Abstract
We analyse the canonical Bogoliubov free energy functional in three dimensions at low temperatures in the dilute limit. We prove existence of a first-order phase transition and, in the limit \({\int V\to 8\pi a}\), we determine the critical temperature to be \({T_{\rm{c}}=T_{\rm{fc}}(1+1.49\rho^{1/3}a)}\) to leading order. Here, \({T_{\rm{fc}}}\) is the critical temperature of the free Bose gas, ρ is the density of the gas and a is the scattering length of the pair-interaction potential V. We also prove asymptotic expansions for the free energy. In particular, we recover the Lee–Huang–Yang formula in the limit \({\int V\to 8\pi a}\).
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Napiórkowski, M., Reuvers, R. & Solovej, J.P. The Bogoliubov Free Energy Functional II: The Dilute Limit. Commun. Math. Phys. 360, 347–403 (2018). https://doi.org/10.1007/s00220-017-3064-x
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DOI: https://doi.org/10.1007/s00220-017-3064-x