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e(\\rho)=\\frac{3}{5}(3\\pi^2)^{\\frac{2}{3}}\\rho^{\\frac{5}{3}}+2\\pi a\\rho^2\n  +\\frac{12}{35}(11-2\\ln2)3^{\\frac{1}{3}}\\pi^{\\frac{2}{3}}a^2\\rho^{\\frac{7}{3}}\n  +o(\\rho^{\\frac{7}{3}}).\n  \\end{equation*}\n  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 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second order Huang-Yang formula to the 3D Fermi gas: the Gross-Pitaevskii regime","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Jiahao Wu, Xuwen Chen, Zhifei Zhang","submitted_at":"2024-10-22T02:08:48Z","abstract_excerpt":"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 $e(\\rho)$ is of the form\n  \\begin{equation*}\n  e(\\rho)=\\frac{3}{5}(3\\pi^2)^{\\frac{2}{3}}\\rho^{\\frac{5}{3}}+2\\pi a\\rho^2\n  +\\frac{12}{35}(11-2\\ln2)3^{\\frac{1}{3}}\\pi^{\\frac{2}{3}}a^2\\rho^{\\frac{7}{3}}\n  +o(\\rho^{\\frac{7}{3}}).\n  \\end{equation*}\n  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 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