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Derivation of the Bogoliubov Time Evolution for a Large Volume Mean-Field Limit

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Abstract

The derivation of mean-field limits for quantum systems at zero temperature has attracted many researchers in the last decades. Recent developments are the consideration of pair correlations in the effective description, which lead to a much more precise description of both spectral properties and the dynamics of the Bose gas in the weak coupling limit. While mean-field results typically lead to convergence for the reduced density matrix only, one obtains norm convergence when considering the pair correlations proposed by Bogoliubov in his seminal 1947 paper. In this article, we consider an interacting Bose gas in the case where both the volume and the density of the gas tend to infinity simultaneously. We assume that the coupling constant is such that the self-interaction of the fluctuations is of leading order, which leads to a finite (nonzero) speed of sound in the gas. In our first main result, we show that the difference between the N-body and the Bogoliubov description is small in \(L^2\) as the density of the gas tends to infinity and the volume does not grow too fast. This describes the dynamics of delocalized excitations of the order of the volume. In our second main result, we consider an interacting Bose gas near the ground state with a macroscopic localized excitation of order of the density. We prove that the microscopic dynamics of the excitation coming from the N-body Schrödinger equation converges to an effective dynamics which is free evolution with the Bogoliubov dispersion relation. The main technical novelty are estimates for all moments of the number of particles outside the condensate for large volume, and in particular control of the tails of their distribution.

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Acknowledgements

We are grateful for the hospitality of the Institute for Advanced Study and Central China Normal University (CCNU), where parts of this work were done. The comments of the referees, which greatly improved this article, are especially acknowledged. We would like to thank Phan Thành Nam for helpful comments on the article and interesting discussions that led to Eq. (48). S. P. gratefully acknowledges support from the German Academic Exchange Service (DAAD) and the National Science Foundation under Agreement No. DMS-1128155 and thanks the University of Washington for hospitality. A. S. is partially supported by NSF DMS grant 01600749 and CNSF grant 11671163.

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Authors and Affiliations

  1. Department of Mathematics and Logistics, Jacobs University, Campus Ring 1, 28759, Bremen, Germany

    Sören Petrat

  2. Mathematisches Institut, Ludwig-Maximilians-Universität, Theresienstr. 39, 80333, Munich, Germany

    Peter Pickl

  3. Duke Kunshan University, No. 8 Duke Avenue, Kunshan, 215316, Jiangsu Province, China

    Peter Pickl

  4. Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ, 08854, USA

    Avy Soffer

  5. School of Mathematics and Statistics, Central China Normal University, No. 152 Luoyu Road, Wuhan, 430079, Hubei, China

    Avy Soffer

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  1. Sören Petrat
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  2. Peter Pickl
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Correspondence to Sören Petrat.

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Communicated by Alain Joye.

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Petrat, S., Pickl, P. & Soffer, A. Derivation of the Bogoliubov Time Evolution for a Large Volume Mean-Field Limit. Ann. Henri Poincaré 21, 461–498 (2020). https://doi.org/10.1007/s00023-019-00878-0

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