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arXiv:1505.00846 · Physical Review A

Dissipative fluid dynamics for the dilute Fermi gas at unitarity: Anisotropic fluid dynamics

Marcus Bluhm, Thomas Schäfer

Published Oct 1, 2015• 38 citations
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Abstract

We consider the time evolution of a dilute atomic Fermi gas after release from a trapping potential. A common difficulty with using fluid dynamics to study the expansion of the gas is that the theory is not applicable in the dilute corona, and that a naive treatment of the entire cloud using fluid dynamics leads to unphysical results. We propose to remedy this problem by including certain nonhydrodynamic degrees of freedom, in particular anisotropic components of the pressure tensor, in the theoretical description. We show that, using this method, it is possible to describe the crossover from fluid dynamics to ballistic expansion locally. We illustrate the use of anisotropic fluid dynamics by studying the expansion of the dilute Fermi gas at unitarity using different functional forms of the shear viscosity, including a shear viscosity that is solely a function of temperature, $\ensuremath{\eta}\ensuremath{\sim}{(mT)}^{3/2}$, as predicted by kinetic theory in the dilute limit.

DOI
10.1103/physreva.92.043602

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We consider the time evolution of a dilute atomic Fermi gas after release from a trapping potential. A common difficulty with using fluid dynamics to study the expansion of the gas is that the theory is not applicable in the dilute corona, and that a naive treatment of the entire cloud using fluid dynamics leads to unphysical results. We propose to remedy this problem by including certain nonhydrodynamic degrees of freedom, in particular anisotropic components of the pressure tensor, in the theoretical description. We show that, using this method, it is possible to describe the crossover from fluid dynamics to ballistic expansion locally. We illustrate the use of anisotropic fluid dynamics by studying the expansion of the dilute Fermi gas at unitarity using different functional forms of the shear viscosity, including a shear viscosity that is solely a function of temperature, \ensuremathη\ensuremath∼(mT)3/2\ensuremath{\eta}\ensuremath{\sim}{(mT)}^{3/2}\ensuremathη\ensuremath∼(mT)3/2, as predicted by kinetic theory in the dilute limit.
Determinations of the shear viscosity of trapped ultracold gases suffer from systematic, uncontrolled uncertainties related to the treatment of the dilute part of the gas cloud. In this work we present an analysis of expansion experiments based on a new method, anisotropic fluid dynamics, that interpolates between Navier-Stokes fluid dynamics at the center of the cloud and ballistic behavior in the dilute corona. We validate the method using a comparison between anisotropic fluid dynamics and numerical solutions of the Boltzmann equation. We then apply anisotropic fluid dynamics to the expansion data reported by Cao et al. In the high-temperature limit we find η=0.282(mT)^{3/2}, which agrees within about 5% with the theoretical prediction η=0.269(mT)^{3/2}.
We investigate the expansion dynamics of a dilute Fermi gas at unitarity in the context of dissipative fluid dynamics. Our aim is to quantify the effects of shear viscosity on the time evolution of the system. We compare exact numerical solutions of the equations of viscous hydrodynamics to various approximations that have been proposed in the literature. Our main findings are (i) shear viscosity leads to characteristic features in the expansion dynamics; (ii) a quantitative description of these effects has to include reheating; (iii) dissipative effects are not sensitive to the equation of state P(n,T)P(n,T)P(n,T) as long as the universal relation P=23EP=\frac{2}{3}\mathcal{E}P=32​E is satisfied; (iv) the expansion dynamics mainly constrains the cloud average of the shear viscosity.
Determinations of the shear viscosity of trapped ultracold gases suffer from systematic, uncontrolled uncertainties related to the treatment of the dilute part of the gas cloud. In this work we present an analysis of expansion experiments based on a new method, anisotropic fluid dynamics, that interpolates between Navier-Stokes fluid dynamics at the center of the cloud and ballistic behavior in the dilute corona. We validate the method using a comparison between anisotropic fluid dynamics and numerical solutions of the Boltzmann equation. We then apply anisotropic fluid dynamics to the expansion data reported by Cao et al. In the high temperature limit we find η=0.282(mT)3/2\eta=0.282(mT)^{3/2}η=0.282(mT)3/2, which agrees within about 5\% with the theoretical prediction η=0.269(mT)3/2\eta=0.269(mT)^{3/2}η=0.269(mT)3/2.
We compute second-order transport coefficients of the dilute Fermi gas at unitarity. The calculation is based on kinetic theory and the Boltzmann equation at second order in the Knudsen expansion. The second-order transport coefficients describe the shear stress relaxation time, nonlinear terms in the strain-stress relation, and nonlinear couplings between vorticity and strain. An exact calculation in the dilute limit gives ${\ensuremath{\tau}}_{R}=\ensuremath{\eta}/P$, where ${\ensuremath{\tau}}_{R}$ is the shear stress relaxation time, $\ensuremath{\eta}$ is the shear viscosity, and PPP is pressure. This relation is identical to the result obtained using the Bhatnagar-Gross-Krook approximation to the collision term, but other transport coefficients are sensitive to the exact collision integral.
We determine the shear viscosity of the ultracold Fermi gas at unitarity in the normal phase using hydrodynamic expansion data. The analysis is based on a generalized fluid dynamic framework which ensures a smooth transition between the fluid dynamic core of the cloud and the ballistic corona. We use expansion data taken by Joseph, Elliott, and Thomas [Shear Viscosity of a Universal Fermi Gas Near the Superfluid Phase Transition, Phys. Rev. Lett. 115, 020401 (2015).PRLTAO0031-900710.1103/PhysRevLett.115.020401] and measurements of the equation of state by Ku et al. [Revealing the superfluid lambda transition in the universal thermodynamics of a unitary Fermi gas, Science 335, 563 (2012).SCIEAS0036-807510.1126/science.1214987]. We find that the shear viscosity to particle density ratio just above the critical temperature is η/n|_{T_{c}}=0.41±0.11. We also obtain evidence that the shear viscosity to entropy density ratio has a minimum slightly above T_{c} with η/s|_{min}=0.50±0.10.
In this work we establish constraints on the temperature dependence of the shear viscosity $\ensuremath{\eta}$ in the superfluid phase of a dilute Fermi gas in the unitary limit. Our results are based on analyzing experiments that measure the aspect ratio of a deformed cloud after release from an optical trap. We discuss how to apply the two-fluid formalism to the unitary gas and provide a suitable parametrization of the equation of state. We show that in expansion experiments the difference between the normal and superfluid velocities remains small and can be treated as a perturbation. We find that expansion experiments favor a shear viscosity that decreases significantly in the superfluid regime. Using an exponential parametrization, we find $\ensuremath{\eta}({T}_{c}/2{T}_{F})\ensuremath{\lesssim}0.37[\ensuremath{\eta}({T}_{c}/{T}_{F})]$, where Tc{T}_{c}Tc​ is the critical temperature and TF{T}_{F}TF​ is the local Fermi temperature of the gas.