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arXiv:1708.01548 · Physical Review A

Hydrodynamic tails and a fluctuation bound on the bulk viscosity

Mauricio Martínez, Thomas Schäfer

Published Dec 5, 2017• 39 citations
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Abstract

We study the small frequency behavior of the bulk viscosity spectral function using stochastic fluid dynamics. We obtain a number of model independent results, including the long-time tail of the bulk stress correlation function and the leading nonanalyticity of the spectral function at small frequency. We also establish a lower bound on the bulk viscosity which is weakly dependent on assumptions regarding the range of applicability of fluid dynamics. The bound on the bulk viscosity $\ensuremath{\zeta}$ scales as ${\ensuremath{\zeta}}_{\mathrm{min}}\ensuremath{\sim}{(P\ensuremath{-}\frac{2}{3}\mathcal{E})}^{2}{\ensuremath{\sum}}_{i}{D}_{i}^{\ensuremath{-}2}$, where ${D}_{i}$ are the diffusion constants for energy and momentum and $P\ensuremath{-}\frac{2}{3}\mathcal{E}$, where $P$ is the pressure and $\mathcal{E}$ is the energy density, is a measure of scale breaking. Applied to the cold Fermi gas near unitarity, $|\ensuremath{\lambda}/{a}_{s}|\ensuremath{\gtrsim}1$, where $\ensuremath{\lambda}$ is the thermal de Broglie wavelength and ${a}_{s}$ is the $s$-wave scattering length, this bound implies that the ratio of bulk viscosity to entropy density satisfies $\ensuremath{\zeta}/s\ensuremath{\gtrsim}0.1\ensuremath{\hbar}/{k}_{B}$. Here, $\ensuremath{\hbar}$ is Planck's constant and ${k}_{B}$ is Boltzmann's constant.

