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

M. Bluhm and T. Schäfer

Phys. Rev. A 92, 043602 – Published 1 October 2015

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, $η \sim {(m T)}^{3 / 2}$ , as predicted by kinetic theory in the dilute limit.

Received 27 May 2015

DOI:https://doi.org/10.1103/PhysRevA.92.043602

Authors & Affiliations

M. Bluhm and T. Schäfer

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

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Vol. 92, Iss. 4 — October 2015

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Images

Figure 1
Comparison between solutions of the Boltzmann equation in the limits of free streaming (blue solid curves) and ideal fluid dynamics (red dashed curves). In the left panel, we show the velocity field component $u_{x}$ and the transverse density profile $x n (x)$ at an early time $t = ω_{⊥}^{- 1}$ . In the right panel, we show these observables at a later time $t = 6 ω_{⊥}^{- 1}$ . The solutions correspond to a trap deformation $λ = 0.045$ . The density $n$ and the velocity $u_{x}$ are given in arbitrary units, but the scales in the left and the right panel are identical.
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Figure 2
This figure shows the evolution of the aspect ratio $A_{R}$ as a function of time $t$ in dimensionless units, as explained in the text. The initial trap deformation is $A_{R} (0) = λ = 0.1$ and the initial energy is $E_{0} / E_{F} = 1$ . The red solid curve shows the evolution in ideal fluid dynamics, and the black dashed curve is the free-streaming limit. The remaining curves were obtained using viscous fluid dynamics with different values of ${\bar{α}}_{n}$ for the shear viscosity $\bar{η} = {\bar{α}}_{n} \bar{n}$ . The dotted curves show Navier-Stokes results for ${\bar{α}}_{n} = 0.1, 1$ , and the data points are the corresponding predictions from anisotropic fluid dynamics. In the case of anisotropic fluid dynamics, we also show the result for ${\bar{α}}_{n} = 1000$ , which is close to the free-streaming limit.
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Figure 3
This figure shows the $x x$ component of the dissipative correction to the stress tensor $δ Π_{x x} (x, 0, 0)$ in Navier-Stokes theory (green solid curve) and in anisotropic fluid dynamics (red squares) for different times $t / t_{0} = 0.75 - 2.0$ in steps of $Δ t = 0.25 t_{0}$ . The magnitude of the viscous stresses decreases with time. The calculation was performed with a density-dependent shear viscosity $\bar{η} = {\bar{α}}_{n} \bar{n}$ and ${\bar{α}}_{n} = 0.15$ . We used a trap deformation $λ = 0.045$ and an initial energy $E_{0} / E_{F} = 3$ .
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Figure 4
Same as Fig. 3 for a temperature-dependent shear viscosity $\bar{η} = {\bar{α}}_{T} {\bar{T}}^{3 / 2}$ with ${\bar{α}}_{T} = 0.06$ . The blue dashed line shows the negative of the anisotropic pressure component $P_{x}$ .
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Figure 5
This figure shows the time evolution of the aspect ratio $A_{R}$ from a simulation using anisotropic fluid dynamics with ${\bar{α}}_{T} = 0.023$ in $\bar{η} = {\bar{α}}_{T} {\bar{T}}^{3 / 2}$ and $λ = 0.1, E_{0} / E_{F} = 1$ (blue squares). For comparison, we also show the result in ideal fluid dynamics (red solid curve) and the case of purely ballistic expansion (black dashed curve). The green dotted curve shows a fit to the ${\bar{α}}_{T} = 0.023$ result based on Navier-Stokes theory with ${\bar{α}}_{n} = 0.066$ in $\bar{η} = {\bar{α}}_{n} \bar{n}$ .
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Figure 6
This figure shows the dependence of $〈 α_{n} 〉$ , the effective trap averaged ratio $η / n$ , on the initial energy per particle in units of the Fermi energy $E_{F}$ . A precise definition of $〈 α_{n} 〉$ is given in the text. The calculation was performed for a gas cloud of $2 \times 10^{5}$ particles with an initial trap deformation of $λ = 0.045$ . The microscopic shear viscosity is given by the kinetic theory result in the high-temperature limit, $η = 15 / (32 \sqrt{π}) {(m T)}^{3 / 2}$ .
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