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arXiv:1207.5068 · Physical Review A

Quasiclassical molecular dynamics for the dilute Fermi gas at unitarity

Kevin Dusling, Thomas Schäfer

Published Dec 26, 2012• 2 citations
arXivHTMLPDF

Abstract

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 $n{\ensuremath{\lambda}}^{3}$, where $n$ is the density and $\ensuremath{\lambda}$ is the thermal de Broglie wavelength. In the case of the contact density, which parameterizes the short distance behavior of the correlation function, we find that the results of molecular dynamics interpolate between the truncated second- and third-order virial expansion and are in excellent agreement with existing $T$-matrix calculations. For the shear viscosity we reproduce the expected scaling behavior at high temperature, $\ensuremath{\eta}\ensuremath{\sim}1/{\ensuremath{\lambda}}^{3}$, and we determine the leading density-dependent correction to this result.

DOI
10.1103/physreva.86.063634

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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 n\ensuremathλ3n{\ensuremath{\lambda}}^{3}n\ensuremathλ3, where nnn is the density and \ensuremathλ\ensuremath{\lambda}\ensuremathλ is the thermal de Broglie wavelength. In the case of the contact density, which parameterizes the short distance behavior of the correlation function, we find that the results of molecular dynamics interpolate between the truncated second- and third-order virial expansion and are in excellent agreement with existing TTT-matrix calculations. For the shear viscosity we reproduce the expected scaling behavior at high temperature, \ensuremathη\ensuremath∼1/\ensuremathλ3\ensuremath{\eta}\ensuremath{\sim}1/{\ensuremath{\lambda}}^{3}\ensuremathη\ensuremath∼1/\ensuremathλ3, and we determine the leading density-dependent correction to this result.
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.
The recently discovered universal thermodynamic behavior of dilute, strongly interacting Fermi gases also implies a universal structure in the many-body pair-correlation function at short distances, as quantified by the contact . Here, we theoretically calculate the temperature dependence of this universal contact for a Fermi gas in free space and in a harmonic trap. At high temperatures above the Fermi degeneracy temperature, T ≳ T F , we obtain a reliable non-perturbative quantum virial expansion up to third order. At low temperatures, we compare different approximate strong-coupling theories. These make different predictions, which need to be tested either by future experiments or by advanced quantum Monte Carlo simulations. We conjecture that in the universal unitarity limit, the contact or correlation decreases monotonically with increasing temperature, unless the temperature is significantly lower than the critical temperature, T ≪ T c ∼0.2 T F . We also discuss briefly how to measure the universal contact in either homogeneous or harmonically trapped 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, 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.
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, $\ensuremath{\zeta}\ensuremath{\sim}(z\ensuremath{\lambda}/a{)}^{2}{\ensuremath{\lambda}}^{\ensuremath{-}3}$.
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 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.