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Physical Review

Low-Temperature Behavior of a Dilute Bose System of Hard Spheres. II. Nonequilibrium Properties

T. D. Lee, Chongming Yang

Published Mar 15, 1959• 59 citations
Publisher

Abstract

The calculation of a previous paper is extended to cover nonequilibrium properties. The phenomena of superfluidity, critical velocity, and "infinite heat conductivity" are given natural explanations. By using classical kinetic theory on the wave packets, hydrodynamical equations for reversible flow are derived and the dependence of the two "sound velocities" on the temperature studied. The relationship between macroscopic sound vibrations and microscopic excitations is analyzed. The work is confined to the model of a dilute hard-sphere Bose system.

DOI
10.1103/physrev.113.1406

Related papers

Low-Temperature Behavior of a Dilute Bose System of Hard Spheres. I. Equilibrium Properties

T. D. Lee, Chongming Yang

By a generalization of the method used in a previous paper the distribution of energy levels of a dilute Bose system of hard spheres is found. These energy levels are then used to compute the statistical properties for the system. A phase transition is found, and the transition point is calculated to the lowest order of $a$, the diameter of the hard spheres. Furthermore, the thermodynamical functions of the system in both the gas phase and the degenerate phase are obtained, provided $(\frac{a}{\ensuremath{\lambda}})\ensuremath{\ll}1$ and $\ensuremath{\rho}{a}^{3}\ensuremath{\ll}1$, where $\ensuremath{\lambda}$ is the thermal wavelength and $\ensuremath{\rho}$ is the particle density.

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Some Nonequilibrium Properties of a Bose System of Hard Spheres at Extremely Low Temperatures

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Conclusion

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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.

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Yves Pomeau A

Abstract The explanation of the phenomenon of superfluidity requires an in‐depth analysis of both equilibrium and nonequilibrium statistics. This can be done in dilute gases by a systematic expansion in powers of the density, as done for classical gases in Prigogine's book on kinetic theory. Following the method of expansion of Prigogine and Résibois, the thermodynamics of a dilute gas of N identical Bosons with condensate can be studied in the case of a superflow, that is, of a finite and uniform velocity difference between the normal and superfluid component (assumed at rest). The microscopic interactions between the two components conserve the momentum of the normal component, making it a thermodynamic parameter. No thermal equilibrium exists if the velocity difference is larger than a critical value far below the onset of dissipation given by Landau's criterion. Consequences of this are discussed. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

International Journal of Quantum Chemistry•Jan 1, 2004•0 citations

Ground-state properties and excitation spectra of non-Galilean-invariant interacting Bose systems

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We study the ground-state properties and the excitation spectrum of bosons which, in addition to a short-range repulsive two body potential, interact through the exchange of some dispersionless bosonic modes. The latter induces a time dependent (retarded) boson-boson interaction which is attractive in the static limit. Moreover the coupling with dispersionless modes introduces a reference frame for the moving boson system and hence breaks the Galilean invariance of this system. The ground state of such a system is depleted linearly in the boson density due to the zero point fluctuations driven by the retarded part of the interaction. Both quasiparticle (microscopic) and compressional (macroscopic) sound velocities of the system are studied. The microscopic sound velocity is calculated up the second order in the effective two body interaction in a perturbative treatment, similar to that of Beliaev for the dilute weakly interacting Bose gas. The hydrodynamic equations are used to obtain the macroscopic sound velocity. We show that these velocities are identical within our perturbative approach. We present analytical results for them in terms of two dimensional parameters---an effective interaction strength and an adiabaticity parameter---which characterize the system. We find that due the presence of several competing effects, which determine the speed of the sound of the system, three qualitatively different regimes can be in principle realized in the parameter space and discuss them on physical grounds.

