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Entropy Chaos and Bose-Einstein Condensation

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Abstract

We prove the entropy-chaos property for the system of N indistinguishable interacting diffusions rigorously associated with the ground state of N trapped Bose particles in the Gross–Pitaevskii scaling limit of infinitely many particles. On the path-space we show that the sequence of probability measures of the one-particle interacting diffusion weakly converges to a limit probability measure, uniquely associated with the minimizer of the Gross-Pitaevskii functional.

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Notes

  1. See the derivation of the solution \(\phi _0\) before Definition 3.1.

  2. In fact if \({\mathbb {Q}}_1\) and \(\mathbb {Q}_2\) are two absolutely continuous measures in \(\mathbb {R}^d\) with densities \(f_1\) and \(f_2\) respectively , then \(d_{VT}(\mathbb {Q}_1,\mathbb {Q}_2):=\sup _{A\in {\mathcal B}(\mathbb {R}^d)}|{{\mathbb {Q}}}_1(A)-{\mathbb {Q}}_2(A)|=\sup _{A\in {\mathcal B}(\mathbb {R}^d)}|\int _{A}f_1(\mathbf{r})d\mathbf{r}-\int _{A}f_2(\mathbf{r})d\mathbf{r}|=\int _{f_1>f_2} (f_1(\mathbf{r})-f_2(\mathbf{r}))d\mathbf{r}=\int _{f_2>f_1} (f_2(\mathbf{r})-f_1(\mathbf{r}))\mathbf{r}=\frac{1}{2}\int |f_1(\mathbf{r})-f_2(\mathbf{r})|d\mathbf{r}\).

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Acknowledgements

The authors have benefited from the gracious hospitality of the Centre Interfacultaire Bernoulli at the EPFL, Lausanne, during the semester: Geometric Mechanics, Variational and Stochastic Methods (1 January, 30 June 2015) organized by S. Albeverio, A.B. Cruzeiro and D. Holm. The second and third authors also thanks the NSF for financial support.

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Authors and Affiliations

  1. Department of Applied Mathematics, University of Bonn, HCM (Germany), BiBoS, IZKS, CERFIM, Locarno, Switzerland

    Sergio Albeverio

  2. Dipartimento di Matematica, Università di Milano, via Saldini 50, Milano, Italy

    Francesco C. De Vecchi & Stefania Ugolini

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  1. Sergio Albeverio
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  2. Francesco C. De Vecchi
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  3. Stefania Ugolini
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Correspondence to Stefania Ugolini.

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Albeverio, S., De Vecchi, F.C. & Ugolini, S. Entropy Chaos and Bose-Einstein Condensation. J Stat Phys 168, 483–507 (2017). https://doi.org/10.1007/s10955-017-1820-0

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