Unitary Thermodynamics from Thermodynamic Geometry II: Fit to a Local-Density Approximation

Abstract

Strongly interacting Fermi gasses at low density possess universal thermodynamic properties that have recently seen very precise PVT measurements by a group at MIT. This group determined local thermodynamic properties of a system of ultracold $^6\text{ Li }$ 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 the 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 “superfluid” segment for $z<z_c$ and a “normal” 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 $Y(z)= \tilde{y}_0\,\text{ exp }[2\gamma z/3]+ \tilde{y}_1\,\text{ exp }[(2\gamma /3-1)z]+ \tilde{y}_2\,\text{ exp }[(2\gamma /3-2)z]+\cdots$, 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 $\xi _B=0.368(5)$. I also discuss the virial expansion in the context of this thermodynamic geometric theory.

Appendices

Appendix 1: Solution for $Y_H(z)$

In this Appendix, I discuss the solution of the geometric equation for $Y_H(z)$ about the singular point $\mathcal {P}_0$ at $z\rightarrow \infty$, where we expect $R\rightarrow 0$, since this is the diffuse gas limit where interactions become weak. Such singular points were discussed in detail in Appendix 1 of the first paper [5], where it was argued that the appropriate form of the geometric equation and background subtraction is

$$\begin{aligned} R = -\kappa \left[ \frac{k_B T}{p}-\left( \frac{k_B T}{p}\right) _0\right] . \end{aligned}$$

(24)

Here, the parentheses $()_0$ on the right-hand side of this equation mean evaluation at $\mathcal {P}_0$.

The solution of Eq. (24) starts with a physically motivated series solution about $\mathcal {P}_0$. In the first paper [5], I used a critical phenomena style Puiseux series and obtained a good fit to DUKE1. However, it was not possible to fit MIT1 at high z with such a solution. MIT1 has tighter error bars and more high z points than DUKE1 and thus poses a greater fitting challenge. I tried another solution to Eq. (24) in the form of a perturbation about the ideal gas:

$$\begin{aligned} Y_H(z)= \tilde{y}_0\,\text{ exp }[2\gamma z/3]+ \tilde{y}_1\,\text{ exp }[(2\gamma /3-1)z]+ \tilde{y}_2\,\text{ exp }[(2\gamma /3-2)z]+\cdots , \end{aligned}$$

(25)

where $\gamma$ and the first two series coefficients $\tilde{y}_0$ and $\tilde{y}_1$ may be picked freely. The remaining series coefficients $\tilde{y}_i$ ($i\ge 2$) are determined in terms of $(\gamma ,\tilde{y}_0,\tilde{y}_1)$ from the series solution of the differential equation for $Y_H(z)$, described below. $\gamma =1$ corresponds to a limiting ($z\rightarrow \infty$) ideal gas solution (the Sackur–Tetrode equation, with an appropriate multiplier).

From the methods of the first paper [5], we may write

$$\begin{aligned} R = \frac{1}{\rho }\left[ \frac{-10 Y(z) Y''(z)^2+5 Y(z) Y'(z)Y^{(3)}(z) + 5 Y'(z)^2 Y''(z)}{4 Y'(z)^3 - 10 Y(z) Y'(z) Y''(z)}\right] . \end{aligned}$$

(26)

Define the series expansion parameter

$$\begin{aligned} x=\text{ e }^{-z}. \end{aligned}$$

(27)

Eqs. (25) and (26) yield

$$\begin{aligned} \begin{array}{lr} R=\displaystyle \left( \frac{15\tilde{y}_1}{8 \rho \gamma ^2\tilde{y}_0}\right) x \\ +\displaystyle \left( \frac{15[-15\tilde{y}_1^2+24\gamma \tilde{y}_1^2-16\gamma ^2\tilde{y}_1^2+32\gamma ^2\tilde{y}_0\tilde{y}_2]}{32\rho \gamma ^4\tilde{y}_0^2}\right) x^2 +O(x^3). \end{array} \end{aligned}$$

(28)

The definitions of T and p yield

$$\begin{aligned} \displaystyle \frac{k_B T}{p}=\frac{\gamma }{\rho }-\left( \frac{3\tilde{y}_1}{2\rho \tilde{y}_0}\right) x -\left( \frac{3[-\tilde{y}_1^2+2\tilde{y}_0\tilde{y}_2]}{2\rho \tilde{y}_0^2}\right) x^2+ O(x^3). \end{aligned}$$

(29)

The series Eqs. (28) and (29) are related by the geometric equation Eq. (24). Matching the zeroth-order terms shows that the subtracter in Eq. (24) must be

$$\begin{aligned} \left( \frac{k_B T}{p}\right) _0=\frac{\gamma }{\rho }. \end{aligned}$$

(30)

The first-order terms in x match, no matter what the values of the constants $\tilde{y}_0$ and $\tilde{y}_1$, provided that

$$\begin{aligned} \kappa =\frac{5}{4\gamma ^2}. \end{aligned}$$

(31)

Matching second-order terms yields a linear algebraic equation for $\tilde{y}_2$, yielding its value uniquely in terms of $(\gamma ,\tilde{y}_0,\tilde{y}_1)$. Matching successively higher-order series terms now yields unique values for all of the remaining series coefficients $\tilde{y}_i\,(i\ge 3)$.

