Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1404.6843 · Physical Review A

Second-order fluid dynamics for the unitary Fermi gas from kinetic theory

Thomas Schäfer

Published Oct 29, 2014• 12 citations
arXivHTMLPDF

Abstract

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 $P$ 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.

DOI
10.1103/physreva.90.043633

Related papers

Thermal relaxation and the complete set of second-order transport coefficients for the unitary Fermi gas from kinetic theory

Christian Hall, Thomas Schäfer

We compute the complete set of second-order transport coefficients of the unitary Fermi gas, a dilute gas of spin-1/2 particles interacting via an $s$-wave interaction tuned to infinite scattering length. The calculation is based on kinetic theory and the Chapman-Enskog method at second order in the Knudsen expansion. We take into account the exact two-body collision integral. We extend previous results on second-order coefficients related to shear stress by including terms related to heat flow and gradients of the fugacity. We confirm that the thermal relaxation time is given by the simple estimate ${\ensuremath{\tau}}_{\ensuremath{\kappa}}=\ensuremath{\kappa}m/({c}_{P}T)$ even if the full collision kernel is taken into account. Here, $\ensuremath{\kappa}$ is the thermal conductivity, $m$ is the mass of the particles, ${c}_{P}$ is the specific heat at constant pressure, and $T$ is the temperature.

arXiv:2507.04202•Physical Review A•Sep 23, 2025•0 citations

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

Marcus Bluhm, Thomas Schäfer

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.

arXiv:1505.00846•Physical Review A•Oct 1, 2015•38 citations

Shear viscosity of the normal phase of a unitary Fermi gas from the ɛ expansion

Andrei B. Kryjevski

Using the $\ensuremath{\varepsilon}$-expansion technique, we compute $\ensuremath{\eta}/s$, where $\ensuremath{\eta}$ is the shear viscosity and $s$ is the entropy density of the normal phase of unitary Fermi gas in $d=4\ensuremath{-}\ensuremath{\varepsilon}$ dimensions to leading order (LO) in $\ensuremath{\varepsilon}$. We use the kinetic theory approach and solve transport equations for a medium perturbed by a shear hydrodynamic flow. The collision integrals are calculated to ${\ensuremath{\varepsilon}}^{2}$, which is LO. The LO result is temperature independent with $\ensuremath{\eta}/s\ensuremath{\simeq}(0.11/{\ensuremath{\varepsilon}}^{2})(\ensuremath{\hbar}/{k}_{B})$. The $d=3$ prediction for $\ensuremath{\eta}/s$ exceeds the $\ensuremath{\hbar}/4\ensuremath{\pi}{k}_{B}$ bound by a factor of about $1.4$.

Physical Review A•Feb 14, 2014•3 citations

Elliptic flow of the dilute Fermi gas: From kinetics to hydrodynamics

Kevin Dusling, Thomas Schäfer

We use the Boltzmann equation in the relaxation-time approximation to study the expansion of a dilute Fermi gas at unitarity. We focus, in particular, on the approach to the hydrodynamic limit. Our main findings are the following: (i) In the regime that has been studied experimentally hydrodynamic effects beyond the Navier-Stokes approximation are small. (ii) Mean-field corrections to the Boltzmann equation are not important. (iii) Experimental data imply that freezeout occurs very late, which means that the relaxation time remains smaller than the expansion time during the entire evolution of the system. (iv) The experimental results also imply that the bulk viscosity is significantly smaller than the shear viscosity of the system.

arXiv:1103.4869•Physical Review A•Jul 28, 2011•10 citations

Dissipative fluid dynamics for the dilute Fermi gas at unitarity: Free expansion and rotation

Thomas Schäfer

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 $P(n,T)$ as long as the universal relation $P=\frac{2}{3}\mathcal{E}$ is satisfied; (iv) the expansion dynamics mainly constrains the cloud average of the shear viscosity.

arXiv:1008.3876•Physical Review A•Dec 27, 2010•27 citations

Medium effects and the shear viscosity of the dilute Fermi gas away from the conformal limit

