The mean-field limit of quantum Bose gases at positive temperature Article Swipe
Jürg Fröhlich
,
Antti Knowles
,
Benjamin Schlein
,
Vedran Sohinger
·
YOU?
·
· 2021
· Open Access
·
· DOI: https://doi.org/10.1090/jams/987
YOU?
·
· 2021
· Open Access
·
· DOI: https://doi.org/10.1090/jams/987
We prove that the grand canonical Gibbs state of an interacting quantum Bose gas converges to the Gibbs measure of a nonlinear Schrödinger equation in the mean-field limit, where the density of the gas becomes large and the interaction strength is proportional to the inverse density. Our results hold in dimensions . For the Gibbs measure is supported on distributions of negative regularity and we have to renormalize the interaction. More precisely, we prove the convergence of the relative partition function and of the reduced density matrices in the -norm with optimal exponent . Moreover, we prove the convergence in the -norm of Wick-ordered reduced density matrices, which allows us to control correlations of Wick-ordered particle densities as well as the asymptotic distribution of the particle number. Our proof is based on a functional integral representation of the grand canonical Gibbs state, in which convergence to the mean-field limit follows formally from an infinite-dimensional stationary phase argument for ill-defined non-Gaussian measures. We make this argument rigorous by introducing a white-noise-type auxiliary field, through which the functional integral is expressed in terms of propagators of heat equations driven by time-dependent periodic random potentials and can, in turn, be expressed as a gas of interacting Brownian loops and paths. When the gas is confined by an external trapping potential, we control the decay of the reduced density matrices using excursion probabilities of Brownian bridges.
Related Topics
Concepts
Gibbs measure
Bose gas
Canonical ensemble
Grand canonical ensemble
Mathematics
Mathematical physics
Brownian motion
Thermodynamic limit
Stationary state
Hamiltonian (control theory)
Mean field theory
Quantum mechanics
Physics
Mathematical analysis
Bose–Einstein condensate
Statistics
Mathematical optimization
Monte Carlo method
Metadata
- Type
- preprint
- Language
- en
- Landing Page
- https://doi.org/10.1090/jams/987
- OA Status
- green
- Cited By
- 8
- References
- 70
- Related Works
- 20
- OpenAlex ID
- https://openalex.org/W2996871071
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https://openalex.org/W2996871071Canonical identifier for this work in OpenAlex
- DOI
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https://doi.org/10.1090/jams/987Digital Object Identifier
- Title
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The mean-field limit of quantum Bose gases at positive temperatureWork title
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preprintOpenAlex work type
- Language
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enPrimary language
- Publication year
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2021Year of publication
- Publication date
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2021-06-16Full publication date if available
- Authors
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Jürg Fröhlich, Antti Knowles, Benjamin Schlein, Vedran SohingerList of authors in order
- Landing page
-
https://doi.org/10.1090/jams/987Publisher landing page
- Open access
-
YesWhether a free full text is available
- OA status
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greenOpen access status per OpenAlex
- OA URL
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https://arxiv.org/pdf/2001.01546Direct OA link when available
- Concepts
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Gibbs measure, Bose gas, Canonical ensemble, Grand canonical ensemble, Mathematics, Mathematical physics, Brownian motion, Thermodynamic limit, Stationary state, Hamiltonian (control theory), Mean field theory, Quantum mechanics, Physics, Mathematical analysis, Bose–Einstein condensate, Statistics, Mathematical optimization, Monte Carlo methodTop concepts (fields/topics) attached by OpenAlex
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8Total citation count in OpenAlex
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2025: 1, 2023: 1, 2022: 2, 2021: 2, 2020: 1Per-year citation counts (last 5 years)
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20Other works algorithmically related by OpenAlex
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