Equation (24) may be written as a third-order differential equation, which may be solved for any z, using the series for $Y_H(z)$ in Eq. (25) to generate initial conditions. In practice, however, there was no need to solve the full differential equation because the series Eq. (25) converges very rapidly for all values $z>z_c\sim 0.7$. Table 2 shows the series to increasing order for a solution corresponding closely to the best overall fit to MIT1.

Table 2 Tabulation of results for the series for $Y_H(z)$, Eq. (25)

Appendix 2: Virial Expansion

In this Appendix, the geometric equation is solved in the context of the virial expansion, which is generally employed for the regime of high T and small $\rho$, a regime where approximately ideal gas behavior might be expected. I show that a virial expansion is consistent with the geometric equation, and I calculate such an expansion for a particularly good limiting ideal gas ($\gamma =1$) fit to MIT1 from Sect. 4.2. Some comparison with virial expansions calculated by other means [7, 19, 23] is given, but a detailed discussion of the broader theoretical context is beyond the scope of this paper.

The virial expansion for the free energy per volume $\phi$ is²

$$\begin{aligned} \phi \equiv \frac{p}{k_B T}=\frac{2}{\lambda ^3}\left( b_1 f + b_2 f^2 + b_3 f^3 + O(f^4)\right) , \end{aligned}$$

(32)

where the expansion parameter

$$\begin{aligned} f=\text{ exp }\left( \beta \mu \right) \end{aligned}$$

(33)

is the fugacity, $\beta =1/k_B T$, and

$$\begin{aligned} \lambda =\frac{h}{\sqrt{2\pi m}}\,\beta ^{1/2} \end{aligned}$$

(34)

is the thermal wavelength. Generally, the coefficients $b_i$, $i\ge 1$, may depend on $\beta$. Here, they are taken to be constant.

We have

$$\begin{aligned} \phi = \frac{s}{k_B} - \beta e + \beta \mu \rho . \end{aligned}$$

(35)

$\phi$ is naturally written as $\phi = \phi (\beta ,h)$, where $h=-\beta \mu$, as in Eq. (32). The energy per volume $e=-\phi _{,\beta }$, and the particle density $\rho =-\phi _{,h}$. It is easy to show that, to leading order in f, Eq. (32) yields standard ideal gas equations of state: $p=\rho k_B T$, $e=3\rho \,k_B T/2$, and $C_V/k_B N=3/2$. Clearly, these equations of state are all independent of the value of $b_1$.

In $(\beta ,h)$ coordinates, the thermodynamic metric elements take the simple form $g_{\alpha \beta } = \phi _{,\alpha \beta }$ [8, 9]. On using the series for $\phi$ in Eq. (32), and using Eq. (6) of reference [5] for R, we get

$$\begin{aligned} R=-\frac{5\beta ^{3/2}h^3}{16\sqrt{2}\,\pi ^{3/2}m^{3/2}}\left[ \frac{\,b_2}{\,b_1^2}+\frac{\,\left( -15\,b_2^2+8\,b_1 b_3\right) }{\,b_1^3}\,f+O\left( f^2\right) \right] . \end{aligned}$$

(36)

Also,

$$\begin{aligned} \displaystyle \frac{1}{\phi } = \displaystyle \frac{\beta ^{3/2}h^3}{4\sqrt{2}\,\pi ^{3/2}m^{3/2}}\left[ \frac{1}{b_1 f}-\frac{\,b_2}{\,b_1^2}+\frac{\,\left( b_2^2-b_1 b_3\right) }{\,b_1^3}\,f+O\left( f^2\right) \right] . \end{aligned}$$

(37)

Subtracting the series for $(1/\phi )_0$ computed from Eq. (30) removes the $1/b_1 f$ term in Eq. (37) and yields the singular part

$$\begin{aligned} \displaystyle \left[ \frac{1}{\phi }-\left( \frac{1}{\phi }\right) _0\right] = \displaystyle \frac{\beta ^{3/2}h^3}{4\sqrt{2}\,\pi ^{3/2}m^{3/2}}\left[ \frac{\,b_2}{\,b_1^2}+\frac{\,\left( -3b_2^2+2b_1 b_3\right) }{\,b_1^3}\,f+O\left( f^2\right) \right] . \end{aligned}$$

