# Universal Hydrodynamic Transport Times
in the Normal Phase of a Unitary Fermi Gas

Xiang Li, J. Huang, and J. E. Thomas  1Department of Physics, North Carolina
State University, Raleigh, NC 27695, USA

###### Abstract

We determine the hydrodynamic relaxation times $\tau_{\eta}$ for the shear
viscosity and $\tau_{\kappa}$ for the thermal conductivity in the normal phase
of a unitary Fermi gas confined in a box potential. Using a kinetic theory
relaxation model, we extract $\tau_{\eta}$ and $\tau_{\kappa}$ from the time-
dependent free-decay of a spatially periodic density perturbation, yielding
the universal density-shift coefficients for the shear viscosity and thermal
conductivity.

Measurements of the universal hydrodynamic transport properties of a unitary
Fermi gas connect ultracold atoms to nuclear matter [1, 2, 3] and provide new
challenges to theoretical predictions [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
15]. A unitary Fermi gas is a strongly interacting, scale-invariant, quantum
many-body system, created by tuning a trapped, two-component cloud near a
collisional (Feshbach) resonance [16]. At resonance, the thermodynamic and
transport properties are universal functions of the density and temperature
[17], permitting parameter-free comparisons with predictions. Early
measurements on expanding Fermi gas clouds with nonuniform density [18, 19]
have made way for new measurements in optical box potentials [20], where the
density is nearly uniform [21, 22, 23, 24, 25, 26]. For a unitary Fermi gas,
the second bulk viscosity vanishes, as predicted for scale-invariant systems
[27, 28] and demonstrated in experiments on conformal symmetry [29]. Hence, at
temperatures above the superfluid transition [30], the normal phase of a
unitary gas affords the simplest universal system for hydrodynamic transport
measurements, as the static transport properties comprise only the shear
viscosity $\eta$ and the thermal conductivity $\kappa_{T}$.

Remarkably, the measured shear viscosity and thermal conductivity in the
normal phase are well fit by the simple expressions [23, 31],

$\eta=\frac{15}{32\sqrt{\pi}}\frac{(mk_{B}T)^{3/2}}{\hbar^{2}}+\alpha_{2\eta}\,\hbar
n_{0}$ (1)

and

$\kappa_{T}=\frac{15}{4}\frac{k_{B}}{m}\,\eta(\alpha_{2\eta}\rightarrow\alpha_{2\kappa})$
(2)

with $k_{B}$ the Boltzmann constant and $m$ the atom mass. The density shift
coefficients $\alpha_{2\eta}$ and $\alpha_{2\kappa}$ are fit parameters. Here,
the temperature $T$ and density $n_{0}$ contributions can be understood by
dimensional analysis. For the shear viscosity, with a dimension of
momentum/area, we expect $\eta\propto\hbar/L^{3}$, with $L$ a length scale. At
high temperature, $L\rightarrow\lambda_{T}$, the thermal de Broglie wavelength
$\propto T^{-1/2}$. At lower temperature, where the cloud is degenerate,
$1/L^{3}=n_{0}$. For both $\eta$ and $\kappa_{T}$, the leading $T^{3/2}$
dependence is obtained by variational calculations for a unitary gas in the
two-body Boltzmann limit [4, 12, 6]. Density shifts can be shown to arise from
Pauli blocking in an ideal Fermi gas, but are reduced by in medium effects in
a unitary Fermi gas [13]. In contrast to the $T^{3/2}$ coefficients, the
density shift coefficients $\alpha_{2\eta}$ and $\alpha_{2\kappa}$ are unknown
universal constants in the early stages of measurement, with measured values
that are sensitive to the finite hydrodynamic relaxation times.

Figure 1: A unitary Fermi gas is loaded into a repulsive box potential created
by two digital micromirror devices DMDs (top,right). A third DMD (bottom)
generates a static spatially periodic perturbation $\delta U$ with an
adjustable wavelength, creating spatially periodic 1D density profiles (left).
After $\delta U$ is abruptly extinguished, the dominant Fourier component
exhibits an oscillatory decay (see Fig. 2).

In this work, we employ a hydrodynamic relaxation model based on kinetic
theory to extract of the universal hydrodynamic relaxation times $\tau_{\eta}$
and $\tau_{\kappa}$ from the measured time-dependent free-decay of a spatially
periodic density perturbation in a unitary Fermi gas. The extracted relaxation
times determine the static shear viscosity and thermal conductivity, yielding
the two universal density shift parameters $\alpha_{2\eta}$,
$\alpha_{2\kappa}$, corrected for the finite response time over which the
viscous force and heat current relax to their Navier-Stokes forms.

The experiments employ ultracold 6Li atoms in a balanced mixture of the two
lowest hyperfine states, which are evaporatively cooled in a CO2 laser trap
and loaded into a box potential, Fig. 1, producing a sample of nearly uniform
density $n_{0}$. The box comprises six sheets of blue-detuned light, created
by two digital micromirror devices (DMDs) [23, 21] (top and right). The box
potential $U_{0}(\mathbf{r})$ yields a rectangular density profile with
typical dimensions $(x,y,z)=(52\times 50\times 150)\,\mu$m. The density varies
slowly in the direction of the long ($z$) axis, due to the harmonic confining
potential $\propto z^{2}$ arising from the curvature of the bias magnetic
field, which has little effect on the shorter $x$ and $y$ axes. The typical
total central density is $n_{0}=3.3\times 10^{11}$ atoms/cm3, with the Fermi
energy $\epsilon_{\\!F}\equiv k_{B}T_{F}=k_{B}\times 0.18\,\mu$K and Fermi
speed $v_{F}\simeq 2.25$ cm/s. The box depth $U_{0}\simeq 0.75\,\mu$K (see
Ref. [23]).

We add a third DMD Fig. 1 (bottom) to independently project a static optical
potential $\delta U$ that spatially varies with an adjustable wavelength
$\lambda$ along one axis, creating a sinusoidal density perturbation. Once
equilibrium is established, the perturbing potential is abruptly extinguished,
causing an oscillatory decay of the measured density perturbation $\delta
n(z,t)=n(z,t)-n_{0}(z)$. By performing a fast Fourier transform (FFT) of
$\delta n(z,t)$ at each time, in a region containing an integer number
(typically 3-4) of spatial periods, we obtain $\delta n(q,t)$, Fig. 2.

Figure 2: Fourier component of the density perturbation $\delta n(q,t)$ with
$q=2\pi/\lambda$, for wavelengths $\lambda=40.0\,\mu$m, $31.3\,\mu$m, and
$22.7\,\mu$m, at reduced temperatures $T/T_{F}=0.42$, $0.36$, and $0.32$,
respectively. Dots (data); Curves: hydrodynamic relaxation time model. The
error bars are the standard deviation of the mean of $\delta n(q,t)$ for 5-8
runs, taken in random time order.