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Theory of a dilute low-temperature trapped Bose condensate

Alexander L. Fetter

This set of four lectures reviews various aspects of the theory of a dilute low-temperature trapped Bose gas, starting with (I) a review of the Bogoliubov description of the elementary excitations in a uniform system. The treatment is then generalized (II) to include the new physical effects of a confining harmonic trap potential on the condensate and its normal modes. An equivalent hydrodynamic description (III) focuses directly on the density and velocity fluctuations. The physics of vortices (IV) in an incompressible fluid is summarized and extended to the case of a trapped Bose condensate.

arXiv:cond-mat/9811366•Nov 26, 1998•12 citations
The calculation of a previous paper is extended to cover nonequilibrium properties. The phenomena of superfluidity, critical velocity, and "infinite heat conductivity" are given natural explanations. By using classical kinetic theory on the wave packets, hydrodynamical equations for reversible flow are derived and the dependence of the two "sound velocities" on the temperature studied. The relationship between macroscopic sound vibrations and microscopic excitations is analyzed. The work is confined to the model of a dilute hard-sphere Bose system.
By a generalization of the method used in a previous paper the distribution of energy levels of a dilute Bose system of hard spheres is found. These energy levels are then used to compute the statistical properties for the system. A phase transition is found, and the transition point is calculated to the lowest order of aaa, the diameter of the hard spheres. Furthermore, the thermodynamical functions of the system in both the gas phase and the degenerate phase are obtained, provided $(\frac{a}{\ensuremath{\lambda}})\ensuremath{\ll}1$ and $\ensuremath{\rho}{a}^{3}\ensuremath{\ll}1$, where $\ensuremath{\lambda}$ is the thermal wavelength and $\ensuremath{\rho}$ is the particle density.
The pseudopotential method is used to study a special type of flow for a Bose system of hard spheres. In the first-order approximation, the wave function of the entire system is assumed to be the product of identical single-particle wave functions, which in general are time-dependent. Such a flow is necessarily irrotational, and the single-particle wave function satisfies a Schrödinger equation with a nonlinear self-coupling term. On the basis of this equation of motion, the following properties of the Bose system are discussed: the effect of the rigid wall, the moment of inertia, the compressional wave, and a type of ``vortex filament.'' In the second-order approximation, the wave function of the system is expressed in terms of two functions such that one of them describes the single-particle state suitable for most of the particles while the other one describes the pair excitations. The much more complicated equations of motion are found, but in this approximation the flow is no longer strictly irrotational. The compressional waves are also studied in the second-order approximation.
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.
Abstract The explanation of the phenomenon of superfluidity requires an in‐depth analysis of both equilibrium and nonequilibrium statistics. This can be done in dilute gases by a systematic expansion in powers of the density, as done for classical gases in Prigogine's book on kinetic theory. Following the method of expansion of Prigogine and Résibois, the thermodynamics of a dilute gas of N identical Bosons with condensate can be studied in the case of a superflow, that is, of a finite and uniform velocity difference between the normal and superfluid component (assumed at rest). The microscopic interactions between the two components conserve the momentum of the normal component, making it a thermodynamic parameter. No thermal equilibrium exists if the velocity difference is larger than a critical value far below the onset of dissipation given by Landau's criterion. Consequences of this are discussed. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004
We study the ground-state properties and the excitation spectrum of bosons which, in addition to a short-range repulsive two body potential, interact through the exchange of some dispersionless bosonic modes. The latter induces a time dependent (retarded) boson-boson interaction which is attractive in the static limit. Moreover the coupling with dispersionless modes introduces a reference frame for the moving boson system and hence breaks the Galilean invariance of this system. The ground state of such a system is depleted linearly in the boson density due to the zero point fluctuations driven by the retarded part of the interaction. Both quasiparticle (microscopic) and compressional (macroscopic) sound velocities of the system are studied. The microscopic sound velocity is calculated up the second order in the effective two body interaction in a perturbative treatment, similar to that of Beliaev for the dilute weakly interacting Bose gas. The hydrodynamic equations are used to obtain the macroscopic sound velocity. We show that these velocities are identical within our perturbative approach. We present analytical results for them in terms of two dimensional parameters---an effective interaction strength and an adiabaticity parameter---which characterize the system. We find that due the presence of several competing effects, which determine the speed of the sound of the system, three qualitatively different regimes can be in principle realized in the parameter space and discuss them on physical grounds.
This set of four lectures reviews various aspects of the theory of a dilute low-temperature trapped Bose gas, starting with (I) a review of the Bogoliubov description of the elementary excitations in a uniform system. The treatment is then generalized (II) to include the new physical effects of a confining harmonic trap potential on the condensate and its normal modes. An equivalent hydrodynamic description (III) focuses directly on the density and velocity fluctuations. The physics of vortices (IV) in an incompressible fluid is summarized and extended to the case of a trapped Bose condensate.