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Unitary thermodynamics from thermodynamic geometry II: Fit to a local density approximation

Strongly interacting Fermi gasses at low density possess universal thermodynamic properties which have recently seen very precise $PVT$ measurements by a group at MIT. This group determined local thermodynamic properties of a system of ultra cold $^6\mbox{Li}$ atoms tuned to Feshbach resonance. In this paper, I analyze the MIT data with a thermodynamic theory of unitary thermodynamics based on ideas from critical phenomena. This theory was introduced in the first paper of this sequence, and characterizes the scaled thermodynamics by the entropy per particle $z= S/N k_B$, and energy per particle $Y(z)$, in units of the Fermi energy. $Y(z)$ is in two segments, separated by a second-order phase transition at $z=z_c$: a "normal" segment for $z>z_c$, and a "superfluid" segment for $z<z_c$. For small $z$, the theory obeys a series $Y(z)=y_0+y_1 z^{α}+y_2 z^{2 α}+\cdots,$ where $α$ is a constant exponent, and $y_i$ ($i\ge 0$) are constant series coefficients. For large $z$, the theory obeys a perturbation of the ideal gas $Y(z)= \tilde{y}_0\,\mbox{exp}[2γz/3]+ \tilde{y}_1\,\mbox{exp}[(2γ/3-1)z]+ \tilde{y}_2\,\mbox{exp}[(2γ/3-2)z]+\cdots$ where $γ$ is a constant exponent, and $\tilde{y}_i$ ($i\ge 0$) are constant series coefficients. This limiting form for large $z$ differs from the series used in the first paper, and was necessary to fit the MIT data. I fit the MIT data by adjusting four free independent theory parameters: $(α,γ,\tilde{y}_0,\tilde{y}_1)$. This fit process was augmented by trap integration and comparison with earlier thermal data taken at Duke University. The overall match to both the data sets was good, and had $α=1.21(3)$, $γ=1.21(3)$, $z_c=0.69(2)$, scaled critical temperature $T_c/T_F=0.161(3)$, where $T_F$ is the Fermi temperature, and Bertsch parameter $ξ_B=0.368(5)$.