We give a detailed derivation of the second-order (local) hydrodynamics for Boltzmann equation with an external force by using the renormalization group method. In this method, we solve the Boltzmann equation faithfully to extract the hydrodynamics without recourse to any ansatz. Our method leads to microscopic expressions of not only all the transport coefficients that are of the same form as those in Chapman-Enskog method but also those of the viscous relaxation times
τi that admit physically natural interpretations. As an example, we apply our microscopic expressions to calculate the transport coefficients and the relaxation times of the cold fermionic atoms in a quantitative way, where the transition probability in the collision term is given explicitly in terms of the
s-wave scattering length
as. We thereby discuss the quantum statistical effects, temperature dependence, and scattering-length dependence of the first-order transport coefficients and the viscous relaxation times: It is shown that as the temperature is lowered, the transport coefficients and the relaxation times increase rapidly because Pauli principle acts effectively. On the other hand, as
as is increased, these quantities decrease and become vanishingly small at unitarity because of the strong coupling. The numerical calculation shows that the relation
τπ=η/P, which is derived in the relaxation-time approximation and used in most of literature without almost any foundation, turns out to be satisfied quite well, while the similar relation for the relaxation time
τJ of the heat conductivity is satisfied only approximately with a considerable error.
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 .
Nonequilibrium Green's functions provide an efficient way to describe the evolution of the energy-momentum tensor during the early-time preequilibrium stage of high-energy heavy-ion collisions. Besides their practical relevance they also provide a meaningful way to address the question when and to what extent a hydrodynamic description of the system becomes applicable. Within the kinetic theory framework we derive a new method to calculate time-dependent nonequilibrium Green's functions describing the evolution of energy and momentum perturbations on top of an evolving far-from-equilibrium background. We discuss the approach toward viscous hydrodynamics along with the emergence of various scaling phenomena for conformal systems. By comparing our results obtained in the relaxation time approximation to previous calculations in Yang-Mills kinetic theory, we further address the question which macroscopic features of the energy-momentum tensor are sensitive to the underlying microscopic dynamics.
At low temperatures the De Broglie wavelength of the gas particles becomes on the order of their average separation, and the effects of their indistinguishability become important. In the absence of a phase transition in the gas, the quantum mechanical Wigner distribution function for a dilute gas of fermions or bosons satisfies the Uehling-Uhlenbeck equation. This equation satisfies an H- theorem with equilibrium solutions being ideal boson or ideal fermion distributions. Navier-Stokes equations can be derived by standard methods. A low temperature gas of weakly interacting bosons undergoes a Bose-Einstein condensation with a macroscopically occupied ground state. A different approach is required to describe the non-equilibrium processes in such a situation. A kinetic equation can be derived for the Bogoliubov excitations in the gas at very low temperatures. The associated hydrodynamic equations are the Landau-Khaltnikov, two fluid equations, and explicit expressions are obtained for the six associated transport coefficients.