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arXiv:1207.6855 · Physical Review E

Transport coefficients from the boson Uehling-Uhlenbeck equation

Erich D. Gust, L. E. Reichl

Published Apr 15, 2013• 11 citations
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Abstract

Expressions for the bulk viscosity, shear viscosity, and thermal conductivity of a quantum degenerate Bose gas above the critical temperature for Bose-Einstein condensation are derived using the Uehling-Uhlenbeck kinetic equation. For contact potentials and hard sphere interactions, the eigenvalues (relaxation rates) of the Uehling-Uhlenbeck collision operator have an upper cutoff. This cutoff requires summation over all discrete eigenvalues and eigenvectors of the collision operator when computing transport coefficients. We numerically compute the shear viscosity and thermal conductivity for any boson gas that interacts via a contact potential. We find that the bulk viscosity of the degenerate boson gas remains identically zero, as it is for the classical gas.

DOI
10.1103/physreve.87.042109

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Cambridge University Press eBooks•Feb 19, 2009•0 citations
Expressions for the bulk viscosity, shear viscosity, and thermal conductivity of a quantum degenerate Bose gas above the critical temperature for Bose-Einstein condensation are derived using the Uehling-Uhlenbeck kinetic equation. For contact potentials and hard sphere interactions, the eigenvalues (relaxation rates) of the Uehling-Uhlenbeck collision operator have an upper cutoff. This cutoff requires summation over all discrete eigenvalues and eigenvectors of the collision operator when computing transport coefficients. We numerically compute the shear viscosity and thermal conductivity for any boson gas that interacts via a contact potential. We find that the bulk viscosity of the degenerate boson gas remains identically zero, as it is for the classical gas.
We linearize the Uehling-Uhlenbeck equation for bosonic gases close to thermal equilibrium under the assumption of a contact interaction characterized by a scattering length a. We show that the spectrum of relaxation rates is similar to that of a classical hard-sphere gas. However, the relaxation rates show a significant dependence on the fugacity z of the gas, increasing by as much as 60% of their classical value for z approaching 1. The relaxation modes are also significantly altered at higher values of z. The relaxation rates and modes are determined by the eigenvalues and eigenvectors of a Fredholm integral operator of the second kind. We derive an analytical form for the kernel of this operator and present numerical results for the first few eigenvalues and eigenvectors.
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.
We obtain microscopic expressions for the six hydrodynamic modes of a dilute Bose-Einstein condensate: two transverse (shear) modes and four longitudinal modes corresponding to the first sound (elastic waves) and second sound (temperature waves). Our microscopic expressions include both the speed of the two types of sound and the rate of relaxation of the sound waves. We obtain numerical values for the shear viscosity of a dilute BEC composed of bosons which interact via a contact potential. Our values for the shear viscosity are obtained using the eigenvalues and eigenvectors of the three types of collision operators that govern the relaxation of the condensate.
Griffin, Wu and Stringari have derived the hydrodynamic equations of a trapped dilute Bose gas above the Bose-Einstein transition temperature. We give the extension which includes hydrodynamic damping, following the classic work of Uehling and Uhlenbeck based on the Chapman-Enskog procedure. Our final result is a closed equation for the velocity fluctuations δv which includes the hydrodynamic damping due to the shear viscosity η and the thermal conductivity κ. Following Kavoulakis, Pethick and Smith, we introduce a spatial cutoff in our linearized equations when the density is so low that the hydrodynamic description breaks down. Explicit expressions are given for η and κ, which are position-dependent through dependence on the local fugacity when one includes the effect of quantum degeneracy of the trapped gas. We also discuss a trapped Bose-condensed gas, generalizing the work of Zaremba, Griffin and Nikuni to include hydrodynamic damping due to the (non-condensate) normal fluid.
Griffin, Wu and Stringari have derived the hydrodynamic equations of a trapped dilute Bose gas above the Bose-Einstein transition temperature. We give the extension which includes hydrodynamic damping, following the classic work of Uehling and Uhlenbeck based on the Chapman-Enskog procedure. Our final result is a closed equation for the velocity fluctuations δvδvδv which includes the hydrodynamic damping due to the shear viscosity ηηη and the thermal conductivity κκκ. Following Kavoulakis, Pethick and Smith, we introduce a spatial cutoff in our linearized equations when the density is so low that the hydrodynamic description breaks down. Explicit expressions are given for ηηη and κκκ, which are position-dependent through dependence on the local fugacity when one includes the effect of quantum degeneracy of the trapped gas. We also discuss a trapped Bose-condensed gas, generalizing the work of Zaremba, Griffin and Nikuni to include hydrodynamic damping due to the (non-condensate) normal fluid.
In Chapter 17, we derived two-fluid hydrodynamic equations that include damping related to transport coefficients. Our entire analysis was based on the coupled ZNG equations for the condensate and in the thermal cloud. These involved a generalized GP equation for the condensate and a kinetic equation for the thermal atoms. A crucial role is played by the C 12 collision term in the kinetic equation, which describes the interactions between atoms in the condensate and in the thermal cloud. Our analysis of the deviation from the diffusive local equilibrium solution of the kinetic equation was based on the Chapman–Enskog approach, extensively developed for classical gases and first applied to Bose-condensed gases by Kirkpatrick and Dorfman (1983, 1985a). This approach required a careful treatment of the novel feature relating to the C 12 collisions both in the kinetic equation describing the thermal atoms and also in the source term Γ 12 in the generalized GP equation for the condensate. Using the Chapman– Enskog approach to solve the kinetic equation for a trapped Bose gas, we obtained explicit expressions for the function ψ( p, r , t ) that describes the deviation from diffusive local equilibrium, as defined by (17.25) and (17.39). This deviation can be related to various transport coefficients, as discussed in Chapter 17. These transport coefficients are determined by the solutions of the three integral equations (17.40)–(17.42) for the three contributions to the deviation function ψ( p, r , t ) in (17.39). In Section 18.1, we will solve these integral equations and obtain explicit expressions for the thermal conductivity k , the shear viscosity η and the four second viscosity coefficients ζ i .