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 C12 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 C12 collisions both in the kinetic equation describing the thermal atoms and also in the source term Γ12 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 ζi.