This chapter considers the behavior of the superfluid Bose liquid at finite temperatures. The thermally-excited quasi-particles are identified with the normal fluid. In general, their motion will be irreversible and may be described, in suitable limit, with the aid of a coefficient of viscosity. Consideration of the rotating bucket experiment provides such a prescription, both experimentally and theoretically. Since condensate motion is irrotational, the superfluid component will not respond to rotation of its container. Collisions with the walls thus represent the physical mechanism by which the normal fluid is singled out, the superfluid component being unaffected by the presence of the pipe. According to the two-fluid model, the velocity of the normal fluid vanishes, since it is "attached" to the walls of the pipe by viscous effects. As a consequence of superfluid flow, the thermal quasi-particle distribution is asymmetric in momentum space.
This chapter considers the behavior of the superfluid Bose liquid at finite temperatures. The thermally-excited quasi-particles are identified with the normal fluid. In general, their motion will be irreversible and may be described, in suitable limit, with the aid of a coefficient of viscosity. Consideration of the rotating bucket experiment provides such a prescription, both experimentally and theoretically. Since condensate motion is irrotational, the superfluid component will not respond to rotation of its container. Collisions with the walls thus represent the physical mechanism by which the normal fluid is singled out, the superfluid component being unaffected by the presence of the pipe. According to the two-fluid model, the velocity of the normal fluid vanishes, since it is "attached" to the walls of the pipe by viscous effects. As a consequence of superfluid flow, the thermal quasi-particle distribution is asymmetric in momentum space.
The Bose-Einstein condensation mechanism is shown to be capable of accounting for the existence of separate hydrodynamic velocity fields for the normal and the superfluid, provided suitable assumptions are made with regard to the single-particle energy spectrum. We consider the effect on the microcanonical gas distribution of imposing a nonzero value of the total momentum P. For an ordinary gas the effect is trivial, the whole distribution being merely shifted in momentum space. However, in the case of a degenerate Bose gas whose single-particle energy spectrum has a sharp minimum (gap or cusp), only the excited part of the gas (normal fluid) participates in the imposed motion, the condensate (superfluid) remaining "frozen" in momentum space. This rigidity of the condensate in momentum space plays the same role as the rigidity of the superelectrons on imposition of a magnetic field in the London theory of superconductivity. P acts as an additional thermodynamic variable, states with P \ensuremath{\ne} 0 being macroscopically metastable and corresponding to the existence of a relative velocity between normal and superfluid. The basic hydrodynamic assumption of the two-fluid model is thus reduced to an assumption concerning the form of an effective single-particle energy spectrum, and the parallelism between the theories of superconductivity and superfluidity is clearly exhibited. The present theory permits, in particular, the introduction of the mathematical form of Landau's phonon and "roton" spectra within the framework of the Bose-Einstein condensation picture. The statistical-thermodynamic formulas are derived and are shown to lead to characteristic two-fluid equations derived previously from a variational principle.
In this chapter, we review the famous Landau theory of superfluidity at finite temperatures. This theory is based on coupled hydrodynamic equations for the superfluid and normal fluid components. Landau's two-fluid description is only valid when collisions among the thermal excitations making up the normal fluid are strong enough to produce local hydrodynamic equilibrium. These two-fluid equations were originally developed for liquid. He but are thought to be generic in form, describing the collision-dominated hydrodynamic region of all Bose superfluids. In this chapter, we will consider the solutions of the two-fluid equations mainly for a uniform superfluid. We discuss the existence of second sound (involving the out-of-phase motion of the superfluid and normal fluid components) as a characteristic feature of a Bose superfluid at finite temperatures. This chapter gives background material needed for Chapters 15–19. In Chapter 15, we will show that, in the appropriate limit, the Landau two-fluid equations can be derived from the ZNG coupled equations given in Chapter 3 for a trapped dilute Bose-condensed gas. In Chapters 17–19, we extend this discussion and derive the Landau–Khalatnikov two-fluid equations, which include hydrodynamic damping associated with various transport coefficients. Useful reviews of the two-fluid equations in the context of dilute spatially uniform Bose-condensed gases are given by Pethick and Smith (2008, Chapter 10) and Pitaevskii and Stringari (2003, Chapter 6). History of two-fluid equations The original discovery of superfluidity in liquid He was dramatically announced with the publication of the famous back-to-back papers of Kapitza (1938) in Moscow and Allen and Misener (1938) in Cambridge. These and subsequent experiments in the next few years showed that superfluid He could exhibit very bizarre hydrodynamic behaviour compared to classical liquids.
