Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Progress of Theoretical Physics

Dynamics of Bose-Einstein Condensate and Two-Fluid Hydrodynamics

Kazumasa Miyake, K. Yamada

Published Dec 1, 1976• 5 citations
Publisher

Abstract

Dynamics of Bose-Einstein Condensate (BEC) is treated on the basis of the Mori theory of non-equilibrium thermodynamics in the hydrodynamic region. The local values of conserved quantities, i.e., particle number, momentum and energy, and Ψ(x), the wave function of BEC, are adopted as a complete set of macrovariables. The set of equations of motion thus obtained is useful to describe the dynamical behavior of superfluid 4He near the λ-point. The density of condensate is not conserved, but its equation of motion has the term which is relevant to the renormalization of condensate to superfluid. The expression for the relaxation time τc of condensate density to its local-equilibrium is given and it is shown that the anomaly of τc near the λ-point coincides with the behavior experimentally determined if nc(∂µc/∂nc) is proportional to (Tλ-T) as in the Landau theory of phase transition. In case ωτc ≪1 the ordinary two-fluid hydrodynamics is derived by eliminating the motion of condensate density.

DOI
10.1143/ptp.56.1689

Related papers

Hydrodynamics of a Superfluid Condensate

Eugene P. Gross

The theory of the condensate of a weakly interacting Bose gas is developed. The condensate is described by a wavefunction ψ(x, t) normalized to the number of particles. It obeys a nonlinear self-consistent field equation. The solution in the presence of a rigid wall with the boundary condition of vanishing wavefunction involves a de Broglie length. This length depends on the mean potential energy per particle. The self-consistent field term keeps the density uniform except in localized spatial regions. In the hydrodynamical version, a key role is played by the quantum potential. A theory of quantized vortices and of general potential flows follows immediately. In contrast to classical hydrodynamics, the cores of vortices are completely determined by the de Broglie length and all energies are finite. Nonstationary disturbances of the condensate correspond to phonons, rotons, vortex waves etc. They can exchange momentum with rigid boundaries. This is compatible with the vanishing of the wavefunction at a boundary. This condition fully determines the dynamics of the system. These points are illustrated by considering the motion of a foreign ion in a Bose gas, a rotating container of fluid, and the Landau criterion for superfluidity.

Journal of Mathematical Physics•Feb 1, 1963•996 citations

Liquid Helium II: Bose-Einstein Condensation and Two-Fluid Model

Paul R. Zilsel

The Bose-Einstein condensation mechanism is shown to be capable of accounting for the existence of separate hydrodynamic velocity fields for the normal and the superfluid, provided suitable assumptions are made with regard to the single-particle energy spectrum. We consider the effect on the microcanonical gas distribution of imposing a nonzero value of the total momentum P. For an ordinary gas the effect is trivial, the whole distribution being merely shifted in momentum space. However, in the case of a degenerate Bose gas whose single-particle energy spectrum has a sharp minimum (gap or cusp), only the excited part of the gas (normal fluid) participates in the imposed motion, the condensate (superfluid) remaining "frozen" in momentum space. This rigidity of the condensate in momentum space plays the same role as the rigidity of the superelectrons on imposition of a magnetic field in the London theory of superconductivity. P acts as an additional thermodynamic variable, states with P \ensuremath{\ne} 0 being macroscopically metastable and corresponding to the existence of a relative velocity between normal and superfluid. The basic hydrodynamic assumption of the two-fluid model is thus reduced to an assumption concerning the form of an effective single-particle energy spectrum, and the parallelism between the theories of superconductivity and superfluidity is clearly exhibited. The present theory permits, in particular, the introduction of the mathematical form of Landau's phonon and "roton" spectra within the framework of the Bose-Einstein condensation picture. The statistical-thermodynamic formulas are derived and are shown to lead to characteristic two-fluid equations derived previously from a variational principle.

Physical Review•Dec 1, 1953•12 citations

Two-Fluid Dynamics for a Bose-Einstein Condensate out of Local Equilibrium with the Noncondensate

Tetsuro Nikuni, E. Zaremba, Adam Griffin

We extend our recent work on the two-fluid hydrodynamics of a Bose-condensed gas by including collisions involving both condensate and noncondensate atoms. These collisions are essential for establishing a state of local thermodynamic equilibrium between the condensate and noncondensate. Our theory is more general than the usual Landau two-fluid theory, to which it reduces in the appropriate limit, in that it allows one to describe situations in which a state of complete local equilibrium between the two components has not been reached. The exchange of atoms between the condensate and noncondensate is associated with a new relaxational mode of the gas.

arXiv:cond-mat/9812320•Physical Review Letters•Jul 5, 1999•67 citations

Hydrodynamic Equations for the Condensate and the Depletion of Helium II

H. Haug, K. Weiss

The equations of motion for the reduced density matrices of an interacting Bose system are treated using the concept of off-diagonal long-range order. The equations are decoupled on the one-particle level. With this approximation, the hydrodynamic equations for the densities and the velocities of the condensate and the depletion are derived. The interaction between the two fluid components depends on density variations and on the relative velocity. The latter interaction terms appear as kinetic pressure terms and a Magnus force term. Via the equation for the total momentum, a connection between the velocities of the condensate and the depletion, and the velocities in Landau's two-fluid model is derived. The excitation spectrum of the system is investigated in the phonon region. A comparison of the resulting velocities for the first and second sound with the experimentally determined ones shows that the functional dependence of the condensate density on temperature is similar to that of the superfluid component.