(38)

Matching the zeroth-order terms in the series in Eqs. (36) and (38) (with the second series multiplied by $-\kappa$, by Eq. (24)) shows that $\kappa =5/4$, consistent with Eq. (31) for $\gamma =1$, as it must be, and that $b_1$ and $b_2$ may be set freely. Matching the first-order series terms yields $b_3=2\,b_2^2/b_1$. Matching higher-order series terms yields $b_4=505\,b_2^3/96\,b_1^2$, as well as all higher-order virial coefficients.

Matching the leading term for Y(z) in Eq. (25) to the leading term for the same quantity calculated with the virial expansion Eq. (32) yields

$$\begin{aligned} b_1=\sqrt{\frac{6}{\pi e^5}} \,\tilde{y}_0^{-3/2}, \end{aligned}$$

(39)

where $e=2.71828$ is here, and in the next equation, the base of the natural logarithms. Matching the leading terms in the two R series Eqs. (28) and (36) leads to

$$\begin{aligned} b_2=-\frac{3}{2}\frac{\,b_1\tilde{y}_1}{\,e^{5/2}\tilde{y}_0}. \end{aligned}$$

(40)

Before calculating numbers, I comment on the sign of $b_2$. For other Fermi systems, R was found to be uniformly positive; see [10] for review. The best overall fit in this paper likewise has R uniformly positive; see Fig. 7. By Eq. (36) for R, we thus expect negative $b_2$. Calculation with statistical mechanics for unitary thermodynamics yields positive $b_2$ [7, 19, 23], at odds with the finding here. A reconciliation in sign could be achieved if there were a zero crossing for R somewhere in the normal phase. However, a general argument has been made [24] that such a zero crossing indicates a fundamental change in the character of the interparticle interactions, and it is not clear that this is in play here. Whether or not a zero crossing by R could possibly be consistent with the experimental data analyzed here is a question beyond the scope of this paper. Let me add, however, that all of my fits in this paper have $\tilde{y}_0$ and $\tilde{y}_1$ both positive, so Eq. (40) also points to negative $b_2$. The noninteracting Fermi gas also has negative $b_2$ [14].

Consider now the fit from Sect. 4.2 with

$$\begin{aligned} \{\alpha , \tilde{y}_0, \tilde{y}_1, z_c, \gamma \} = \{1.22, 0.137866, 0.087832, 0.841487, 1\}. \end{aligned}$$

(41)

This fit to all of MIT1 ($\chi ^2=1.67$) corresponds to a limiting ideal gas case ($\gamma =1$) and is near the best fit to MIT1. Eqs. (39) and (40) yield $\{b_1, b_2\}=\{2.21604, -0.173830\}$, and the series solution to the geometric equation in $(\beta ,h)$ coordinates yields $b_3=0.027271$. Figure 6 shows MIT1, the curve from the fit in Eq. (41), and the curve generated by this fit on using just the first three terms in its virial series.

Fig. 6

A limiting ideal gas curve ($\gamma =1$) fit to all of MIT1, and the third-order geometric virial expansion resulting from this curve, with $\{b_1, b_2,b_3\}=\{2.21604, -0.173830, 0.027271\}$. Also shown (red solid curve) is the curve from the third-order virial expansion from statistical mechanics, with $\{b_1, b_2, b_3\} = \{1, 3\sqrt{2}/8, -0.29095295\}$ (Color figure online)

Fig. 7

$\rho R$ as a function of z. R is uniformly positive and diverges to $+\infty$ as $T\rightarrow 0$. R shows a discontinuous jump at the phase transition $z_c=0.652$; $\rho R: 0.370{-}0.948$ (Color figure online)

The virial expansion coefficients have been calculated to third order with statistical mechanics [7, 19, 23]: $\{b_1, b_2, b_3\} = \{1, 3\sqrt{2}/8, -0.29095295\}$. The resulting curve is also shown in Fig. 6, and it shows slightly better agreement with the very high-temperature MIT1 data than does the curve from the thermodynamic geometric theory. The later curve is stressed at high temperature trying to fit all of MIT1. However, as the temperature is decreased, the thermodynamic geometric theory appears to yield a better converging third-order virial series than the one from statistical mechanics.