As shown previously [23], $\delta n(q,t)$ contains a thermally diffusive mode
($\simeq 30$%) that decays at a rate $\propto\kappa_{T}$ and an oscillating
first sound mode, which decays at a rate dependent on both $\eta$ and
$\kappa_{T}$. The relaxation model determines the response of the viscous
force and heat current at the first sound frequency in the sound mode, as well
as the response of the heat current at the decay rate in the thermally
diffusive mode.

We derive the relaxation model in the linear response regime by constructing
four coupled equations: two describe the changes in the density $\delta
n(z,t)$ and temperature $\delta T(z,t)$, and two describe the relaxation of
the viscous force and heat current. We ignore the box potential, since we
measure the free-decay over time scales that avoid perturbing $\delta n(z,t)$
in the measured central region by reflections from the boundaries. After the
perturbing potential is extinguished, the density change obeys [31],

$\delta\ddot{n}=c_{T}^{2}\,\partial_{z}^{2}(\delta n+\delta\tilde{T})+\delta
Q_{\eta}\,.$ (3)

The first term in Eq. 3 arises from the pressure change with $c_{T}$ the
isothermal sound speed, and

$\delta\dot{Q}_{\eta}+\frac{1}{\tau_{\eta}}\,\delta
Q_{\eta}=\frac{4}{3}\,\frac{p}{mn_{0}}\,\partial_{z}^{2}\delta\dot{n}$ (4)

describes the relaxation of the viscous damping force [31]. Here the pressure
is $p=\frac{2}{5}n\epsilon_{F}\\!f_{E}(\theta)$, where the universal function
$f_{E}(\theta)$ has been measured [30], $n_{0}$ is the background density, and
we have used the continuity equation to eliminate the velocity field. For fast
relaxation, Eq. 4 yields the usual Navier-Stokes form for $\delta Q_{\eta}$ in
Eq. 3 with $\eta=\tau_{\eta}p$ the static shear viscosity, independent of the
single particle phase space distribution [31].

In Eq. 3, we have defined a scaled temperature,
$\delta\tilde{T}=n_{0}\beta\,\delta T$ with a dimension of density, where
$\beta=-1/n(\partial n/\partial T)_{P}$ is the thermal expansivity [31]. We
find

$\delta\dot{\tilde{T}}=\epsilon_{LP}\,\delta\dot{n}+\delta Q_{\kappa}\,.$ (5)

The first term describes the adiabatic change in the temperature with
$\epsilon_{LP}\equiv c_{P_{1}}/c_{V_{1}}-1$ the Landau-Placzek parameter. The
heat capacities per particle at constant volume $c_{V_{1}}$ and at constant
pressure $c_{P_{1}}$ are determined by the measured equation of state
$f_{E}(\theta)$ [30, 31] and

$\delta\dot{Q}_{\kappa}+\frac{1}{\tau_{\kappa}}\,\delta
Q_{\kappa}=\frac{5}{2}\frac{k_{B}}{m}\frac{p}{n_{0}c_{V_{1}}}\,\partial_{z}^{2}\delta\tilde{T}$
(6)

describes the relaxation of the heat current. For fast relaxation, Eq. 6
yields the usual heating rate $\delta Q_{\kappa}$ in Eq. 5 with
$\kappa_{T}=\frac{5}{2}\frac{k_{B}}{m}\tau_{\kappa}p$ the static thermal
conductivity. Here, the factor $5/2$ is dependent on a Maxwell-Boltzmann
approximation for the single particle phase space distribution [31].

A spatial Fourier transform of Eqs. 3-6 yields coupled linear equations for
$\delta\ddot{n}(q,t)$, $\delta\dot{\tilde{T}}(q,t)$,
$\delta\dot{Q}_{\eta}(q,t)$, and $\delta\dot{Q}_{\kappa}(q,t)$ with
$q=2\pi/\lambda$. As the system is initially in mechanical equilibrium and
isothermal, only $\delta n(q,0)\equiv A\neq 0$. Fig. 2 shows fits of the
relaxation model (solid curves) to typical data (scaled by $1/A$), where the
fit parameters are $c_{T}q,\tau_{\eta},\tau_{\kappa}$, and the amplitude $A$.
As in our previous work [23], the wavelength of the perturbation and the fit
frequency $c_{T}q$ self-consistently determine the sound speed $c_{T}$ and the
corresponding reduced temperature $T/T_{F}=\theta(c_{T}/v_{F})$ from
$f_{E}(\theta)$, with the Fermi speed $v_{F}$ given for the average central
density $n_{0}$ [31].

Figure 3: Hydrodynamic relaxation times in units of the Fermi time
$\tau_{F}\equiv\lambda_{F}/v_{F}=\pi\hbar/\epsilon_{F}$ versus reduced
temperature $T/T_{F}$.
a) $\tau_{\eta}$ for the shear viscosity (blue circles). b) $\tau_{\kappa}$
for the thermal conductivity (blue squares). Red curves include the density
shift coefficients $\alpha_{2\eta}=0.45$ and $\alpha_{2\kappa}=0.22$. Red
dashed curves: High temperature limits, where $\alpha_{2\eta}=0$ and
$\alpha_{2\kappa}=0$ and $\tau_{\kappa}/\tau_{\eta}=3/2$. (c,d) Wavelength
dependence for $\theta\simeq 0.30$. Error bars are statistical [32]. (color
online)

Our extracted relaxation times $\tau_{\eta}$ for the shear viscosity and
$\tau_{\kappa}$ for the thermal conductivity are shown as functions of
$\theta=T/T_{F}$ in Fig. 3. The relaxation times are given in units of the
Fermi time $\tau_{F}\equiv\lambda_{F}/v_{F}=\pi\hbar/\epsilon_{F}\simeq
120\,\mu$s. The wavelength dependence of $\tau_{\eta}$ and $\tau_{\kappa}$ is
shown for $\theta\simeq 0.30$, demonstrating negligible $\lambda$-dependence
as expected. As discussed below, the fitted density shift coefficients
$\alpha_{\eta}>\alpha_{\kappa}$ make $\tau_{\kappa}/\tau_{\eta}\simeq 1.2$ for
$\theta\simeq 0.4$, deviating significantly from the high temperature limit
(dashed red curves), where $\tau_{\kappa}/\tau_{\eta}=3/2$ [13, 31].