Marcus Bluhm, Thomas Schäfer

We study the shear viscosity of a dilute Fermi gas as a function of the scattering length in the vicinity of the unitarity limit. The calculation is based on kinetic theory, which provides a systematic approach to transport properties in the limit in which the fugacity $z=n{\ensuremath{\lambda}}^{3}/2$ is small. Here, $n$ is the density of the gas and $\ensuremath{\lambda}$ is the thermal wavelength of the fermions. At leading order in the fugacity expansion, the shear viscosity is independent of density, and the minimum shear viscosity is achieved at unitarity. At the next order, medium effects modify the scattering amplitude as well as the quasiparticle energy and velocity. We show that these effects shift the minimum of the shear viscosity to the Bose-Einstein condensation side of the resonance, in agreement with the result of recent experiments.

arXiv:1410.2827•Physical Review A•Dec 8, 2014•11 citations
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 \ensuremathτR=\ensuremathη/P{\ensuremath{\tau}}_{R}=\ensuremath{\eta}/P\ensuremathτR​=\ensuremathη/P, where \ensuremathτR{\ensuremath{\tau}}_{R}\ensuremathτR​ is the shear stress relaxation time, \ensuremathη\ensuremath{\eta}\ensuremathη 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 compute the complete set of second-order transport coefficients of the unitary Fermi gas, a dilute gas of spin-1/2 particles interacting via an sss-wave interaction tuned to infinite scattering length. The calculation is based on kinetic theory and the Chapman-Enskog method at second order in the Knudsen expansion. We take into account the exact two-body collision integral. We extend previous results on second-order coefficients related to shear stress by including terms related to heat flow and gradients of the fugacity. We confirm that the thermal relaxation time is given by the simple estimate ${\ensuremath{\tau}}_{\ensuremath{\kappa}}=\ensuremath{\kappa}m/({c}_{P}T)$ even if the full collision kernel is taken into account. Here, $\ensuremath{\kappa}$ is the thermal conductivity, mmm is the mass of the particles, cP{c}_{P}cP​ is the specific heat at constant pressure, and TTT is the temperature.
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.
Using the $\ensuremath{\varepsilon}$-expansion technique, we compute $\ensuremath{\eta}/s$, where $\ensuremath{\eta}$ is the shear viscosity and sss is the entropy density of the normal phase of unitary Fermi gas in $d=4\ensuremath{-}\ensuremath{\varepsilon}$ dimensions to leading order (LO) in $\ensuremath{\varepsilon}$. We use the kinetic theory approach and solve transport equations for a medium perturbed by a shear hydrodynamic flow. The collision integrals are calculated to ${\ensuremath{\varepsilon}}^{2}$, which is LO. The LO result is temperature independent with $\ensuremath{\eta}/s\ensuremath{\simeq}(0.11/{\ensuremath{\varepsilon}}^{2})(\ensuremath{\hbar}/{k}_{B})$. The d=3d=3d=3 prediction for $\ensuremath{\eta}/s$ exceeds the $\ensuremath{\hbar}/4\ensuremath{\pi}{k}_{B}$ bound by a factor of about 1.41.41.4.
We use the Boltzmann equation in the relaxation-time approximation to study the expansion of a dilute Fermi gas at unitarity. We focus, in particular, on the approach to the hydrodynamic limit. Our main findings are the following: (i) In the regime that has been studied experimentally hydrodynamic effects beyond the Navier-Stokes approximation are small. (ii) Mean-field corrections to the Boltzmann equation are not important. (iii) Experimental data imply that freezeout occurs very late, which means that the relaxation time remains smaller than the expansion time during the entire evolution of the system. (iv) The experimental results also imply that the bulk viscosity is significantly smaller than the shear viscosity of the system.
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.
We study the shear viscosity of a dilute Fermi gas as a function of the scattering length in the vicinity of the unitarity limit. The calculation is based on kinetic theory, which provides a systematic approach to transport properties in the limit in which the fugacity $z=n{\ensuremath{\lambda}}^{3}/2$ is small. Here, nnn is the density of the gas and $\ensuremath{\lambda}$ is the thermal wavelength of the fermions. At leading order in the fugacity expansion, the shear viscosity is independent of density, and the minimum shear viscosity is achieved at unitarity. At the next order, medium effects modify the scattering amplitude as well as the quasiparticle energy and velocity. We show that these effects shift the minimum of the shear viscosity to the Bose-Einstein condensation side of the resonance, in agreement with the result of recent experiments.