Version: |
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