The measured $\tau_{\eta}$ determines the static shear viscosity
$\eta=\tau_{\eta}\,p$ shown in Fig. 4 a (Blue circles). Eq. 1, in units of
$\hbar n_{0}$, gives

$\eta(\theta)=\alpha_{3/2}\,\theta^{3/2}+\alpha_{2\eta}\,,$ (7)

where $\alpha_{3/2}=\frac{45\pi^{3/2}}{64\sqrt{2}}\simeq 2.77$ [4, 12, 6]. The
red curve in Fig. 4 a shows the fit for $\eta(\theta)$ with the density shift
coefficient $\alpha_{2\eta}$ as the only fit parameter, yielding
$\alpha_{2\eta}=0.45(04)$. Fig. 3 a shows the corresponding curve for
$\tau_{\eta}=\eta/p$, with $p=\frac{2}{5}n\epsilon_{F}\\!f_{E}(\theta)$. The
red-dashed curve in Fig. 4 a is the high temperature limit,
$\alpha_{2\eta}=0$.

Figure 4: Transport properties obtained from the measured transport times
$\tau_{\eta}$ and $\tau_{\kappa}$ versus reduced temperature $\theta=T/T_{F}$.
a) Shear viscosity (Blue circles). b) Thermal conductivity (Blue squares). c)
First sound diffusivity (Blue Triangles). Red solid curves include the density
shift coefficients $\alpha_{2\eta}=0.45$ in Eq. 7 and $\alpha_{2\kappa}=0.22$
in Eq. 8. Red-dashed curves are the high temperature limits, where
$\alpha_{2\eta}=0$ and $\alpha_{2\kappa}=0$. Error bars are statistical.
(color online)

Similarly, the measured $\tau_{\kappa}$ determines the static thermal
conductivity $\kappa_{T}=\frac{5}{2}\frac{k_{B}}{m}\tau_{\kappa}\,p$ shown in
Fig. 4 b (Blue squares). In the high temperature two-body Boltzmann limit, one
can show that $\tau_{\kappa}/\tau_{\eta}=3/2$ for any isotropic collision
cross section $d\sigma/d\Omega$, so that
$\kappa_{T}=\frac{15}{4}\frac{k_{B}}{m}\,\eta$. Eq. 2, in units of $n_{0}\hbar
k_{B}/m$, gives

$\kappa_{T}(\theta)=\frac{15}{4}\,(2.77\,\theta^{3/2}+\alpha_{2\kappa})\,.$
(8)

The red curve in Fig. 4 b shows the fit of $\kappa_{T}(\theta)$ with the
density shift coefficient $\alpha_{2\kappa}$ as the only fit parameter,
yielding $\alpha_{2\kappa}=0.22(03)$, i.e., the shift is $15/4\times 0.22$ in
units of $n_{0}\hbar k_{B}/m$. Fig. 3 b shows the corresponding curve for
$\tau_{\kappa}=\frac{2}{5}\frac{m}{k_{B}}\kappa_{T}/p$. The red-dashed curve
in Fig. 4 b is the high temperature limit, $\alpha_{2\kappa}=0$.

Finally, Fig. 4 c shows the corresponding first sound diffusivity [33, 31]
$D_{1}$ in units of $\hbar/m$. The measured transport times determine
$D_{1}=\frac{8}{15}\,\tau_{\eta}\frac{\epsilon_{F}}{\hbar}\,f_{E}(\theta)+\frac{2}{3}\tau_{\kappa}\frac{\epsilon_{F}}{\hbar}\,\theta$
(Blue triangles). The red-solid curve gives $D_{1}$ in terms of the fits for
the static shear viscosity and thermal conductivity
$D_{1}=4/3\,(2.77\,\theta^{3/2}\\!\\!+0.45)+(n\,k_{B}T/p)\,(2.77\,\theta^{3/2}\\!\\!+0.22)$.
Here, the thermal conductivity contribution includes a factor $(n\,k_{B}T/p)$
for the unitary gas, where $p$ is the pressure [31]. The red-dashed curve is
the high temperature limit $D_{1}=7/3\times 2.77\,\theta^{3/2}$.

In conclusion, we have used a kinetic theory model to extract the transport
relaxation times for the shear viscosity and thermal conductivity of a normal-
phase unitary Fermi gas in a box potential. The relaxation times of the
thermal current and viscous force are found to be comparable to the Fermi time
$\lambda_{F}/v_{F}$, small, but not negligible, compared to the time scales
for the oscillatory decay of the density perturbation. The measured transport
times enable new estimates of the universal density shift corrections for both
the static shear viscosity, $\Delta\eta=n_{0}\hbar\,\alpha_{2\eta}$ with
$\alpha_{2\eta}=0.45(04)$ and the static thermal conductivity
$\Delta\kappa_{T}=n_{0}\hbar\frac{k_{B}}{m}\frac{15}{4}\,\alpha_{2\kappa}$,
where $\alpha_{2\kappa}=0.22(03)$. For the measured range of $\theta$, our
$\tau_{\eta}$ and $\eta$ are in reasonable agreement with the predictions of
Enss et al., see Figs. 6 (where $\tau_{F}\equiv\hbar/\epsilon_{F}$) and 7 of
Ref. [8]. However, the measured values of $\kappa_{T}$ are significantly
smaller than predicted in [13] and larger than predicted in [15]. It can be
shown that density shifts of $\eta$ and $\kappa_{T}$, i.e., $\alpha_{2\eta}$
and $\alpha_{2\kappa}$, arise from Pauli blocking in an ideal Fermi gas.
However, the results for the ideal gas are too large and are mitigated by
strong interactions in a unitary Fermi gas [13]. Our measurements emphasize
the need for improved calculations of the leading density-dependent
corrections to the hydrodynamic transport properties as well as more
sophisticated relaxation models.

We thank Thomas Schäfer for stimulating discussions and Ilya Arakelyan for
help with the new optical system. Primary support for this research is
provided by the National Science Foundation (PHY-2006234 and PHY-2307107).
Additional support is provided by the Air Force Office of Scientific Research
(FA9550-22-1-0329).

∗Corresponding author<EMAIL_ADDRESS>
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##  Appendix A Supplemental Material

In this supplemental material, we derive a kinetic theory relaxation model,
which is used to extract the hydrodynamic transport times $\tau_{\eta}$ for
the shear viscosity and $\tau_{\kappa}$ for the thermal conductivity from the
measured free oscillatory decay of a spatially periodic density perturbation
in a normal fluid unitary Fermi gas.

###  A.1 Hydrodynamic linear response for a normal fluid.

We consider a normal phase unitary Fermi gas, which is a single component
fluid with a mass density $\rho\equiv n\,m$, where $n$ is the total particle
density (we assume a 50-50 mixture of two components) and $m$ is the atom
mass. $\rho({\mathbf{r}},t)$ satisfies the continuity equation,

$\partial_{t}\rho+\partial_{i}(\rho\,v_{i})=0,$ (S1)

where a sum over $i=x,y,z$ is implied. The mass flux (momentum density) is
$\rho\,v_{i}$, with $v_{i}({\mathbf{r}},t)$ the velocity field.