DOI
10.1103/physreva.96.063607

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We study the small frequency behavior of the bulk viscosity spectral function using stochastic fluid dynamics. We obtain a number of model independent results, including the long-time tail of the bulk stress correlation function and the leading nonanalyticity of the spectral function at small frequency. We also establish a lower bound on the bulk viscosity which is weakly dependent on assumptions regarding the range of applicability of fluid dynamics. The bound on the bulk viscosity \ensuremathζ\ensuremath{\zeta}\ensuremathζ scales as \ensuremathζmin\ensuremath∼(P\ensuremath−23E)2\ensuremath∑iDi\ensuremath−2{\ensuremath{\zeta}}_{\mathrm{min}}\ensuremath{\sim}{(P\ensuremath{-}\frac{2}{3}\mathcal{E})}^{2}{\ensuremath{\sum}}_{i}{D}_{i}^{\ensuremath{-}2}\ensuremathζmin​\ensuremath∼(P\ensuremath−32​E)2\ensuremath∑i​Di\ensuremath−2​, where Di{D}_{i}Di​ are the diffusion constants for energy and momentum and P\ensuremath−23EP\ensuremath{-}\frac{2}{3}\mathcal{E}P\ensuremath−32​E, where PPP is the pressure and E\mathcal{E}E is the energy density, is a measure of scale breaking. Applied to the cold Fermi gas near unitarity, ∣\ensuremathλ/as∣\ensuremath≳1|\ensuremath{\lambda}/{a}_{s}|\ensuremath{\gtrsim}1∣\ensuremathλ/as​∣\ensuremath≳1, where \ensuremathλ\ensuremath{\lambda}\ensuremathλ is the thermal de Broglie wavelength and as{a}_{s}as​ is the sss-wave scattering length, this bound implies that the ratio of bulk viscosity to entropy density satisfies \ensuremathζ/s\ensuremath≳0.1\ensuremathℏ/kB\ensuremath{\zeta}/s\ensuremath{\gtrsim}0.1\ensuremath{\hbar}/{k}_{B}\ensuremathζ/s\ensuremath≳0.1\ensuremathℏ/kB​. Here, \ensuremathℏ\ensuremath{\hbar}\ensuremathℏ is Planck's constant and kB{k}_{B}kB​ is Boltzmann's constant.
The dilute Fermi gas at unitarity is scale invariant and its bulk viscosity vanishes. We compute, in the high temperature limit, the leading contribution to the bulk viscosity when the scattering length is not infinite. A measure of scale breaking is provided by the ratio $(P\ensuremath{-}2/3\mathcal{E})/P$, where PPP is the pressure and E\mathcal{E}E is the energy density. At high temperature this ratio scales as $z\ensuremath{\lambda}/a$, where zzz is the fugacity, $\ensuremath{\lambda}$ is the thermal wavelength, and aaa is the scattering length. We show that the bulk viscosity $\ensuremath{\zeta}$ scales as the second power of this parameter, $\ensuremath{\zeta}\ensuremath{\sim}(z\ensuremath{\lambda}/a{)}^{2}{\ensuremath{\lambda}}^{\ensuremath{-}3}$.
The viscosity of strongly interacting systems is a topic of great interest in diverse fields. We focus here on the bulk and shear viscosities of nonrelativistic quantum fluids, with particular emphasis on strongly interacting ultracold Fermi gases. We use Kubo formulas for the bulk and shear viscosity spectral functions, $\ensuremath{\zeta}(\ensuremath{\omega})$ and $\ensuremath{\eta}(\ensuremath{\omega})$, respectively, to derive exact, nonperturbative results. Our results include a microscopic connection between the shear viscosity $\ensuremath{\eta}$ and the normal-fluid density ${\ensuremath{\rho}}_{n}$; sum rules for $\ensuremath{\zeta}(\ensuremath{\omega})$ and $\ensuremath{\eta}(\ensuremath{\omega})$ and their evolution through the BCS-BEC crossover (where BEC denotes Bose-Einstein condensate); and universal high-frequency tails for $\ensuremath{\eta}(\ensuremath{\omega})$ and the dynamic structure factor $S(\mathbf{q},\ensuremath{\omega})$. We use our sum rules to show that, at unitarity, $\ensuremath{\zeta}(\ensuremath{\omega})$ is identically zero and thus relate $\ensuremath{\eta}(\ensuremath{\omega})$ to density-density correlations. We predict that frequency-dependent shear viscosity $\ensuremath{\eta}(\ensuremath{\omega})$ of the unitary Fermi gas can be experimentally measured using Bragg spectroscopy.
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 $\ensuremath{\eta}/s$ as a function of the temperature. The minimum provides a bound on $\ensuremath{\eta}/s$ which is independent of the conjectured bound in string theory, $\ensuremath{\eta}/s\ensuremath{\ge}\ensuremath{\hbar}/(4\ensuremath{\pi}{k}_{B})$, where sss is the entropy density. For the dilute Fermi gas at unitarity, we find $\ensuremath{\eta}/s\ensuremath{\gtrsim}0.2\ensuremath{\hbar}$. 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 $\ensuremath{\omega}$ diverges as $1/\sqrt{\ensuremath{\omega}}$, and that the shear viscosity in two dimensions diverges as $\mathrm{ln}(1/\ensuremath{\omega})$.
The dilute Fermi gas at unitarity is scale invariant and its bulk viscosity vanishes. We compute the leading contribution to the bulk viscosity when the scattering length is not infinite. A measure of scale breaking is provided by the ratio (P−23E)/P(P-\frac{2}{3}{\cal E})/P(P−32​E)/P, where PPP is the pressure and E{\cal E}E is the energy density. In the high temperature limit this ratio scales as zλa\frac{zλ}{a}azλ​, where zzz is the fugacity, λλλ is the thermal wave length, and aaa is the scattering length. We show that the bulk viscosity ζζζ scales as the second power of this parameter, ζ∼(zλa)2λ−3ζ\sim (\frac{zλ}{a})^2 λ^{-3}ζ∼(azλ​)2λ−3.
The bulk viscosity determines dissipation during hydrodynamic expansion. It vanishes in scale invariant fluids, while a nonzero value quantifies the deviation from scale invariance. For the dilute Fermi gas the bulk viscosity is given exactly by the correlation function of the contact density of local pairs. As a consequence, scale invariance is broken purely by pair fluctuations. These fluctuations give rise also to logarithmic terms in the bulk viscosity of the high-temperature nondegenerate gas. For the quantum degenerate regime I report numerical Luttinger-Ward results for the contact correlator and the dynamical bulk viscosity throughout the BEC-BCS crossover. The ratio of bulk to shear viscosity ζ/η is found to exceed the kinetic theory prediction in the quantum degenerate regime. Near the superfluid phase transition the bulk viscosity is enhanced by critical fluctuations and has observable effects on dissipative heating, expansion dynamics, and sound attenuation.
We compute the shear viscosity of the unitary Fermi gas above the superfluid transition temperature, using a diagrammatic technique that starts from the exact Kubo formula. The formalism obeys a Ward identity associated with scale invariance which guarantees that the bulk viscosity vanishes identically. For the shear viscosity, vertex corrections and the associated Aslamazov-Larkin contributions are shown to be crucial to reproduce the full Boltzmann equation result in the high-temperature, low fugacity limit. The frequency dependent shear viscosity η(ω)\eta(\omega)η(ω) exhibits a Drude-like transport peak and a power-law tail at large frequencies which is proportional to the Tan contact. The weight in the transport peak is given by the equilibrium pressure, in agreement with a sum rule due to Taylor and Randeria. Near the superfluid transition the peak width is of the order of 0.5TF0.5 T_F0.5TF​, thus invalidating a quasiparticle description. The ratio η/s\eta/sη/s between the static shear viscosity and the entropy density exhibits a minimum near the superfluid transition temperature whose value is larger than the string theory bound ℏ/(4πkB)\hbar/(4\pi k_B)ℏ/(4πkB​) by a factor of about seven.