In Chapter 17, we derived the Landau–Khalatnikov two-fluid hydrodynamics which describes the collision-dominated region of a trapped Bose condensate interacting with a thermal cloud. In this chapter, we use these equations to discuss the damping of the hydrodynamic collective modes in a trapped Bose gas at finite temperatures. We derive variational expressions based on these equations for both the frequency and the damping of collective modes. This extends the analysis in Chapter 16 in which a variational approach was developed to calculate the hydrodynamic two-fluid oscillation frequencies in the non-dissipative limit. A novel feature of our treatment is the introduction of frequency-dependent transport coefficients, which produce a natural cutoff eliminating the collisionless region in the low-density tail of the thermal cloud. Our expression for the damping in trapped superfluid Bose gases is a natural generalization of the approach used by Landau and Lifshitz (1959) for uniform classical fluids. This chapter is mainly based on Nikuni and Griffin (2004). In Chapters 15 and 17, we derived a closed set of two-fluid hydrodynamic equations for a trapped Bose-condensed gas starting from the simplified microscopic model describing the coupled dynamics of the condensate and noncondensate atoms given in Chapter 3. These hydrodynamic two-fluid equations include dissipative terms associated with the shear viscosity, the thermal conductivity and the four second viscosity coefficients. Explicit formulas for these transport coefficients were derived in Section 18.1. Our goal in this chapter is to find a general expression for the damping of the two-fluid modes in terms of these transport coefficients. We emphasize that the damping of hydrodynamic two-fluid oscillations is completely different in nature from the Landau and Beliaev damping of oscillations in the collisionless region which is treated in Chapters 12 and 13.
In Chapter 17, we derived the Landau–Khalatnikov two-fluid hydrodynamics which describes the collision-dominated region of a trapped Bose condensate interacting with a thermal cloud. In this chapter, we use these equations to discuss the damping of the hydrodynamic collective modes in a trapped Bose gas at finite temperatures. We derive variational expressions based on these equations for both the frequency and the damping of collective modes. This extends the analysis in Chapter 16 in which a variational approach was developed to calculate the hydrodynamic two-fluid oscillation frequencies in the non-dissipative limit. A novel feature of our treatment is the introduction of frequency-dependent transport coefficients, which produce a natural cutoff eliminating the collisionless region in the low-density tail of the thermal cloud. Our expression for the damping in trapped superfluid Bose gases is a natural generalization of the approach used by Landau and Lifshitz (1959) for uniform classical fluids. This chapter is mainly based on Nikuni and Griffin (2004). In Chapters 15 and 17, we derived a closed set of two-fluid hydrodynamic equations for a trapped Bose-condensed gas starting from the simplified microscopic model describing the coupled dynamics of the condensate and noncondensate atoms given in Chapter 3. These hydrodynamic two-fluid equations include dissipative terms associated with the shear viscosity, the thermal conductivity and the four second viscosity coefficients. Explicit formulas for these transport coefficients were derived in Section 18.1. Our goal in this chapter is to find a general expression for the damping of the two-fluid modes in terms of these transport coefficients. We emphasize that the damping of hydrodynamic two-fluid oscillations is completely different in nature from the Landau and Beliaev damping of oscillations in the collisionless region which is treated in Chapters 12 and 13.
Recently it was shown that the Landau-Khalatnikov two-fluid hydrodynamics describes the collision-dominated region of a trapped Bose condensate interacting with a thermal cloud. We use these equations to discuss the low frequency hydrodynamic collective modes in a trapped Bose gas at finite temperatures. We derive variational expressions based on these equations for both the frequency and damping of collective modes. A new feature is our use of frequency-dependent transport coefficients, which produce a natural cutoff by eliminating the collisionless low-density tail of the thermal cloud. Above the superfluid transition, our expression for the damping in trapped inhomogeneous gases is analogous to the result first obtained by Landau and Lifshitz for uniform classical fluids. We also use the moment method to discuss the crossover from the collisionless to the hydrodynamic region. Recent data for the monopole-quadrupole mode in the hydrodynamic region of a trapped gas of metastable 4He is discussed. We also present calculations for the damping of the analogous m=0 monopole-quadrupole condensate mode in the superfluid phase.
In Chapter 15, we showed that in the limit of short collision times the coupled equations of motion for the condensate and noncondensate atoms lead to Landau's non-dissipative two-fluid hydrodynamics. However the approach used in Chapter 15 was not based on a small expansion parameter, in contrast with the more systematic Chapman–Enskog procedure used to derive hydrodynamic damping in the kinetic theory of classical gases. In the present chapter, we generalize the procedure of Chapter 15 to trapped Bose-condensed gases, in order to derive two-fluid hydrodynamic equations that include dissipation due to transport processes. We solve the kinetic equation by expanding the nonequilibrium single-particle distribution function f ( p, r , t ) around the distribution function f (0) ( p, r , t ) that describes complete local equilibrium between the condensate and the noncondensate components. All hydrodynamic damping effects are included by taking into account deviations from the local equilibrium distribution function f (0) . Our discussion for a trapped Bose gas is a natural extension of the pioneering work of Kirkpatrick and Dorfman (1983, 1985a) for a uniform Bose-condensed gas. This chapter is mainly based on their work as well as on Nikuni and Griffin (2001a,b). We will prove that, with appropriate definitions of various thermodynamic variables, our two-fluid hydrodynamic equations including damping have precisely the structure of those first derived by Landau and Khalatnikov for superfluid He. In particular, the damping associated with the collisional exchange of atoms between the condensate and noncondensate components, which is discussed at length in Chapter 15, is now expressed in terms of frequency-dependent second viscosity coefficients. This special type of damping is a characteristic signature of a dilute Bose superfluid and exists in addition to the hydrodynamic damping associated with the shear viscosity and thermal conductivity of the normal fluid.