Our experiments measure the response of the density in the central region of
the box over short enough time scales that forces arising from the walls of
the box potential can be neglected. As the perturbing potential $\delta U=0$
during the measured evolution, the momentum density and corresponding momentum
flux $\rho\,v_{i}v_{j}$ obey

$\partial_{t}(\rho\,v_{i})+\partial_{j}(\rho\,v_{i}v_{j})=-\partial_{i}p-\partial_{j}p^{1}_{ij},$
(S2)

where $-\partial_{i}p$ is the force per unit volume arising from the scalar
pressure $p$ and $\partial_{j}p^{1}_{ij}$ is the viscous force per unit
volume, which we determine using a kinetic theory relaxation model in S A.2.
Taking the divergence of eq. S2, and using eq. S1, we immediately obtain

$-\partial_{t}^{2}\rho+\partial_{i}\partial_{j}(\rho\,v_{i}v_{j})=-\partial_{i}^{2}p-\partial_{i}\partial_{j}p^{1}_{ij}.$
(S3)

In the linear response regime, the second term on the left hand side is second
order in small quantities and can be dropped. Specializing to one dimension,
and taking $\delta n=n-n_{0}$ with $n_{0}$ is the background density, the
change in density $\delta n(z,t)$ obeys

$\partial_{t}^{2}\delta n=\frac{1}{m}\partial_{z}^{2}\,\delta
p\,+\frac{1}{m}\,\partial_{z}^{2}p^{1}_{zz}\,.$ (S4)

As our initial conditions are isothermal, we find the pressure change $\delta
p=p-p_{0}$ in terms of the changes in the density $\delta n$ and temperature
$\delta T$. In this case, the pressure change can be written in the form [23]
$\delta p=mc_{T}^{2}(\delta n+\delta\tilde{T})$, so that

$\displaystyle\delta\ddot{n}=c_{T}^{2}\,\partial_{z}^{2}(\delta
n+\delta\tilde{T})+\frac{1}{m}\,\partial_{z}^{2}p^{1}_{zz}\,.$ (S5)

where $\delta\tilde{T}\equiv n_{0}\,\beta\,\delta{T}$ with
$\beta=-1/n\,(\partial n/\partial T)_{p}$ the expansivity.

Next, we require the evolution equation for $\delta T$, which obeys [23],

$\delta\dot{T}=\epsilon_{LP}\,\frac{\delta\dot{n}}{\beta\,n_{0}}+\frac{\delta\dot{q}}{n_{0}c_{V_{1}}},$
(S6)

where $\epsilon_{LP}\equiv c_{P_{1}}/c_{V_{1}}-1$ is the Landau-Placzek
parameter, which describes the adiabatic change in the temperature arising
from the change in density. Here, $c_{V_{1}}$, $c_{P_{1}}$ are the heat
capacities per particle. The heat flow per unit volume
$\delta\dot{q}=-\partial_{z}J_{E}$, where $J_{E}$ is the energy current, which
we determine using a kinetic theory relaxation model in S A.2. Multiplying eq.
S6 by $n_{0}\,\beta$, we obtain

$\delta\dot{\tilde{T}}=\epsilon_{LP}\,\delta\dot{n}-\frac{\beta}{c_{V_{1}}}\,\partial_{z}J_{E}.$
(S7)

###  A.2 Kinetic theory relaxation model

In this section, we derive the relaxation model equations for a normal phase
unitary Fermi gas, which determine how the viscous force and heat current
relax to their Navier-Stokes forms. To proceed, we rewrite Eq. S5 as

$\displaystyle\delta\ddot{n}=c_{T}^{2}\,\partial_{z}^{2}(\delta
n+\delta\tilde{T})+\delta Q_{\eta}\,,$ (S8)

where

$\delta Q_{\eta}\equiv\frac{1}{m}\,\partial_{z}^{2}p^{1}_{zz}.$ (S9)

Similarly, Eq. S7 is rewritten as

$\delta\dot{\tilde{T}}=\epsilon_{LP}\,\delta\dot{n}+\delta Q_{\kappa}\,,$
(S10)

with

$\delta Q_{\kappa}\equiv-\frac{\beta}{c_{V_{1}}}\,\partial_{z}J_{E}\,.$ (S11)

We derive the evolution equations for $\delta Q_{\eta}$ (see Eq. S37) and
$\delta Q_{\kappa}$ (see Eq. S53) using a relaxation time approximation for
the Boltzmann equation. In this case, the single particle phase space
distribution $f(\mathbf{r},\mathbf{v},t)$ obeys

$\partial_{t}f+\mathbf{v}\cdot\nabla
f=-\frac{1}{\tau}\,(f-f_{0})\equiv-\frac{1}{\tau}\,f_{1}$ (S12)

with $\tau$ is the relaxation time and $f=f_{0}+f_{1}$, with $f_{0}$ the
equilibrium distribution.

In the high-temperature Maxwell-Boltzmann limit, the equilibrium distribution
is

$f_{0}=n_{0}\,W_{0}(\mathbf{U}),$ (S13)

where $\mathbf{U}=\mathbf{v}-\mathbf{u}$ is the particle velocity relative to
the stream velocity $\mathbf{u}(\mathbf{r},t)$ and
$\int\\!d^{3}\mathbf{U}\,W(\mathbf{U})=1$. Here,

$W_{0}(\mathbf{U})=\frac{e^{-\mathbf{U}^{2}/v_{0}^{2}}}{(v_{0}\sqrt{\pi})^{3}}$
(S14)

and $v_{0}=\sqrt{2k_{B}T/m}$ is the thermal speed. In general, the background
temperature $T_{0}$ spatially varies $T\equiv T(\mathbf{r})$. For convenience,
we drop the subscript 0 and use $T$ for the temperature in deriving the
relaxation equations.

Without specifying the phase space distribution $f$, the pressure tensor is
given by

$p_{ij}=m\int\\!d^{3}\mathbf{U}\,U_{i}U_{j}\,f(\mathbf{r},\mathbf{v},t).$
(S15)

Taking $p_{ij}=p^{0}_{ij}+p^{1}_{ij}$, the scalar pressure $p_{0}\equiv p$, is
immediately obtained from $p^{0}_{ij}=\delta_{ij}\,p$ with $f=f_{0}$ and
$i=j=x$,

$p=m\int d^{3}\mathbf{U}\,U^{2}_{x}\,f_{0}.$ (S16)

Writing $\int\\!d^{3}\mathbf{U}\,U^{2}_{x}\,f_{0}=n_{0}\langle
U^{2}_{x}\rangle\equiv n_{0}\,\overline{U^{2}_{x}}$, we have

$p=n_{0}m\,\overline{U^{2}_{x}},$ (S17)

which gives $p\rightarrow n_{0}\,k_{B}T_{0}$ in the Maxwell-Boltsmann limit,
Eq. S14.

