We compute the viscosity spectral function of the dilute Fermi gas for different values of the s-wave scattering length $a$, including the unitarity limit $a\to\infty$. We perform the calculation in kinetic theory by studying the response to a non-trivial background metric. We find the expected structure consisting of a diffusive peak in the transverse shear channel and a sound peak in the longitudinal channel. At zero momentum the width of the diffusive peak is $ω_0\simeq (2ε)/(3η)$ where $ε$ is the energy density and $η$ is the shear viscosity. At finite momentum the spectral function approaches the collisionless limit and the width is of order $ω_0\sim k(T/m)^{1/2}$.
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 $z=nλ^3/2$ is small. Here, $n$ is the density of the gas and $λ$ is the thermal wave length 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 …
We confirm and expand on work by Chafin and Schaefer on hydrodynamic fluctuations in the unitary Fermi gas. Using the result for the equation of state from a recent MIT experiment, we derive lower bounds for η/n and η/s as a function of temperature. Re-analyzing recent quantum Monte Carlo data for the shear-viscosity spectral function we point out a possible resolution for the tension between the viscosity bound η/n> 0.3 from Chafin and Schaefer and the quantum Monte Carlo results η/n<0.2$ from Wlazlowski et al. near the critical temperature
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 $(P-\frac{2}{3}{\cal E})/P$, where $P$ is the pressure and ${\cal E}$ is the energy density. In the high temperature limit this ratio scales as $\frac{zλ}{a}$, where $z$ is the fugacity, $λ$ is the thermal wave length, and $a$ is the scattering length. We show that the bulk viscosity $ζ$ scales as the second power of this parameter, $ζ\sim (\frac{zλ}{a})^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 stat…
We study hydrodynamic fluctuations in a non-relativistic 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, $η/s \geq \hbar/(4πk_B)$, where $s$ is the entropy density. For the dilute Fermi gas at unitarity we find $η/s\gsim 0.2\hbar$. This bound is not universal -- it depends on thermodynamic properties of the unitary Fermi gas, and on empirical information about the range of validity of…
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 non-hydrodynamic 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…
We study two observables related to the anomalous breaking of scale invariance in a dilute two dimensional Fermi gas, the frequency shift and damping rate of the monopole mode in a harmonic confinement potential. For this purpose we compute the speed of sound and the bulk viscosity of the two dimensional gas in the high temperature limit. We show that the anomaly in the speed of sound scales as $(2P-ρc_s^2)/P\sim z/[\log(T/E_B)]^2$, and that the bulk viscosity $ζ$ scales as $ζ/η\sim z^2/[\log(T/E_B)]^6$. Here, $P$ is the pressure, $c_s^2$ is the speed of sound, $η$ is the shear viscosity, $z$ …
We review recent work on a kinetic model for very low temperature dilute Bose gases. The brief derivation, expressions for hydrodynamics modes, and the comparison with a experiment on a BEC of $^{87}{\rm Rb}$ atoms are presented.
