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Unitary thermodynamics from thermodynamic geometry

Degenerate Fermi gases of atoms near a Feshbach resonance show universal thermodynamic properties, which are here calculated with the geometry of thermodynamics, and the thermodynamic curvature $R$. Unitary thermodynamics is expressed as the solution to a pair of ordinary differential equations, a "superfluid" one valid for small entropy per atom $z\equiv S/N k_B$, and a "normal" one valid for high $z$. These two solutions are joined at a second-order phase transition at $z=z_c$. Define the internal energy per atom in units of the Fermi energy as $Y=Y(z)$. For small $z$, $Y(z)=y_0+y_1 z^α+y_2 z^{2 α}+\cdots,$ where $α$ is a constant exponent, $y_0$ and $y_1$ are scaling factors, and the series coefficients $y_i$ ($i\ge 2$) are determined uniquely in terms of $(α, y_0, y_1)$. For large $z$ the solution follows if we also specify $z_c$, with $Y(z)$ diverging as $z^{5/3}$ for high $z$. The four undetermined parameters $(α,y_0,y_1,z_c)$ were determined by fitting the theory to experimental data taken by a Duke University group on $^6$Li in an optical trap with a Gaussian potential. The very best fit of this theory to the data had $α=2.1$, $z_c=4.7$, $y_0=0.277$, and $y_1=0.0735$, with $χ^2=0.95$. The corresponding Bertsch parameter is $ξ_B=0.462(40)$.