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arXiv:0902.2384 · WORLD SCIENTIFIC eBooks

Transport Properties of a Non-Relativistic Delta-Shell Gas with Long Scattering Lengths

Sergey Postnikov, Madappa Prakash

Published Jul 1, 2011• 1 citation
arXivHTMLPDF

Abstract

The coefficients of diffusion, thermal conductivity, and shear viscosity are calculated for a system of non-relativistic particles interacting via a delta-shell potential V(r)=-v δ(r-R) when the average distance between particles is smaller than R. The roles of resonances and long scattering lengths including the unitary limit are examined. Results for ratios of diffusion to viscosity and viscosity to entropy density are presented for varying scattering lengths.

DOI
10.1142/9789814329880_0016

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The coefficients of diffusion, thermal conductivity, and shear viscosity are calculated for a system of non-relativistic particles interacting via a delta-shell potential V(r)=-v δ(r-R) when the average distance between particles is smaller than R. The roles of resonances and long scattering lengths including the unitary limit are examined. Results for ratios of diffusion to viscosity and viscosity to entropy density are presented for varying scattering lengths.
This work extends the seminal work of Gottfried on the two-body quantum physics of particles interacting through a delta-shell potential to many-body physics by studying a system of nonrelativistic particles when the thermal De-Broglie wavelength of a particle is larger than the range of the potential and the density is such that average distance between particles is larger than the above range. The ability of the delta-shell potential to reproduce some basic properties of the deuteron are examined. Relations for moments of bound-states are derived. The virial expansion is used to calculate the first quantum correction to the ideal gas pressure in the form of the second virial coefficient. Additionally, all thermodynamical functions are calculated up to the first-order quantum corrections. For small departures from equilibrium, the net flows of mass, energy and momentum, characterized by the coefficients of diffusion, thermal conductivity and shear viscosity, respectively, are calculated. Properties of the gas are examined for various values of physical parameters including the case of infinite scattering length when the unitary limit is achieved.
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.
We derive the Boltzmann equation for relaxation rate, viscosity and thermal conductivity in ultra cold normal dipolar bose gas. We use anisotropic dipole- dipole interaction. We obtain the temperature dependence of the relaxation rate and give formal expressions for the shear viscosity and thermal conductivity coefficients. These coefficients are proportional to . The different form of potentials in calculations show themselves in the coefficients of the viscosity and thermal conductivity. The ratio of the viscosity coefficient for dipole-dipole interaction to contact potential is three order and this ratio for thermal conductivity is two order.
The scattering properties of spin-polarized Fermi gases are dominated by ppp-wave interactions. Besides their inherent angular dependence, these interactions differ from their sss-wave counterparts as they also require the presence of a finite effective range in order to understand the low-energy properties of the system. In this article we examine how the shear viscosity and thermal conductivity of a three-dimensional spin-polarized Fermi gas in the normal phase depend on the effective range and the scattering volume in both the weakly and the strongly interacting limits. We show that, although the shear viscosity and the thermal conductivity both explicitly depend on the effective range near resonance, the Prandtl number which parametrizes the ratio of momentum to thermal diffusivity does not have an explicit interaction dependence both at resonance and for weak interactions in the low-energy limit. In contrast to sss-wave systems, ppp-wave scattering exhibits an additional resonance at weak attraction from a quasi-bound state at positive energies, which leads to a pronounced dip in the shear viscosity at specific temperatures.
Feshbach resonances of trapped ultracold alkali-metal atoms allow to vary the atomic scattering length aaa. At very large values of aaa the system enters an universal strongly coupled regime in which its properties---the ground-state energy, pressure, etc.---become independent of aaa. We discuss the transport properties of such systems. In particular, the universality arguments imply that the shear viscosity of ultracold Fermi atoms at the Feschbach resonance is proportional to the particle number density nnn and the Plank constant $\ensuremath{\hbar}$: $\ensuremath{\eta}=\ensuremath{\hbar}n{\ensuremath{\alpha}}_{\ensuremath{\eta}}$, where ${\ensuremath{\alpha}}_{\ensuremath{\eta}}$ is a universal constant. Using Heisenberg uncertainty principle and Einstein's relation between diffusion and viscosity we argue that the viscosity has the lower bound given by ${\ensuremath{\alpha}}_{\ensuremath{\eta}}\ensuremath{\leqslant}{(6\ensuremath{\pi})}^{\ensuremath{-}1}$. We relate the damping of low-frequency density oscillations of ultracold optically trapped 6Li^{6}\mathrm{Li}6Li atoms to viscosity and find that the value of the coefficient ${\ensuremath{\alpha}}_{\ensuremath{\eta}}$ is about 0.3. We also show that such a small viscosity cannot be explained by kinetic theory based on binary scattering. We conclude that the system of ultracold atoms near the Feshbach resonance is a near-ideal liquid.