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

TL;DR

This paper applies a thermodynamic geometric theory to fit precise experimental data on ultra-cold Fermi gases at unitarity, revealing critical behavior and providing accurate thermodynamic parameters.

Contribution

It introduces a refined large-z asymptotic form for the thermodynamic function and successfully fits MIT data with this model, improving understanding of unitary Fermi gases.

Findings

Good fit to MIT and Duke data sets

Determined critical parameters and exponents

Confirmed universal thermodynamic behavior

Abstract

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 $Y(z)=y_0+y_1 z^{\alpha }+y_2 z^{2 \alpha}+\cdots,$ where $\alpha$ 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 $Y(z)= \tilde{y}_0\,\mbox{exp}[2\gamma z/3]+ \tilde{y}_1\,\mbox{exp}[(2\gamma/3-1)z]+ \tilde{y}_2\,\mbox{exp}[(2\gamma/3-2)z]+\cdots$ where $\gamma$ 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: $(\alpha,\gamma,\tilde{y}_0,\tilde{y}_1)$. 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 $\alpha=1.21(3)$, $\gamma=1.21(3)$, $z_c=0.69(2)$, scaled critical temperature $T_c/T_F=0.161(3)$, where $T_F$ is the Fermi temperature, and Bertsch parameter $\xi_B=0.368(5)$.