####  A.2.1 Shear viscosity

We find the relaxation equation for $\delta Q_{\eta}$ of Eq. S9 from that of
$p^{1}_{ij}$, assuming that the stream velocity $u(\mathbf{r},t)$ is position
dependent, producing a shear stress. Here, we assume that the background
temperature $T_{0}$ is spatially constant. To proceed, we first consider Eq.
S15 for $i\neq j$. The equilibrium distribution $f_{0}$ is a symmetric
function of $\mathbf{U}$, so that

$\int\\!d^{3}\mathbf{U}U_{i}U_{j}\,f_{0}=0.$ (S18)

Then Eq. S15 with $f=f_{0}+f_{1}$ and Eq. S18 yield

$p^{1}_{ij}=m\\!\int\\!d^{3}\mathbf{U}\,U_{i}U_{j}\,f=m\\!\int\\!d^{3}\mathbf{U}\,U_{i}U_{j}\,f_{1}.$
(S19)

Multiplying Eq. S12 by $\int\\!d^{3}\mathbf{U}\,U_{i}U_{j}$ and using Eq. S19,
we obtain

$\dot{p}^{1}_{ij}+m\int\\!d^{3}\mathbf{U}\,U_{i}U_{j}\,v^{k}\partial_{k}f=-\frac{1}{\tau_{\eta}}\,p^{1}_{ij},$
(S20)

where we define $\tau\equiv\tau_{\eta}$ for the shear viscosity. Then,

$\dot{p}^{1}_{ij}+\frac{1}{\tau_{\eta}}p^{1}_{ij}=-I_{ij},$ (S21)

where

$I_{ij}=m\int\\!d^{3}\mathbf{U}\,U_{i}U_{j}\,v^{k}\partial_{k}f=m\int\\!d^{3}\mathbf{U}\,U_{i}U_{j}\,v^{k}\frac{\partial
U^{l}}{\partial x^{k}}\frac{\partial f}{\partial U^{l}}.$ (S22)

For fast relaxation, where $f_{1}\rightarrow\tau_{\eta}v_{k}\partial_{k}f$ in
Eq. S12, we see that $p^{1}_{ij}\simeq\tau_{\eta}I_{ij}$ is already first
order in $\tau_{\eta}$, so that we can take $f\rightarrow f_{0}$ in Eq. S22.
Using $\mathbf{U}=\mathbf{v}-\mathbf{u}$,

$\frac{\partial U^{l}}{\partial x^{k}}=-\frac{\partial u^{l}}{\partial
x^{k}}.$ (S23)

Then with $v_{k}=U_{k}+u_{k}$ in Eq. S22, we obtain

$I_{ij}=-m\frac{\partial u^{l}}{\partial
x^{k}}\int\\!d^{3}\mathbf{U}\,U_{i}U_{j}(U_{k}+u_{k})\frac{\partial
f_{0}}{\partial U^{l}}.$ (S24)

Integrating by parts, we then have

$I_{ij}=m\frac{\partial u^{l}}{\partial
x^{k}}\int\\!d^{3}\mathbf{U}\,\frac{\partial}{\partial
U^{l}}[U_{i}U_{j}(U_{k}+u_{k})]\,f_{0},$ (S25)

Using $\partial U_{i}/\partial U_{l}=\delta_{il}$ and defining $\int
d^{3}\mathbf{U}\,g(\mathbf{U})f_{0}(\mathbf{U})=n_{0}\langle
g(\mathbf{U})\rangle$, Eq. S25 can be rewritten as

$I_{ij}=m\frac{\partial u^{l}}{\partial
x^{k}}\,n_{0}\left\\{\delta_{il}\langle
U_{j}(U_{k}+u_{k})\rangle+\delta_{jl}\langle
U_{i}(U_{k}+u_{k})\rangle+\delta_{kl}\langle U_{i}U_{j}\rangle\right\\}$ (S26)

Eq. S26 is simplified with

$n_{0}\langle
U_{j}(U_{k}+u_{k})\rangle=\int\\!d^{3}\mathbf{U}\,U_{j}(U_{k}+u_{k})\,f_{0}=n_{0}\,\delta_{jk}\overline{U^{2}_{x}},$
(S27)

where the $u_{k}$ term vanishes since $f_{0}$ is symmetric in $U_{j}$. Hence,
we can write $\langle U_{i}U_{j}\rangle=\delta_{ij}\overline{U^{2}_{x}}$. With
$p=n_{0}m\,\overline{U^{2}_{x}}$ from Eq. S17, we have

$I_{ij}=p\,\frac{\partial u^{l}}{\partial
x^{k}}\,\\{\delta_{il}\delta_{jk}+\delta_{jl}\delta_{ik}+\delta_{kl}\delta_{ij}\\}\,.$
(S28)

Carrying out the sums over repeated indices, we obtain

$I_{ij}=p\left(\frac{\partial u^{i}}{\partial x^{j}}+\frac{\partial
u^{j}}{\partial x^{i}}+\delta_{ij}\nabla\cdot\mathbf{u}\right),$ (S29)

where the $\delta_{ij}$ term vanishes for $i\neq j$.

To determine $p^{1}_{ij}$ for all $i,j$, we consider the symmetric second rank
traceless pressure tensor,

$p^{1}_{ij}\rightarrow
m\\!\int\\!d^{3}\mathbf{U}\,\left(U_{i}U_{j}-\frac{1}{3}\delta_{ij}\mathbf{U}^{2}\right)\,f_{1}\,.$
(S30)

where the $f_{0}$ part of $f$ vanishes because it is scalar function of
$\mathbf{U}$. Since $\mathbf{U}^{2}=Tr\\{U_{i}U_{j}\\}$, evaluating Eq. S30
just changes $I_{ij}$ in Eq. S21 and Eq. S29 to

$I_{ij}\rightarrow I_{ij}-\frac{1}{3}\delta_{ij}Tr\\{I_{ij}\\}\equiv
p\,\sigma_{ij}\,.$ (S31)

The $\delta_{ij}$ term in Eq. S29 makes no contribution to the symmetric
traceless tensor, yielding

$\sigma_{ij}=\frac{\partial u^{i}}{\partial x^{j}}+\frac{\partial
u^{j}}{\partial x^{i}}-\frac{2}{3}\,\delta_{ij}\nabla\cdot\mathbf{u}\,.$ (S32)

With Eqs. S31 and S32, Eq. S21 determines the relaxation equation for the
shear stress tensor,

$\dot{p}^{1}_{ij}+\frac{1}{\tau_{\eta}}\,p^{1}_{ij}=-p\,\sigma_{ij}\,.$ (S33)

For small $\tau_{\eta}$, $\dot{p}^{1}_{ij}\ll p^{1}_{ij}/\tau_{\eta}$, we see
that

