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The free energy of the two-dimensional dilute Bose gas. II. Upper bound

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We prove an upper bound on the free energy of a two-dimensional homogeneous Bose gas in the thermodynamic limit. We show that for $a^2 \rho \ll 1$ and $\beta \rho… Click to show full abstract

We prove an upper bound on the free energy of a two-dimensional homogeneous Bose gas in the thermodynamic limit. We show that for $a^2 \rho \ll 1$ and $\beta \rho \gtrsim 1$ the free energy per unit volume differs from the one of the non-interacting system by at most $4 \pi \rho^2 |\ln a^2 \rho|^{-1} (2 - [1 - \beta_{\mathrm{c}}/\beta]_+^2)$ to leading order, where $a$ is the scattering length of the two-body interaction potential, $\rho$ is the density, $\beta$ the inverse temperature and $\beta_{\mathrm{c}}$ is the inverse Berezinskii--Kosterlitz--Thouless critical temperature for superfluidity. In combination with the corresponding matching lower bound proved in \cite{DMS19} this shows equality in the asymptotic expansion.

Keywords: two dimensional; bound; upper bound; energy two; free energy

Journal Title: Journal of Mathematical Physics
Year Published: 2020

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