The second order Huang-Yang formula to the 3D Fermi gas: the Gross-Pitaevskii regime
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913596297969664 |
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| author | Chen, Xuwen Wu, Jiahao Zhang, Zhifei |
| author_facet | Chen, Xuwen Wu, Jiahao Zhang, Zhifei |
| contents | For a system of $N$ Fermions of spin $1/2$, with its interaction potential of scattering length $a$, the classical Huang-Yang formula states that the energy density $e(ρ)$ is of the form
\begin{equation*}
e(ρ)=\frac{3}{5}(3π^2)^{\frac{2}{3}}ρ^{\frac{5}{3}}+2πaρ^2
+\frac{12}{35}(11-2\ln2)3^{\frac{1}{3}}π^{\frac{2}{3}}a^2ρ^{\frac{7}{3}}
+o(ρ^{\frac{7}{3}}).
\end{equation*}
We consider a general system of $N$ Fermions of spin $1/\mathbf{q}$ with the scattering length of the interaction potential at the scale $a\propto N^{-1}$, that is, in the Gross-Pitaevskii regime. We prove the 2nd order ground state energy approximation corresponding to the Huang-Yang formula. The thermodynamic limit case shares a similar logic, and could be dealt with in a separate paper. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_16620 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The second order Huang-Yang formula to the 3D Fermi gas: the Gross-Pitaevskii regime Chen, Xuwen Wu, Jiahao Zhang, Zhifei Mathematical Physics For a system of $N$ Fermions of spin $1/2$, with its interaction potential of scattering length $a$, the classical Huang-Yang formula states that the energy density $e(ρ)$ is of the form \begin{equation*} e(ρ)=\frac{3}{5}(3π^2)^{\frac{2}{3}}ρ^{\frac{5}{3}}+2πaρ^2 +\frac{12}{35}(11-2\ln2)3^{\frac{1}{3}}π^{\frac{2}{3}}a^2ρ^{\frac{7}{3}} +o(ρ^{\frac{7}{3}}). \end{equation*} We consider a general system of $N$ Fermions of spin $1/\mathbf{q}$ with the scattering length of the interaction potential at the scale $a\propto N^{-1}$, that is, in the Gross-Pitaevskii regime. We prove the 2nd order ground state energy approximation corresponding to the Huang-Yang formula. The thermodynamic limit case shares a similar logic, and could be dealt with in a separate paper. |
| title | The second order Huang-Yang formula to the 3D Fermi gas: the Gross-Pitaevskii regime |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2410.16620 |