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.
The goal of this chapter will be to briefly describe the remarkable properties of helium at low temperatures. After stating some of these properties we will see how they can be understood in terms of the phenomenon of Bose–Einstein condensation described in Chapter 7. We will give the main argument in two different formulations, once using the quasi-particle method of Bogoliubov, and then using a Green function approach. We start with some experimental facts. Helium is a remarkable element. It was predicted to exist from observations of the Sun before it was found on Earth. It is the only element which remains a liquid at zero temperature and atmospheric pressure. Experimentally the phase diagram of 4 He is shown in Figure 10.1. Helium I is a normal fluid and has a normal gas-liquid critical point. Helium II is a mixture of a normal fluid and a superfluid. The superfluid is characterized by the vanishing of its viscosity. Helium I and helium II are separated by a line known as the λ -transition line. At T λ = 2.18 K, P λ = 2.29 Pa, helium I, helium II, and helium gas coexist. The specific heat of liquid helium along the vapor transition line forms a logarithmic discontinuity shown in Figure 10.2. The form of this diagram resembles the Greek letter λ and is the reason for calling the transition a λ -transition. The lack of viscosity of helium II leads to some remarkable experimental consequences, one of which we briefly describe. Let two containers A and B be linked by a thin capillary through which only a fluid with zero (or very low) viscosity can pass freely.
The phenomena of superfluidity and superconductivity are intimately connected with the existence of a condensate, a macroscopically occupied quantum state. Such condensates occur in a variety of different physical systems, as described in Chapter 1. The foundation for the description of superfluidity is a picture of the system as being comprised of a condensate and elementary excitations. In Chapter 8 we have seen how physical properties such as the energy and the density of a Bose–Einstein condensed system may be expressed in terms of a contribution from the condensate, plus one from the elementary excitations, and in this chapter we shall consider further developments of this basic idea to other situations. As a first application, we determine the critical velocity for creation of an excitation in a homogeneous system (Sec. 10.1). Following that, we show how to express the momentum density in terms of the velocity of the condensate and the distribution function for excitations. This provides the basis for a two-component description, the two components being the condensate and the thermal excitations (Sec. 10.2). In the past, this framework has proved to be very effective in describing the properties of superfluids and superconductors, and in Sec. 10.3 we apply it to dynamical processes. To describe the state of a superfluid, one must specify the condensate velocity, in addition to the variables needed to characterize the state of an ordinary fluid. As a consequence, the collective behaviour of a superfluid is richer than that of an ordinary one. Collective modes are most simply examined when excitations collide frequently enough that they are in local thermodynamic equilibrium.
We now consider the behavior of the superfluid Bose liquid at finite temperatures. We have remarked in Chapter 1 that the two-fluid theory of Tisza and Landau provides an excellent macroscopic description of such superfluid behavior. It is natural to inquire whether the Bose liquid theory we have developed in Chapters 2 - 5 permits a precise definition of the basic quantities which appear in the two-fluid model. The present chapter is devoted primarily to answering that question; in so doing we shall be led to discuss the extent to which the usual two-fluid equations may be expected to be applicable. We shall confine our attention to equilibrium situations, postponing to the following chapter consideration of a non-equilibrium case.