The Second Order 2D Behaviors of a 3D Bose Gases in the Gross-Pitaevskii Regime
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arXiv
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| Format: | Preprint |
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2024
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| author | Chen, Xuwen Wu, Jiahao Zhang, Zhifei |
| author_facet | Chen, Xuwen Wu, Jiahao Zhang, Zhifei |
| contents | We consider a system of $N$ bosons interacting in a three-dimensional box endowed with periodic boundary condition that is strongly confined in one direction such that the normalized thickness of the box $d\ll1$. We assume particles to interact through a repulsive, radially symmetric and short-range interaction potential with scattering length scale $a\ll d$. We present a comprehensive study of such system in the Gross-Pitaevskii regime, up to the second order ground state energy, starting from proving optimal Bose-Einstein condensation results which were not previously available. The fine interplay between the parameters $N$, $a$ and $d$ generates three regions. Our result in one region on the one hand, is compatible with the classical three-dimensional Lee-Huang-Yang formula. On the other hand, it reveals a new mechanism exhibiting how the second order correction compensates and modifies the first order energy, which was previously thought of as containing a jump, and thus explains how a three-dimensional Bose gas system smoothly transits into two-dimensional system. Moreover, delving into the analysis of this new mechanism exclusive to the second order, we discover a dimensional coupling correlation effect, deeply buried away from the expected 3D and quasi-2D renormalizations, and calculate a new second order correction to the ground state energy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_15540 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Second Order 2D Behaviors of a 3D Bose Gases in the Gross-Pitaevskii Regime Chen, Xuwen Wu, Jiahao Zhang, Zhifei Mathematical Physics Analysis of PDEs We consider a system of $N$ bosons interacting in a three-dimensional box endowed with periodic boundary condition that is strongly confined in one direction such that the normalized thickness of the box $d\ll1$. We assume particles to interact through a repulsive, radially symmetric and short-range interaction potential with scattering length scale $a\ll d$. We present a comprehensive study of such system in the Gross-Pitaevskii regime, up to the second order ground state energy, starting from proving optimal Bose-Einstein condensation results which were not previously available. The fine interplay between the parameters $N$, $a$ and $d$ generates three regions. Our result in one region on the one hand, is compatible with the classical three-dimensional Lee-Huang-Yang formula. On the other hand, it reveals a new mechanism exhibiting how the second order correction compensates and modifies the first order energy, which was previously thought of as containing a jump, and thus explains how a three-dimensional Bose gas system smoothly transits into two-dimensional system. Moreover, delving into the analysis of this new mechanism exclusive to the second order, we discover a dimensional coupling correlation effect, deeply buried away from the expected 3D and quasi-2D renormalizations, and calculate a new second order correction to the ground state energy. |
| title | The Second Order 2D Behaviors of a 3D Bose Gases in the Gross-Pitaevskii Regime |
| topic | Mathematical Physics Analysis of PDEs |
| url | https://arxiv.org/abs/2401.15540 |