$p^{1}_{ij}\rightarrow-\tau_{\eta}p\,\delta_{ij}=-\eta\,\sigma_{ij},$ (S34)

where the static shear viscosity is

$\eta=\tau_{\eta}\,p.$ (S35)

Now we can evaluate the relaxation equation for $\delta Q_{\eta}$ of Eq. S9.
We evaluate Eq. S32 for $\sigma_{zz}$ and eliminate the velocity field using
current conservation Eq. S1. To first order in small quantities,
$\delta\dot{n}+n_{0}\,\partial_{z}v_{z}=0$, yielding

$\sigma_{zz}=\frac{4}{3}\,\partial_{z}v_{z}=-\frac{4}{3}\,\frac{\delta\dot{n}}{n_{0}}\,.$
(S36)

We find $\delta\dot{Q_{\eta}}$ using Eq. S9, $\delta
Q_{\eta}\equiv\frac{1}{m}\,\partial_{z}^{2}p^{1}_{zz}$. With Eq. S36 and Eq.
S33 we obtain finally,

$\delta\dot{Q_{\eta}}+\frac{1}{\tau_{\eta}}\delta
Q_{\eta}=\frac{4}{3}\,\frac{p}{mn_{0}}\,\partial^{2}_{z}\delta\dot{n}\,.$
(S37)

For fast relaxation, $\delta\dot{Q_{\eta}}\ll\delta Q_{\eta}/\tau_{\eta}$, we
find,

$\delta
Q_{\eta}\rightarrow\frac{4}{3}\,\frac{\tau_{\eta}p}{mn_{0}}\,\partial^{2}_{z}\delta\dot{n}=\frac{4}{3}\,\frac{\eta}{mn_{0}}\,\partial^{2}_{z}\delta\dot{n}\,.$
(S38)

In this limit, Eq. S8 with Eq. S38 reproduces the Navier-Stokes form used in
Ref. [23]. We note that Eq. S35 for $\eta$ and Eq. S37 for
$\delta\dot{Q_{\eta}}$ are independent of the form of the single particle
phase space distribution, $f_{0}(\mathbf{r},\mathbf{v})$, which has not been
explicitly used to obtain the background pressure $p$ from Eq. S17.

####  A.2.2 Thermal conductivity

Next, we find the relaxation equation for $\delta Q_{\kappa}$ of Eq. S11 from
that of the 1D energy current, $J_{E}$,

$J_{E}=\int d^{3}\mathbf{v}\,v_{z}\frac{m}{2}\,\mathbf{v}^{2}\,f.$ (S39)

To find the relaxation equation of the energy current, we assume that the
system is in mechanical equilibrium with a stream velocity $\mathbf{u}=0$, so
that $\mathbf{U}=\mathbf{v}$ in Eq. S14, and we include a temperature gradient
$T(\mathbf{r})$. Eq. S39 shows that $J_{E}$ vanishes for $f=f_{0}$, since the
integrand would be odd in $v_{z}$.

Using Eq. S39 and Eq. S12 with $\tau\equiv\tau_{\kappa}$, we have

$\dot{J}_{E}+\frac{m}{2}\,\partial_{z}\\!\\!\int
d^{3}\mathbf{v}\,v_{z}^{2}\mathbf{v}^{2}\,f_{0}=-\frac{1}{\tau_{\kappa}}\,J_{E}$
(S40)

or

$\dot{J}_{E}+\frac{1}{\tau_{\kappa}}\,J_{E}=-\partial_{z}I_{zz}.$ (S41)

Here

$I_{zz}\equiv\frac{m}{2}\,\int
d^{3}\mathbf{v}\,v_{z}^{2}\,\mathbf{v}^{2}\,f_{0}.$ (S42)

To evaluate Eq. S42, we require an explicit form for the equilibrium phase
space distribution, which we take to be a Maxwell-Boltzmann distribution,
$f_{0}(\mathbf{r},\mathbf{v})=n(\mathbf{r})\,W_{0}(\mathbf{v})$, where

$W_{0}(\mathbf{v})=\frac{e^{-\mathbf{v}^{2}/v_{0}^{2}}}{(v_{0}\sqrt{\pi})^{3}}$
(S43)

with $v_{0}=\sqrt{2k_{B}T/m}$ the thermal speed. As discussed above, we assume
that $T=T(\mathbf{r})$ spatially varies, producing a temperature gradient, as
discussed further below. Then,

$I_{zz}=\frac{m}{2}\,\frac{1}{3}\,\langle v^{4}\rangle\,n(\mathbf{r}).$ (S44)

Here,

$\langle v^{4}\rangle=\int
d^{3}\mathbf{v}\,v^{4}\,W_{0}(\mathbf{v})=\frac{4}{\sqrt{\pi}}v_{0}^{4}\int
d\left(\frac{v}{v_{0}}\right)\left(\frac{v}{v_{0}}\right)^{6}\,e^{-(v/v_{0})^{2}},$
(S45)

which yields

$\langle
v^{4}\rangle=\frac{15}{4}\,v_{0}^{4}=15\,\left(\frac{k_{B}T}{m}\right)^{2}.$
(S46)

Using Eq. S46 in Eq. S44, and the pressure
$p(\mathbf{r})=n(\mathbf{r})\,k_{B}T(\mathbf{r})$ from Eqs. S17 and S43, we
have

$I_{zz}=\frac{5}{2}\frac{k_{B}}{m}\,n(\mathbf{r})\,k_{B}T^{2}(\mathbf{r})=\frac{5}{2}\frac{k_{B}}{m}\,p(\mathbf{r})\,T(\mathbf{r}).$
(S47)

With Eq. S41, we then obtain

$\dot{J}_{E}+\frac{1}{\tau_{\kappa}}\,J_{E}=-\frac{5}{2}\frac{k_{B}}{m}\,\partial_{z}[\,p(\mathbf{r})\,T(\mathbf{r})\,].$
(S48)

For pure heat flow, mechanical equilibrium requires

$\nabla p(\mathbf{r})=0.$ (S49)

Therefore, Eq. S48 becomes

$\dot{J}_{E}+\frac{1}{\tau_{\kappa}}J_{E}=-\frac{5}{2}\frac{k_{B}}{m}\,p\,\,\partial_{z}\delta
T,$ (S50)

where we suppress the argument $\mathbf{r}$. Here, the temperature
$T=T_{0}+\delta T$, with $T_{0}$ the uniform background temperature, so that
$\partial_{z}T=\partial_{z}\delta T$.

