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CRC Press eBooks

Microscopic Theory: Non-Uniform Condensate

Philippe Nozières, David Pines

Published Mar 5, 2018• 0 citations
Publisher

Abstract

153 e consider in this chapter the extension of the microscopic theory of Chapter 9 to the case of a condensate structure which varies in space and time. The theory displays clearly the microscopic origin of the macroscopic theory of condensate motion presented in Chapter 5 . It provides as well a description of states in which the condensate motion varies so rapidly that the methods of Chapter 5 are inapplicable.

DOI
10.1201/9780429495717-17

Related papers

Microscopic Theory: Non-Uniform Condensate

Philippe Nozières, David Pines

This chapter considers the extension of the microscopic theory to the case of a condensate structure which varies in space and time. The theory displays clearly the microscopic origin of the macroscopic theory of condensate motion. The chapter considers the problem of arbitrary condensate motion. The condensate motion is influenced by the degree of excitation of the system, so that the two stages of solution are not distinct. What is required is a self-consistent solution, which takes into account the coupling between the condensate and the elementary excitations. The chapter discusses whether the wave functions we have proposed for various simple physical situations are, in fact, consistent with the non-linear condensate equation of motion. It describes a moving condensate, with a uniform density and a slowly varying velocity. The chapter comments briefly on the validity of the factorization approximation.

CRC Press eBooks•Mar 5, 2018•0 citations

Microscopic Theory: Uniform Condensate 1

Philippe Nozières, David Pines

In this chapter, the authors turn their attention to microscopic theories of interacting Bose systems, that class of theories which begin with a given law of interaction between the particles, and derive the various properties of the system therefrom. Microscopic theory does not provide a quantitative description of liquid He II. Indeed, there exists a satisfactory microscopic theory of an interacting boson system in only two limiting cases, those of: Weak repulsive interactions between the bosons; and a dilute gas with arbitrary repulsive interaction between the particles. One sees how the macroscopic occupation of a single quantum state dictates the structure of the microscopic description of system behavior. The Hamiltonian serves as a starting Hamiltonian in authors consideration of microscopic theory; they show that the existence of macroscopic occupation of a single quantum state leads to an effective Hamiltonian which looks rather different.

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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.

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Conclusion

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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.

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Basis for the Two-Fluid Model 1

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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.

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Macroscopic World and Microscopic World

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A physical picture of macroscopic and microscopic worlds is presented. In the space-time description, both the macroscopic and microscopic worlds are described simultaneously in space-time being a single boundless four dimensional continuum in which the motion of the observer is represented as a world line. In our picture, the space-time description is valid only for the macroscopic world and the microscopic world is local in the sense that it is described in a finite space associated with each point of the macroscopic space-time. In this case, space-time does not exist in the microscopic world in which the observer can not move. For the description of our picture, the language of the non-space-time description is necessary in which space-time is not presupposed. We can utilize the space-time description for the macroscopic world and, at the same time, we have a theory of finite degree of freedom describing creations and annihilations of elementary particles in the microscopic world free from the divergence difficulty, instead of the divergent quantum field theory in the space-time description which is considered in our picture a limit of an approximation like the ideal gas in gas theories. The finite degree of freedom is given by a one-particle state-vector space of a finite dimension from which the second quantization leads to a state-vector space of a countable dimension (a Hilbert space) for the theory of creations and annihilations of elementary particles describing a local microscopic world. The relation between microscopic worlds associated with different macroscopic space-time points is given by a dynamical principle up to a phase factor due to possible fluctuations of phases of local state-vector spaces. The experiment on interference of light performed recently by Franson and Potocki is interpreted in terms of fluctuation of the local phases and is considered to support the theory of finite degree of freedom which leads to the idea of local microscopic worlds.

Annals of the Japan Association for Philosophy of Science•Jan 1, 1986•4 citations
153 e consider in this chapter the extension of the microscopic theory of Chapter 9 to the case of a condensate structure which varies in space and time. The theory displays clearly the microscopic origin of the macroscopic theory of condensate motion presented in Chapter 5 . It provides as well a description of states in which the condensate motion varies so rapidly that the methods of Chapter 5 are inapplicable.
This chapter considers the extension of the microscopic theory to the case of a condensate structure which varies in space and time. The theory displays clearly the microscopic origin of the macroscopic theory of condensate motion. The chapter considers the problem of arbitrary condensate motion. The condensate motion is influenced by the degree of excitation of the system, so that the two stages of solution are not distinct. What is required is a self-consistent solution, which takes into account the coupling between the condensate and the elementary excitations. The chapter discusses whether the wave functions we have proposed for various simple physical situations are, in fact, consistent with the non-linear condensate equation of motion. It describes a moving condensate, with a uniform density and a slowly varying velocity. The chapter comments briefly on the validity of the factorization approximation.
In this chapter, the authors turn their attention to microscopic theories of interacting Bose systems, that class of theories which begin with a given law of interaction between the particles, and derive the various properties of the system therefrom. Microscopic theory does not provide a quantitative description of liquid He II. Indeed, there exists a satisfactory microscopic theory of an interacting boson system in only two limiting cases, those of: Weak repulsive interactions between the bosons; and a dilute gas with arbitrary repulsive interaction between the particles. One sees how the macroscopic occupation of a single quantum state dictates the structure of the microscopic description of system behavior. The Hamiltonian serves as a starting Hamiltonian in authors consideration of microscopic theory; they show that the existence of macroscopic occupation of a single quantum state leads to an effective Hamiltonian which looks rather different.
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.
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.
A physical picture of macroscopic and microscopic worlds is presented. In the space-time description, both the macroscopic and microscopic worlds are described simultaneously in space-time being a single boundless four dimensional continuum in which the motion of the observer is represented as a world line. In our picture, the space-time description is valid only for the macroscopic world and the microscopic world is local in the sense that it is described in a finite space associated with each point of the macroscopic space-time. In this case, space-time does not exist in the microscopic world in which the observer can not move. For the description of our picture, the language of the non-space-time description is necessary in which space-time is not presupposed. We can utilize the space-time description for the macroscopic world and, at the same time, we have a theory of finite degree of freedom describing creations and annihilations of elementary particles in the microscopic world free from the divergence difficulty, instead of the divergent quantum field theory in the space-time description which is considered in our picture a limit of an approximation like the ideal gas in gas theories. The finite degree of freedom is given by a one-particle state-vector space of a finite dimension from which the second quantization leads to a state-vector space of a countable dimension (a Hilbert space) for the theory of creations and annihilations of elementary particles describing a local microscopic world. The relation between microscopic worlds associated with different macroscopic space-time points is given by a dynamical principle up to a phase factor due to possible fluctuations of phases of local state-vector spaces. The experiment on interference of light performed recently by Franson and Potocki is interpreted in terms of fluctuation of the local phases and is considered to support the theory of finite degree of freedom which leads to the idea of local microscopic worlds.