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CRC Press eBooks

Superfluid Flow: Macroscopic Limit

Philippe Nozières, David Pines

Published Mar 5, 2018• 0 citations
Publisher

Abstract

In the preceding chapters, we have been mostly concerned with the ground state, |0〉, of a Bose liquid; we have discussed at length its structure, its excitation spectrum, and its response to a weak external probe. We have seen that all these properties were deeply influenced by the existence of a condensate , consisting in the n o particles which occupy the single state k = 0. Such a condensate is clearly at rest; it possesses a uniform density and carries no current.

DOI
10.1201/9780429495717-12

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CRC Press eBooks•Mar 5, 2018•0 citations
In the preceding chapters, we have been mostly concerned with the ground state, |0〉, of a Bose liquid; we have discussed at length its structure, its excitation spectrum, and its response to a weak external probe. We have seen that all these properties were deeply influenced by the existence of a condensate , consisting in the n o particles which occupy the single state k = 0. Such a condensate is clearly at rest; it possesses a uniform density and carries no current.
We conclude our study of superfluidity in Bose liquids with a brief summary of the strengths, and of the inadequacies, of the theory. The cornerstone of our fundamental understanding of superfluidity has been the recognition of the central importance of the condensate, a single quantum state which is macroscopically occupied. For the Bose liquid one has a particle condensate, characterized by a non-vanishing expectation value of ψ (r). The existence of the condensate explains all major aspects of superfluid behavior; in particular, we now possess: A clear understanding of the static and the long wavelength dynamic properties of an equilibrium Bose liquid, including a microscopic basis for the two-fluid model. A precise microscopic theory at T = 0 for a specific model, the dilute Bose gas. Variational calculations of the excitation spectrum of liquid He II (and of the ground state properties as well), which are in good quantitative agreement with experiment. A satisfactory phenomenological treatment of the interactions between elementary excitations which give rise to various irreversible phenomena. A qualitative understanding of the role which is played by quantized vortices in both equilibrium and non-equilibrium phenomena.
In the previous chapter we described quantized vortex lines, which are one of the characteristic features of superfluids. In a classical fluid, the circulation of vortex lines is not quantized and, in addition, vortex lines decay because of viscous processes. Another feature of a superfluid, the lack of response to rotation for a small enough angular velocity, was also demonstrated. This is analogous to the Meissner effect for superconductors. One characteristic common to superfluids and superconductors is the ability to carry currents without dissipation. Such current-carrying states are not the lowest-energy state of the system. They are metastable states, the existence of which is intimately connected to the nature of the low-lying elementary excitations. The word ‘superfluidity’ does not refer to a single property of the system, but it is used to describe a variety of different phenomena (see Ref. [1]). Historically, the connection between superfluidity and the existence of a condensate, a macroscopically occupied quantum state, dates back to Fritz London's suggestion in 1938, as we have described in Chapter 1. However, the connection between Bose–Einstein condensation and superfluidity is a subtle one. A Bose–Einstein condensed system does not necessarily exhibit superfluidity, an example being the ideal Bose gas for which the critical velocity vanishes, as demonstrated in Sec. 10.1 below. Also lower-dimensional systems may exhibit superfluid behaviour in the absence of a true condensate, as we shall see in Chapter 15.
This chapter concludes the study of superfluidity in Bose liquids with a brief summary of the strengths, and of the inadequacies, of the theory. A clear understanding of the static and the long wavelength dynamic properties of an equilibrium Bose liquid, including a microscopic basis for the two-fluid model. Variational calculations of the excitation spectrum of liquid He II, which are in good quantitative agreement with experiment. A satisfactory phenomenological treatment of the interactions between elementary excitations which give rise to various irreversible phenomena. A microscopic theory of the ground state and elementary excitation spectrum of liquid He II; in fact, even the theory of the simple dilute Bose gas has not yet been extended to treat the system at finite temperatures, and under non-equilibrium situations. A clear understanding of a number of "vortex" properties, including the critical current for superfluid flow in thin capillaries.
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 begin our study of the superfluid Bose liquid with a brief review of the experimental and theoretical background of the liquid 4 He, the only Bose liquid found naturally in the laboratory. We then consider, in Chapter 2 , the quasi-particle excitations and show how at T = 0 the macroscopic occupation of the zero momentum state leads, in the long wavelength limit, to an identical excitation spectrum for quasi-particles and density fluctuations. The spectrum consists in phonons with a velocity given by the macroscopic velocity of sound. We turn in Chapter 3 to a consideration of the observed elementary excitation spectrum of He II. We discuss both the experimental measurements and the theories that have thus far been proposed to explain them.