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 expansi…
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 finding are: 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, that means that the relaxation time remains smaller than the expansion time during the entire evolution of the system, iv) th…
We study the dilute Fermi gas at unitarity using molecular dynamics with an effective quantum potential constructed to reproduce the quantum two-body density matrix at unitarity. Results for the equation of state, the pair correlation function and the shear viscosity are presented. These quantities are well understood in the dilute, high temperature, limit. Using molecular dynamics we determine higher order corrections in the diluteness parameter $nλ^3$, where $n$ is the density and $λ$ is the thermal de Broglie wave length. In the case of the contact density, which parameterizes the short dis…
We compute the thermal conductivity and sound attenuation length of a dilute atomic Fermi gas in the framework of kinetic theory. Above the critical temperature for superfluidity, T_c, the quasi-particles are fermions, whereas below T_c, the dominant excitations are phonons. We calculate the thermal conductivity in both cases. We find that at unitarity the thermal conductivity κin the normal phase scales as κ~ T^{3/2}. In the superfluid phase we find κ~ T^{2}. At high temperature the Prandtl number, the ratio of the momentum and thermal diffusion constants, is 2/3. The ratio increases as the t…
We describe recent attempts to extract the shear viscosity of the dilute Fermi gas at unitarity from experiments involving scaling flows. A scaling flow is a solution of the hydrodynamic equations that preserves the shape of the density distribution. The scaling flows that have been explored in the laboratory are the transverse expansion from a deformed trap ("elliptic flow"), the expansion from a rotating trap, and collective oscillations. We discuss advantages and disadvantages of the different experiments, and point to improvements of the theoretical analysis that are needed in order to ach…
We compute the shear viscosity of a two dimensional Fermi gas interacting via a short range potential with scattering length $a_{2d}$ in kinetic theory. We find that kinetic theory predicts that the shear viscosity to entropy density ratio of a strongly interacting two dimensional gas is comparable to that of the three dimensional unitary gas. We use our results to compute the damping of collective modes in a trapped Fermi gas, and compare to experimental data recently obtained in E. Vogt et al., arXiv:1111.1173.
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 corr…
We have considered the orbital-free approximation of the kinetic energy functional to investigate the zero temperature properties of dilute harmonically trapped two component Fermi gas at unitarity. It is shown that our approach provides a realible and inexpensive method to study superfluid strongly interacting dilute Fermi gases.
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, non-linear terms in the strain-stress relation, and non-linear couplings between vorticity and strain. An exact calculation in the dilute limit gives $τ_R=η/P$, where $τ_R$ is the shear stress relaxation time, $η$ is the shear viscosity, and $P$ is pressure. This relation is identical to the result obtained using the B…
We present an ab initio calculation of the shear viscosity as a function of interaction strength in a two-component unpolarized Fermi gas near the unitary limit, within a finite temperature quantum Monte Carlo (QMC) framework and using the Kubo linear-response formalism. The shear viscosity decreases as we tune the interaction strength 1/ak_F from the Bardeen-Cooper-Schrieffer side of the Feshbach resonance towards Bose-Einstein condensation limit and it acquires the smallest value for 1/ak_F approx 0.4, with a minimum value of (eta/s)_min approx 0.2 hbar/k_B, which is about twice as small as …
We derive the mean field kinetic equation for the momentum distribution of Bogoliubov excitations (bogolons) in a spatially uniform Bose-Einstein condensate (BEC), with a focus on the collision integrals. We use the method of Peletminksii and Yatsenko rather than the standard non-equilibrium Green's function formalism. This method produces three collision integrals ${\cal G}^{12}$, ${\cal G}^{22}$ and ${\cal G}^{31}$. Only ${\cal G}^{12}$ and ${\cal G}^{22}$ have been considered by previous authors. The third collision integral ${\cal G}^{31}$ contains the effects of processes where one bogolo…
We develop a microscopic approach to the consistent construction of the kinetic theory of dilute weakly ionized gases of hydrogen-like atoms. The approach is based on the framework of the second quantization method in the presence of bound states of particles and the method of reduced description of relaxation processes. Within the approach we developed the first-order perturbation theory over the weak interaction for a system of kinetic equations for the Wigner distribution functions of free fermions of both kinds and their bound states, the hydrogen-like atoms. It is shown that the condition…
The numerical method toward the fast and accurate calculation of the dilute quantum gas is studied by proposing the Uehing-Uhlenbeck (U-U) model equation. In particular, the direct simulation Monte Carlo (DSMC) method is used to solve the U-U model equation. The DSMC analysis of the U-U model equation surely enables us to obtain the accurate thermalization using a small number of sample particles and calculate the dilute quantum gas dynamics in practical time. Finally, the availability of the DSMC analysis of the U-U model equation toward the fast and accurate calculation of the dilute quantum…