For fast relaxation, where $\dot{J}_{E}\ll J_{E}/\tau_{\kappa}$, Eq. S50 gives

$J_{E}\rightarrow-\frac{5}{2}\frac{k_{B}}{m}\,\tau_{\kappa}\,p\,\,\partial_{z}\delta
T\equiv-\kappa_{T}\,\partial_{z}\delta T,$ (S51)

where the static thermal conductivity is

$\kappa_{T}=\frac{5}{2}\frac{k_{B}}{m}\,\tau_{\kappa}\,p\,.$ (S52)

We find $\delta\dot{Q}_{\kappa}$ using Eq. S11, $\delta
Q_{\kappa}\equiv-(\beta/c_{V_{1}})\,\partial_{z}\,J_{E}$. Then, operating on
Eq. S50 with $-(\beta/c_{V_{1}})\,\partial_{z}$, we have

$\delta\dot{Q}_{\kappa}+\frac{1}{\tau_{\kappa}}\delta{Q}_{\kappa}=\frac{5}{2}\frac{k_{B}}{m}\,\frac{p}{n_{0}c_{V_{1}}}\,\,\partial^{2}_{z}\,\delta\tilde{T},$
(S53)

where $\delta\tilde{T}=n_{0}\beta\,\delta T$ (see Eq. S5). For fast
relaxation, where
$\delta\dot{Q}_{\kappa}\ll\frac{1}{\tau_{\kappa}}\delta{Q}_{\kappa}$, Eq. S53
gives

$\delta{Q}_{\kappa}\rightarrow\frac{5}{2}\frac{k_{B}}{m}\,\frac{\tau_{\kappa}\,p}{n_{0}c_{V_{1}}}\,\partial^{2}_{z}\delta\tilde{T}=\frac{\kappa_{T}}{n_{0}c_{V_{1}}}\,\partial^{2}_{z}\,\delta\tilde{T}.$
(S54)

In this limit, Eq. S10 with Eq. S54 reproduces the Navier-Stokes form used in
Ref. [23]. We note that the coefficient $5/2$ in Eq. S52 for $\kappa_{T}$ and
in Eq. S53 for $\delta\dot{Q}_{\kappa}$ is dependent on the assumed Maxwell-
Boltzmann approximation for the phase space distribution $f_{0}$, which was
needed to evaluate Eq. S42, yielding Eq. S47.

###  A.3 Relaxation of the Fourier Components

In the experiments, as described in the main text, a spatially periodic
perturbing potential $\delta U$ is used to create a spatially periodic density
perturbation $\delta n(z,t=0)$, with a wavelength $\lambda$ and corresponding
wavevector $q=2\pi/\lambda$. After the system reaches mechanical and thermal
equilibrium, $\delta U$ is abruptly extinguished and $\delta n(z,t)$ is
measured. A spatial Fourier transform of the relaxing density perturbation
$\delta n(z,t)$ yields the time dependence of the Fourier component, $\delta
n(q,t)$. The evolution of $\delta n(q,t)$ is readily determined from the
Fourier transforms of Eqs. S8, S37, S10 and S53. Defining $\delta
n(q,t)\equiv\delta n$, $\delta\tilde{T}(q,t)\equiv\delta\tilde{T}$, $\delta
Q_{\eta}(q,t)\equiv\delta Q_{\eta}$, and $\delta Q_{\kappa}(q,t)\equiv\delta
Q_{\kappa}$, we find

$\displaystyle\delta\ddot{n}=-\omega_{T}^{2}\,\delta
n-\omega_{T}^{2}\,\delta\tilde{T}+\delta Q_{\eta}\hskip 18.06749pt{\rm
with}\hskip 18.06749pt\omega_{T}\equiv c_{T}q\,,$ (S55)
$\delta\dot{Q_{\eta}}=-\frac{1}{\tau_{\eta}}\delta
Q_{\eta}-\Omega_{\eta}^{2}\,\delta\dot{n}\hskip 18.06749pt{\rm with}\hskip
18.06749pt\Omega_{\eta}^{2}\equiv\frac{4}{3}\,\frac{p}{mn_{0}}\,q^{2}\,,$
(S56)

and

$\delta\dot{\tilde{T}}=\epsilon_{LP}\,\delta\dot{n}+\delta Q_{\kappa}\,,\hskip
18.06749pt{\rm where}$ (S57)
$\delta\dot{Q}_{\kappa}=-\frac{1}{\tau_{\kappa}}\delta{Q}_{\kappa}-\Omega_{\kappa}^{2}\,\delta\tilde{T}\hskip
18.06749pt{\rm with}\hskip
18.06749pt\Omega_{\kappa}^{2}=\frac{5}{2}\frac{k_{B}}{m}\,\frac{p}{n_{0}c_{V_{1}}}\,\,q^{2}.$
(S58)

For a unitary Fermi gas, the pressure $p$ and internal energy density ${\cal
E}$ have the universal forms

$p=\frac{2}{5}\,n\epsilon_{F}(n)\,f_{E}(\theta)=\frac{2}{3}\,{\cal E},$ (S59)

where the universal function $f_{E}(\theta)$ has been measured [30] as a
function of the reduced temperature $\theta\equiv T/T_{F}$. Here, the local
Fermi energy $\epsilon_{F}(n)=\frac{\hbar^{2}}{2m}(3\pi^{2}n)^{2/3}$ with $n$
the total density for a 50-50 mixture of two spin states. Taking
$\epsilon_{F}(n)=mv_{F}^{2}/2$, which defines the Fermi speed $v_{F}$, we can
write in Eq. S56

$\Omega_{\eta}^{2}\equiv\frac{4}{15}\,\omega_{F}^{2}\,f_{E}(\theta),$ (S60)

where we define $\omega_{F}\equiv v_{F}q$. Similarly, we can write in Eq. S58

$\Omega_{\kappa}^{2}\equiv\frac{1}{2}\,\frac{k_{B}}{c_{V_{1}}}\,\omega_{F}^{2}\,f_{E}(\theta).$
(S61)

In the short relaxation time limit, we see that $\delta
Q_{\eta}\rightarrow-\gamma_{\eta}\delta\dot{n}$, with
$\gamma_{\eta}\equiv\tau_{\eta}\Omega_{\eta}^{2}=\frac{4}{3}\frac{\eta}{n_{0}m}\,q^{2}$
and $\delta Q_{\kappa}\rightarrow-\gamma_{\kappa}\delta\tilde{T}$, with
$\gamma_{\kappa}\equiv\tau_{\kappa}\Omega_{\kappa}^{2}=\frac{\kappa_{T}}{n_{0}c_{V_{1}}}\,q^{2}$,
reproducing the results in the supplement of Ref. [23].

