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Progress of Theoretical Physics

On the Landau Theory of Fermi Fluids. A: Uniform and Non-Uniform Equilibrium Systems

T. Morita

Published Feb 1, 1962• 1 citation
Publisher

Abstract

The formal structure of the Landau theory for uniform and nonuniform equilibrium Fermi fluids is presented using techniques given in a general theory of classical fluids (T. Morita and K. Hiroike, Progr. Theoret. Phys. (Kyoto), 25: 537(l961)). The uniform case is presented, and results are generalized to the nonuniform case. (L.N.N.)
DOI
10.1143/ptp.27.423

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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.
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.
Renormalized expressions for the transport coefficients of a normal quantum fluid are derived from a nonlocal kinetic equation. As was shown in the classical case by Forster and Martin and by R\'esibois, the expressions for the transport coefficients separate naturally into "kinetic" and "direct" parts. The kinetic terms are proportional to matrix elements of the inverse of the kinetic kernel, and have the same general structure as the "bare" transport coefficients obtained from a local Boltzmann equation. The direct terms are proportional to matrix elements of the kernel itself, and have no counterpart in calculations based on a local kinetic equation, as in the kinetic theory of gases. We evaluate the transport coefficients using the weak-coupling approximation to the kernel derived in a previous paper. Results are given, first, for both Bose and Fermi fluids at arbitrary temperature, and then for the Fermi fluid near T=0T=0T=0, where complete solutions are obtained. It is found that the direct parts of the shear viscosity and thermal conductivity are of higher order in TTT than the kinetic parts, and are therefore negligible at very low temperature. The kinetic parts have the same leading temperature dependence as the predictions of the Landau theory. For the bulk viscosity, however, the direct and kinetic contributions begin at the same order (T2{T}^{2}T2) in the temperature.