Starting from the quantum kinetic equation for the noncondensate atoms and the generalized Gross--Pitaevskii equation for the condensate, we derive the two-fluid hydrodynamic equations of a trapped Bose gas at finite temperatures. We follow the standard Chapman--Enskog procedure, starting from a solution of the kinetic equation corresponding to the complete local equilibrium between the condensate and the noncondensate components. Our hydrodynamic equations are shown to reduce to a form identical to the well-known Landau-Khalatnikov two-fluid equations, with hydrodynamic damping due to the deviation from local equilibrium. The deviation from local equilibrium within the thermal cloud gives rise to dissipation associated with shear viscosity and thermal conduction. In addition, we show that effects due to the deviation from the diffusive local equilibrium between the condensate and the noncondensate (recently considered by Zaremba, Nikuni, and Griffin) can be described by four frequency-dependent second viscosity transport coefficients. We also derive explicit formulas for all the transport coefficients. These results are used to introduce two new characteristic relaxation times associated with hydrodynamic damping. These relaxation times give the rate at which local equilibrium is reached and hence determine whether one is in the two-fluid hydrodynamic region.
In the collisional region at finite temperatures, the collective modes of superfluids are described by the Landau two-fluid hydrodynamic equations reviewed in Chapter 14. In the case of trapped Bose gases, these are coupled differential equations with position-dependent coefficients associated with the local thermodynamic functions. Building on the approach initiated by Zaremba et al. (1999) for trapped atomic Bose gases, in this chapter we develop an alternative variational formulation of two-fluid hydrodynamics. This is based on the work of Zilsel (1950), originally developed to deal with superfluid He. Assuming a simple variational ansatz for the superfluid and normal fluid velocity fields, this approach reduces the problem of finding the hydrodynamic collective mode frequencies to solving coupled algebraic equations for a few variational parameters. These equations contain constants that involve spatial integrals over various equilibrium thermodynamic derivatives. Such a variational approach is both simpler and more physical than a direct attempt to solve the Landau two-fluid equations numerically. This chapter is mainly based on Taylor and Griffin (2005), Taylor (2008) and Zilsel (1950). In it, we discuss the normal modes of the non-dissipative Landau two-fluid equations for a trapped superfluid. In Section 16.3, we illustrate this formalism by deriving expressions for the frequencies of the dipole and breathing modes of a trapped Bose superfluid. In Chapters 17 and 18, we discuss an extended version of the two-fluid equations that includes hydrodynamic damping. The hydrodynamic damping of the collective modes is calculated in Chapter 19 using a generalized version of the variational approach developed in this chapter. Zilsel's variational formulation Since two-fluid hydrodynamics only describes a system in local equilibrium, all thermodynamic quantities are functions of position and time. Even in static equilibrium, in the presence of a trapping potential, most thermodynamic quantities will be position dependent.
We extend our recent work on the two-fluid hydrodynamics of the condensate and noncondensate in a trapped Bose gas by including the dissipation associated with viscosity and thermal conduction in the thermal cloud. For purposes of illustration, we consider the hydrodynamic modes in the case of a uniform Bose gas. A finite thermal conductivity and shear viscosity give rise to a damping of the first and second sound modes, in addition to the damping found previously due to the lack of diffusive equilibrium between the condensate and noncondensate. The relaxational mode associated with this equilibration process is strongly coupled to thermal fluctuations and reduces to the usual thermal diffusion mode above the Bose-Einstein transition. In contrast to the standard Landau two-fluid hydrodynamics, we predict a damped mode centered at zero frequency, in addition to the usual second sound doublet.PACS Nos.: 03.75.Fi, 05.30Jp, 67.40.Db
This is based on 4 lectures given at the 13th Australian Physics Summer School, Australia National University, Canberra, Jan 17-28, 2000. The main topic is the theory of collective modes in a trapped Bose gas at finite temperatures. A generalized Gross-Pitaevskii equation is derived at finite temperatures, which is used to discuss a new mechanism for damping in the collisionless region arising from interactions with a static thermal cloud of non-condensate atoms. Next, introducing a kinetic equation for the thermal cloud, we derive two-fluid equations of motion for the condensate and non-condensate components in the collision-dominated hydrodynamic region. We show that these are precisely the equivalent of the Landau two-fluid equations in the limit that the two components are in diffusive local equilibrium. However, our equations also predict the existence of a new zero frequency relaxational mode, in addition to the usual Landau hydrodynamic modes (such as first and second sound). The special importance and simplicity of two-fluid hydrodynamics is stressed.