The linear time-dependent equations S55, S56, S57, and S58 are easily solved
to determine $\delta n(q,t)$, with the initial conditions $\delta n(q,0)=A$,
$\delta\tilde{T}(q,0)=0$, $\delta Q_{\eta}(q,0)=0$, and $\delta
Q_{\kappa}(q,0)=0$. The data are fit using the amplitude $A$, isothermal
frequency $\omega_{T}$, and relaxation times $\tau_{\eta}$ and $\tau_{\kappa}$
as free parameters. As in our previous experiments [23], the ratio
$\omega_{T}/\omega_{F}$ self consistently determines the reduced temperature
$\theta$ in the fits.

In the main text, the relaxation times are given in units of the “Fermi time,”
which we take to be $\tau_{F}\equiv\lambda_{F}/v_{F}$, i.e., the time for an
atom to move a Fermi wavelength at the Fermi speed. With
$\lambda_{F}=2\pi\hbar/(mv_{F})$, and $mv_{F}^{2}=2\epsilon_{F}$, the Fermi
time is

$\tau_{F}=\frac{\pi\hbar}{\epsilon_{F}}.$ (S62)

###  A.4 Static Transport Properties

With the relaxation times determined from the fits, the static shear viscosity
Eq. S35 is determined in units of $\hbar n_{0}$ by

$\eta=\tau_{\eta}\,p=\alpha_{\eta}\,\hbar n_{0}\,,$ (S63)

where Eq. S59 for $p$ gives the dimensionless shear viscosity coefficient,

$\alpha_{\eta}=\frac{2\pi}{5}\,\frac{\tau_{\eta}}{\tau_{F}}\,f_{E}(\theta).$
(S64)

Similarly, the static thermal conductivity Eq. S52 is determined in units of
$\hbar n_{0}\,k_{B}/m$ by

$\kappa_{T}=\frac{5}{2}\frac{k_{B}}{m}\,\tau_{\kappa}\,p=\alpha_{\kappa}\,\frac{k_{B}}{m}\,\hbar
n_{0}\,,$ (S65)

where the dimensionless thermal conductivity coefficient is

$\alpha_{\kappa}=\pi\,\frac{\tau_{\kappa}}{\tau_{F}}\,f_{E}(\theta).$ (S66)

Finally, the first sound diffusivity [23, 33] is determined from the static
transport properties,

$D_{1}=\frac{4}{3}\frac{\eta}{n_{0}m}+\left(\frac{1}{c_{V_{1}}}-\frac{1}{c_{P_{1}}}\right)\frac{\kappa_{T}}{n_{0}}.$
(S67)

For a unitary Fermi gas in a 50-50 mixture of two spin states and total
density $n$ [23, 22],

$\frac{1}{c_{V_{1}}}-\frac{1}{c_{P_{1}}}=\frac{4}{15}\frac{nT}{p}=\frac{1}{k_{B}}\frac{2}{3}\frac{\theta}{f_{E}(\theta)}.$
(S68)

Then

$\frac{D_{1}}{\hbar/m}=\frac{4}{3}\,\alpha_{\eta}+\frac{2}{3}\frac{\theta}{f_{E}(\theta)}\,\alpha_{\kappa}.$
(S69)

Using Eqs. S64 and S66, we then obtain $D_{1}$ in units of $\hbar/m$ from the
measured relaxation times,

$\frac{D_{1}}{\hbar/m}=\frac{8\pi}{15}\,\frac{\tau_{\eta}}{\tau_{F}}\,f_{E}(\theta)+\frac{2\pi}{3}\,\frac{\tau_{\kappa}}{\tau_{F}}\,\theta.$
(S70)

For comparison with the static transport properties shown in the main text,
which are obtained from the measured relaxation times using Eqs. S64, S66, and
S70, we parameterize $\alpha_{\eta}$ as

$\alpha_{\eta}=\alpha_{3/2}\,\theta^{3/2}+\alpha_{2\eta}.$ (S71)

Here, the first term is the high temperature limit, which is obtained from a
variational calculation [4],
$\alpha_{3/2}=45\pi^{3/2}/(64\sqrt{2})=2.76849\simeq 2.77$ and
$\alpha_{2\eta}$ is the universal density shift coefficient, which is used as
a fit parameter. Similarly, we parameterize $\alpha_{\kappa}$ as

$\alpha_{\kappa}=\frac{15}{4}\,\left(\alpha_{3/2}\,\theta^{3/2}+\alpha_{2\kappa}\right).$
(S72)

Here, the leading factor of $15/4$ is chosen to yield the correct high
temperature limit for the ratio $\kappa_{T}/\eta$. In this limit, Eqs. S65 and
S63 yield $\kappa_{T}=\frac{15}{4}\frac{k_{B}}{m}\,\eta$, where we have used
$\tau_{\kappa}/\tau_{\eta}=3/2$, which can be shown to hold for any isotropic
collision cross section.

With Eqs. S71 and S72, the relaxation times are then parameterized in the main
text by inverting Eqs. S64 and S66 as

$\frac{\tau_{\eta}}{\tau_{F}}=\alpha_{\eta}\,\frac{5}{2\pi}\frac{1}{f_{E}(\theta)}$
(S73)

and

$\frac{\tau_{\kappa}}{\tau_{F}}=\alpha_{\kappa}\,\frac{1}{\pi}\frac{1}{f_{E}(\theta)}.$
(S74)

Fig. S1 shows the sound diffusivity obtained from the measured relaxation
times, Eq. S70, which is compared to the sound diffusivity measured for sound
attenuation by Patel et al., in Ref. [22]. We see that the high temperature
behavior obtained from Eq. S69 using the fitted density shift coefficients for
the shear viscosity and thermal conductivity in Eqs. S71 and S72 is in good
agreement with the high temperature sound attenuation measurements. However,
the lower temperature sound diffusivity measurements of Ref. [22] exhibit a
nearly constant upward shift relative to the sound diffusivity obtained from
the relaxation time experiments.

Figure S1: Sound diffusivity from measured $\tau_{\eta}$, $\tau_{\kappa}$,
$D_{1}[\hbar/m]=\frac{8\pi}{15}\,\frac{\tau_{\eta}}{\tau_{F}}\,f_{E}(\theta)+\frac{2\pi}{3}\frac{\tau_{\kappa}}{\tau_{F}}\,\theta$
(Blue dots), in units of $\hbar/m$ versus reduced temperature
$\theta=T/T_{F}$. Red solid curve:
$D_{1}[\hbar/m]=\frac{4}{3}\,(2.77\,\theta^{3/2}\\!\\!+0.44)+\frac{5}{2}\frac{\theta}{f_{E}(\theta)}\,(2.77\,\theta^{3/2}\\!\\!+0.22)$.
Red-dashed curve (high temperature limit):
$f_{E}(\theta)\rightarrow\frac{5}{2}\,\theta$, $D_{1}[\hbar/m]=7/3\times
2.77\,\theta^{3/2}$. Sound diffusivity data of Ref. [22] (Orange dots). Error
bars (blue dots) are statistical. (color online)