Hydrodynamic fluctuations and the minimum shear viscosity of the dilute Fermi gas at unitarity

Clifford Chafin and Thomas Schäfer

Phys. Rev. A 87, 023629 – Published 27 February 2013

Abstract

We study hydrodynamic fluctuations in a nonrelativistic fluid. We show that in three dimensions, fluctuations lead to a minimum in the shear viscosity to entropy density ratio $η / s$ as a function of the temperature. The minimum provides a bound on $η / s$ which is independent of the conjectured bound in string theory, $η / s \geq ℏ / (4 π k_{B})$ , where $s$ is the entropy density. For the dilute Fermi gas at unitarity, we find $η / s ≳ 0.2 ℏ$ . This bound is not universal—it depends on the thermodynamic properties of the unitary Fermi gas and on empirical information about the range of validity of hydrodynamics. We also find that the viscous relaxation time of a hydrodynamic mode with frequency $ω$ diverges as $1 / \sqrt{ω}$ , and that the shear viscosity in two dimensions diverges as $\ln (1 / ω)$ .

Received 15 October 2012

DOI:https://doi.org/10.1103/PhysRevA.87.023629

Authors & Affiliations

Clifford Chafin and Thomas Schäfer

Department of Physics, North Carolina State University, Raleigh, North Carolina 27695, USA

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Issue

Vol. 87, Iss. 2 — February 2013

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Images

Figure 1
Diagrammatic representation of the leading contribution of thermal fluctuations to the stress tensor correlation function. Solid lines labeled $v_{T}$ denote the transverse velocity correlator, dominated by the shear pole, and wavy lines labeled $v_{L}$ denote the longitudinal velocity correlator, governed by the sound pole and the diffusive heat mode.Reuse & Permissions
Figure 2
Shear viscosity to density ratio $η / n$ as a function of $T / T_{F}$ , where $T_{F}$ is the local Fermi temperature. The left panel shows $η / n$ for the unitary gas in three dimensions. The solid line is the result in kinetic theory and the dashed line includes fluctuations. The band shows the uncertainty if the cutoff is varied in the regime $Λ = (0.25 - 0.75) K_{hyd}$ . The right panel shows the two-dimensional gas at the crossover point $T_{a, 2 D} = T_{F}$ . The solid line is the kinetic result. The dashed and dotted lines include fluctuations where we have used $Λ = K_{hyd}$ and $ω$ was taken to be the frequency of the quadrupole mode in a harmonic trap with $N = 10^{4}$ and $10^{5}$ particles.Reuse & Permissions
Figure 3
Bound on the shear viscosity to entropy density ratio $η / s$ as a function of $T / T_{F}$ , where $T_{F}$ is the local Fermi temperature. This figure shows the bound given in Eq. (35) evaluated using measurements of thermodynamic properties reported in [28]. The band around the dotted line shows the sensitivity to variations in the cutoff in the range $Λ = (0.5 \pm 0.25) K_{hyd}$ . The dashed line shows the string theory bound $η / s = 1 / (4 π)$ .Reuse & Permissions
Figure 4
Trap averaged shear viscosity to density ratio $〈 α_{n} 〉$ . We show $〈 α_{n} 〉$ as a function of $T / T_{F}^{trap}$ , where $T_{F}^{trap} = {(3 λ N)}^{1 / 3} ω_{⊥}$ is the Fermi temperature of the trap. We have chosen $N = 2 \times 10^{5}$ and $λ = 0.045$ as in [29]. The solid line shows the kinetic theory result, and the dashed line includes fluctuation corrections to the shear viscosity. The data are from [44], which is a reanalysis of the results reported in [29]. We do not show data in the superfluid regime $T ≪ T_{c}$ .Reuse & Permissions
Figure 5
Diagrammatic representation of higher-order fluctuation contributions to the stress tensor correlation function. Solid lines labeled $v_{T}$ denote the transverse velocity correlator, and wavy lines labeled $v_{L}$ denote the longitudinal velocity correlator. Temperature fluctuations are shown as double-dashed lines, and density fluctuations are shown as dotted lines. Vertices shown as dots contain no derivatives, whereas vertices labeled by squares contain spatial derivatives.Reuse & Permissions

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