Quantum Fluctuations of Many-Body Dynamics around the Gross-Pitaevskii Equation
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866909248960593920 |
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| author | Caraci, Cristina Oldenburg, Jakob Schlein, Benjamin |
| author_facet | Caraci, Cristina Oldenburg, Jakob Schlein, Benjamin |
| contents | We consider the evolution of a gas of $N$ bosons in the three-dimensional Gross-Pitaevskii regime (in which particles are initially trapped in a volume of order one and interact through a repulsive potential with scattering length of the order $1/N$). We construct a quasi-free approximation of the many-body dynamics, whose distance to the solution of the Schrödinger equation converges to zero, as $N \to \infty$, in the $L^2 (\mathbb{R}^{3N})$-norm. To achieve this goal, we let the Bose-Einstein condensate evolve according to a time-dependent Gross-Pitaevskii equation. After factoring out the microscopic correlation structure, the evolution of the orthogonal excitations of the condensate is governed instead by a Bogoliubov dynamics, with a time-dependent generator quadratic in creation and annihilation operators. As an application, we show a central limit theorem for fluctuations of bounded observables around their expectation with respect to the Gross-Pitaevskii dynamics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_11687 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Quantum Fluctuations of Many-Body Dynamics around the Gross-Pitaevskii Equation Caraci, Cristina Oldenburg, Jakob Schlein, Benjamin Mathematical Physics Analysis of PDEs We consider the evolution of a gas of $N$ bosons in the three-dimensional Gross-Pitaevskii regime (in which particles are initially trapped in a volume of order one and interact through a repulsive potential with scattering length of the order $1/N$). We construct a quasi-free approximation of the many-body dynamics, whose distance to the solution of the Schrödinger equation converges to zero, as $N \to \infty$, in the $L^2 (\mathbb{R}^{3N})$-norm. To achieve this goal, we let the Bose-Einstein condensate evolve according to a time-dependent Gross-Pitaevskii equation. After factoring out the microscopic correlation structure, the evolution of the orthogonal excitations of the condensate is governed instead by a Bogoliubov dynamics, with a time-dependent generator quadratic in creation and annihilation operators. As an application, we show a central limit theorem for fluctuations of bounded observables around their expectation with respect to the Gross-Pitaevskii dynamics. |
| title | Quantum Fluctuations of Many-Body Dynamics around the Gross-Pitaevskii Equation |
| topic | Mathematical Physics Analysis of PDEs |
| url | https://arxiv.org/abs/2308.11687 |