The free energy of the two-dimensional dilute Bose gas. I. Lower bound

Abstract

We prove a lower bound for the free energy (per unit volume) of the two-dimensional Bose gas in the thermodynamic limit. We show that the free energy at density $\rho$ and inverse temperature $\beta$ differs from the one of the non-interacting system by the correction term $4\pi {\rho }^2|\ln a^2\rho {|}^{-1}(2-[1-{\beta }_{\mathrm{c}}/\beta {]}_{+}^2)$. Here $a$ is the scattering length of the interaction potential, $[\cdot {]}_{+}=\max \{0,\cdot \}$ and ${\beta }_{\mathrm{c}}$ is the inverse Berezinskii--Kosterlitz--Thouless critical temperature for superfluidity. The result is valid in the dilute limit $a^2\rho \ll 1$ and if $\beta \rho \gtrsim 1$.

arXiv comment61 pages, 1 figure; final version, to appear in Forum of Mathematics, Sigma

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• The free energy of the two-dimensional dilute Bose gas. II. Upper bound
• Semiclassical approximation and critical temperature shift for weakly interacting trapped bosons
• Scaling limits of bosonic ground states, from many-body to nonlinear Schr{ö}dinger
• Upper bound for the grand canonical free energy of the Bose gas in the Gross-Pitaevskii limit for general interaction potentials
• Pressure of a dilute spin-polarized Fermi gas: Lower bound