Unitary thermodynamics calculated from thermodynamic geometry

Degenerate atomic 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 particle $z\equiv S/N k_B$, and a ''normal'' one valid for large $z$. These two solutions are joined at a second-order phase transition at $z=z_c$. Define the internal energy per particle in units of the Fermi energy as $Y=Y(z)$. For small $z$, $Y(z)=y_0+y_1 z^{\alpha }+y_2 z^{2 \alpha}+\cdots,$ where $\alpha$ 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 $(\alpha, y_0, y_1)$. For large $z$ the solution follows uniquely if, in addition, we specify $z_c$, with $Y(z)$ diverging as $z^{5/3}$. The four undetermined parameters $(\alpha,y_0,y_1,z_c)$ 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 best fit of this theory to the data has $\chi^2\sim1$.