Quantitative Derivation of the Two-Component Gross--Pitaevskii Equation in the Hard-Core Limit with Uniform-in-Time Convergence Rate
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| Format: | Preprint |
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2025
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| _version_ | 1866915314986385408 |
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| author | Chong, Jacky Lee, Jinyeop Sun, Zhiwei |
| author_facet | Chong, Jacky Lee, Jinyeop Sun, Zhiwei |
| contents | We derive the time-dependent two-component Gross--Pitaevskii (GP) equation as an effective description of the dynamics of a dilute two-component Bose gas near its ground state, which exhibits a two-component Bose-Einstein condensate, in the GP limit. Our main result establishes a uniform-in-time bound on the convergence rate between the many-body dynamics and the effective description, explicitly quantified in terms of the particle number $N$, and also implies a uniform-in-time bound for the one-component case. This improves upon the works of Michelangeli and Olgliati [77, 89] by providing a sharper, $N$-dependent, time-independent convergence rate. Our approach further extends the framework of Benedikter, de Oliveira, and Schlein [10] to the multi-component Bose gas in the hard-core limit setting. More specifically, we develop the necessary Bogoliubov theory to analyze the dynamics of multi-component Bose gases in the GP regime. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_18787 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantitative Derivation of the Two-Component Gross--Pitaevskii Equation in the Hard-Core Limit with Uniform-in-Time Convergence Rate Chong, Jacky Lee, Jinyeop Sun, Zhiwei Mathematical Physics Analysis of PDEs 82C10(Primary), 35Q55(Secondary) We derive the time-dependent two-component Gross--Pitaevskii (GP) equation as an effective description of the dynamics of a dilute two-component Bose gas near its ground state, which exhibits a two-component Bose-Einstein condensate, in the GP limit. Our main result establishes a uniform-in-time bound on the convergence rate between the many-body dynamics and the effective description, explicitly quantified in terms of the particle number $N$, and also implies a uniform-in-time bound for the one-component case. This improves upon the works of Michelangeli and Olgliati [77, 89] by providing a sharper, $N$-dependent, time-independent convergence rate. Our approach further extends the framework of Benedikter, de Oliveira, and Schlein [10] to the multi-component Bose gas in the hard-core limit setting. More specifically, we develop the necessary Bogoliubov theory to analyze the dynamics of multi-component Bose gases in the GP regime. |
| title | Quantitative Derivation of the Two-Component Gross--Pitaevskii Equation in the Hard-Core Limit with Uniform-in-Time Convergence Rate |
| topic | Mathematical Physics Analysis of PDEs 82C10(Primary), 35Q55(Secondary) |
| url | https://arxiv.org/abs/2501.18787 |