Fitting curves with $\gamma =1$ to just to the normal phase MIT1 data yields good fits ($\chi ^2\sim 1.7$) for a broad range of $b_1$ values, including the statistical mechanical value $b_1=1$. This insensitivity to $b_1$ results from the fact that the quantities plotted in Fig. 6 are independent of $b_1$ in the high-temperature limit. This fitting uncertainty for $b_1$ results in corresponding uncertainties in the values of the other virial coefficients and makes a meaningful comparison with the statistical mechanical values difficult to do. A detailed comparison is well beyond the scope of this paper.

Appendix 3: Best Fit Function

In this Appendix, I write an explicit expression for the best overall fit to the energy Y (z) per particle in units of the Fermi energy, presented in Sect. 4.1. I also write explicit expressions for a number of thermodynamic functions.

Y(z) is given in two analytic parts, $Y_S(z)$ and $Y_H(z)$, separated by a second-order phase transition at $z=z_c$, characterized by continuous Y(z) and $Y'(z)$:

$$\begin{aligned} Y(z) = \left\{ \begin{array}{llllll} Y_{S}(z) \quad \text{ for } 0<z< z_c,\\ Y_{H}(z) \quad \text{ for } z_c<z. \end{array}\right. \end{aligned}$$

(42)

$Y_S(z)$ and $Y_H(z)$ may each be written as a series:

$$\begin{aligned} Y_S(z)=y_0 + y_1 \, z^{\alpha } + y_2 \, z^{2 \alpha } + \cdots , \end{aligned}$$

(43)

and

$$\begin{aligned} Y_H(z)= \tilde{y}_0\,\text{ exp }[2\gamma z/3] + \tilde{y}_1\,\text{ exp }[(2\gamma /3-1)z] + \tilde{y}_2\,\text{ exp }[(2\gamma /3-2)z] + \cdots . \end{aligned}$$

(44)

The first seven coefficients for each series are shown in Table 3, along with $(\alpha ,\gamma ,z_c)$. Both these series have excellent convergence over the respective ranges of their functions.

Table 3 The leading series coefficients for $Y_S(z)$ and $Y_H(z)$, which correspond to $\{\alpha ,\gamma ,z_c\}=\{1.19, 1.21,0.651793\}$, characterizing the best overall fit

Below, I list explicit expressions for a number of functions encountered in this paper:

The reduced pressure:

$$\begin{aligned} \tilde{p}=\frac{p}{p_0}=\frac{5}{3}\,Y(z). \end{aligned}$$

(45)

The reduced temperature:

$$\begin{aligned} \tilde{T}=\frac{T}{T_F}=Y'(z). \end{aligned}$$

(46)

The reduced chemical potential:

$$\begin{aligned} \tilde{\mu }=\frac{\mu }{\epsilon _F}= \frac{5}{3}\,Y(z) - z Y'(z). \end{aligned}$$

(47)

The reduced inverse compressibility:

$$\begin{aligned} \tilde{\kappa }^{-1}=\frac{\kappa _0}{\kappa }=\frac{5}{3}\,Y(z) - \frac{2}{3}\frac{Y'(z)^2}{Y''(z)}. \end{aligned}$$

(48)

The entropy per particle:

$$\begin{aligned} \frac{S}{N k_B}=z. \end{aligned}$$

(49)

The heat capacity:

$$\begin{aligned} \frac{C_V}{N k_B}=\frac{Y'(z)}{Y''(z)}. \end{aligned}$$

(50)

The reduced energy:

$$\begin{aligned} \frac{E}{E_0}=\frac{5}{3}\,Y(z). \end{aligned}$$

(51)

The reduced Helmholtz free energy:

$$\begin{aligned} \frac{F}{E_0} = \frac{5}{3}\,\frac{F}{N \epsilon _{F}}= \frac{5}{3} \left[ Y(z)-z Y'(z)\right] . \end{aligned}$$

(52)

The thermodynamic curvature:

$$\begin{aligned} R=\frac{1}{\rho }\left[ \frac{-10 Y(z) Y''(z)^2+5 Y(z) Y'(z) Y^{(3)}(z) +5 Y'(z)^2 Y''(z)}{4 Y'(z)^3-10 Y(z) Y'(z) Y''(z)}\right] . \end{aligned}$$

(53)

With these expressions in terms of Y(z), it is possible to relate one property to any other. Simply tabulate both properties in pairs (parametrized by z), interpolate one property versus the other, and plot.

I conclude with a graph for the thermodynamic curvature R, shown in Fig. 7. R is seen to be uniformly positive for unitary thermodynamics over the full range of z and diverges to $+\infty$ at $T\rightarrow 0$. R for the ideal Fermi gas is likewise uniformly positive and diverges at $T\rightarrow 0$ [25, 26].