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CRC Press eBooks

Superfluid Flow: Macroscopic Limit

Philippe Nozières, David Pines

Published Mar 5, 2018• 0 citations
Publisher

Abstract

Actually, the fundamental feature of Bose condensation is the macroscopic occupation of a single quantum state, whatever that state may be. As long as such a macroscopically-occupied state exists, the system will exhibit superfluid behavior. Superfluid flow thus appears as a "collective" phenomenon, in which the particles move together in order to preserve the macroscopic occupation of a single state. The equilibrium is metastable, because obviously there exists a state of lower energy, that of the fluid at rest; the essential point is that no transitions to the latter state will take place. Metastable superfluid flow is only possible because it involves condensate motion. The superfluid velocity is thus proportional to the gradient of the phase of the condensate wave-function. The superfluid motion is thus a uniform translation, in contrast with ordinary viscous flow, for which the fluid velocity varies from the walls to the center of the tube.

DOI
10.1201/9780429492679-5

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Physical Review A•Sep 26, 2008•2 citations
Actually, the fundamental feature of Bose condensation is the macroscopic occupation of a single quantum state, whatever that state may be. As long as such a macroscopically-occupied state exists, the system will exhibit superfluid behavior. Superfluid flow thus appears as a "collective" phenomenon, in which the particles move together in order to preserve the macroscopic occupation of a single state. The equilibrium is metastable, because obviously there exists a state of lower energy, that of the fluid at rest; the essential point is that no transitions to the latter state will take place. Metastable superfluid flow is only possible because it involves condensate motion. The superfluid velocity is thus proportional to the gradient of the phase of the condensate wave-function. The superfluid motion is thus a uniform translation, in contrast with ordinary viscous flow, for which the fluid velocity varies from the walls to the center of the tube.
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.
The two-fluid theory for superfluidity is not entirely consistent with the evidence for condensation into multiple highly occupied momentum states. On the far side of the -transition wave function symmetrisation is in the form of pure momentum loops and momentum state occupancy, the non-local nature of which gives inviscid plug flow, and the entropy of which gives the superfluid critical velocity. Equations of motion for condensed bosons conserve entropy. A quantum stochastic molecular dynamics algorithm based on a continuous form of occupancy gives the viscosity of liquid 4He, which is less than the classical calculation, increasingly so as the temperature is decreased through the condensation transition. Molecular trajectories show the suppression of collisions that transfer longitudinal momentum laterally in the superfluid regime. A second algorithm that treats condensed bosons in the same momentum state as a rigid body with non-local forces also yields a reduced viscosity but with greater uncertainty.
What is a superfluid? The word “superfluid” was coined to describe a qualitatively different state of a fluid that can occur at low temperatures, in which the resistance to flow is identically zero, so that flow round a closed path lasts for ever – a persistent current. Superfluidity can occur either for uncharged particles such as helium atoms or for charged particles such as the electrons in a metal. In the latter case the flow constitutes an electric current and we have a superconductor. Since an electric current is accompanied by a magnetic field, it is much easier to demonstrate the presence of a persistent current in a superconductor than in a neutral superfluid. In this chapter I shall describe the properties of superfluids, starting with the simplest and working up to more complicated examples. But in this introductory section I shall depart from the historical order even further by turning to the last page of the detective story so as to catch a glimpse of the conclusion from almost a century of experimental and theoretical research; we shall then have an idea of where we are heading. The conclusion is that superfluidity is more than just flow without resistance. It is an example of a transition to a more ordered state as the temperature is lowered, like the transition to ferromagnetism. At such a transition new macroscopically measurable quantities appear: in the case of a ferromagnet, the spontaneous magnetization. Such new measurable quantities are known as the order parameter of the low-temperature phase. The new quantities in the case of a superfluid are more subtle and more surprising: the amplitude and phase of the de Broglie wave associated with the motion of the superfluid particles. A superfluid can therefore exhibit quantummechanical effects on a macroscopic scale.
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.
It is shown that any Bose-Einstein-condensed fluid in its ground state will exhibit macroscopic single-particle quantum behavior (MSPQB). That is, (1) the many-particle wave function $\ensuremath{\Psi}({\mathbf{r}}_{1},\dots{},{\mathbf{r}}_{n})$ factors into a single-particle product ${\ensuremath{\prod}}_{n}\ensuremath{\eta}({\mathbf{r}}_{n})$; (2) the function $\ensuremath{\eta}(\mathbf{r})$ extends over macroscopic length scales and obeys the usual quantum equations for particle flux in a single-particle system; and (3) $\ensuremath{\eta}(\mathbf{r})$ obeys a nonlinear single-particle Schr\"odinger equation. The latter equation reduces to the Gross-Pitaevskii equation when interactions are weak and determines the density distribution of the fluid and the time development of this distribution. The arguments used rely only on elementary concepts of probability theory and many-particle wave mechanics and are valid even in strongly interacting fluids such as superfluid 4He^{4}\mathrm{He}4He. It is shown that Bose-Einstein condensation implies that the NNN-particle wave function $\ensuremath{\Psi}$ is delocalized. That is, if one considers a single-particle coordinate r\mathbf{r}r, then for all values that occur of the other $N\ensuremath{-}1$ coordinates, $\ensuremath{\Psi}$ is a nonzero function of r\mathbf{r}r over a region of space proportional to VVV, where VVV is the total volume within which the fluid is contained. MSPQB is a consequence of this delocalization and the absence of long-range correlations between particle positions in fluids. The results are accurate provided that only averages over regions of space containing many particles are considered. For averages over volumes of space containing ${N}_{\ensuremath{\Omega}}$ particles, inaccuracies due to quantum fluctuations are $\ensuremath{\sim}1∕\sqrt{{N}_{\ensuremath{\Omega}}}$.