Physical Review A•Feb 1, 1971•14 citations

1 Time-dependent linear response of an inhomogeneous Bose superfluid: Microscopic theory and connection to current-density functional theory

M. L. Chiofalo, Anna Minguzzi, Monica Tosi

The dynamics of a confined fluid of Bose atoms is treated within the linear response regime, with a view to establishing a current-density functional formalism for an inhomogeneous superfluid state. After evaluating in full detail a simplified case of an external coupling to the density and phase of the condensate, the theory is extended to include the coupling to the total current density. The Kohn-Sham response functions of the condensate and all the exchange-correlation kernels for the superfluid are introduced from the microscopic equations of motion and are expressed in a physically transparent way through functional derivatives of correlation functions. A microscopic formula for the superfluid density is derived and used to introduce a generalized hydrodynamic approach for a weakly inhomogeneous two-fluid model in isothermal conditions. Local-density expressions are thereby derived for the velocities of first and second sound in the weakly inhomogeneous superfluid and for visco-elastic functions describing the transition from the hydrodynamic to the collisionless regime. Landau's hydrodynamic theory and known results in Green's functions language are recovered in the limiting case of a homogeneous superfluid.

•Jan 1, 1998•4 citations

Quantum vortex dynamics in the counterflowing Bose–Einstein condensate

Sergey K. Nemirovskii

The aim of this paper is to derive the equations of motion of quantum vortices in a weakly non-ideal Bose–Einstein condensate (BEC) at finite temperatures and in the presence of macroscopic superfluid and normal velocities. The motivation for this work is that the dynamics of quantum vortices in superfluid helium has been studied at the phenomenological level, and a number of issues require clarification, which can be addressed by studying the corresponding problem in BEC. In BEC, quantum vortices are located at points where the macroscopic wave function (MWF) ψ(r,T), which describes the BEC dynamics, vanishes ψ(r,T)=0. The intersection of the surfaces Reψ(r,T)=0 and Imψ(r,T)=0 forms a one-dimensional line, which is just the quantum vortex line. During the evolution of the MWF, the position of the zero surfaces changes, that is, the vortex lines move. Conversely, the motion of these vortex filaments determines the evolution of the MWF. Thus, there is a mutual correspondence, and the quantity ψ(r,T) can be considered as a functional of the entire vortex configuration, i.e., {s(ξ,t)}. This viewpoint allows us to derive the equations of motion for vortices from the Gross–Pitaevskii equation (GPE), which describes the dynamics of MWF. The key finding is the confirmation that the equation derived from the fundamental quantum description (the GPE) for a BEC has the same mathematical structure as the equation derived from macroscopic, phenomenological reasoning for superfluid helium.

Physics of Fluids•Feb 1, 2026•0 citations
Dynamics of Bose-Einstein Condensate (BEC) is treated on the basis of the Mori theory of non-equilibrium thermodynamics in the hydrodynamic region. The local values of conserved quantities, i.e., particle number, momentum and energy, and Ψ(x), the wave function of BEC, are adopted as a complete set of macrovariables. The set of equations of motion thus obtained is useful to describe the dynamical behavior of superfluid 4He near the λ-point. The density of condensate is not conserved, but its equation of motion has the term which is relevant to the renormalization of condensate to superfluid. The expression for the relaxation time τc of condensate density to its local-equilibrium is given and it is shown that the anomaly of τc near the λ-point coincides with the behavior experimentally determined if nc(∂µc/∂nc) is proportional to (Tλ-T) as in the Landau theory of phase transition. In case ωτc ≪1 the ordinary two-fluid hydrodynamics is derived by eliminating the motion of condensate density.
The theory of the condensate of a weakly interacting Bose gas is developed. The condensate is described by a wavefunction ψ(x, t) normalized to the number of particles. It obeys a nonlinear self-consistent field equation. The solution in the presence of a rigid wall with the boundary condition of vanishing wavefunction involves a de Broglie length. This length depends on the mean potential energy per particle. The self-consistent field term keeps the density uniform except in localized spatial regions. In the hydrodynamical version, a key role is played by the quantum potential. A theory of quantized vortices and of general potential flows follows immediately. In contrast to classical hydrodynamics, the cores of vortices are completely determined by the de Broglie length and all energies are finite. Nonstationary disturbances of the condensate correspond to phonons, rotons, vortex waves etc. They can exchange momentum with rigid boundaries. This is compatible with the vanishing of the wavefunction at a boundary. This condition fully determines the dynamics of the system. These points are illustrated by considering the motion of a foreign ion in a Bose gas, a rotating container of fluid, and the Landau criterion for superfluidity.
The Bose-Einstein condensation mechanism is shown to be capable of accounting for the existence of separate hydrodynamic velocity fields for the normal and the superfluid, provided suitable assumptions are made with regard to the single-particle energy spectrum. We consider the effect on the microcanonical gas distribution of imposing a nonzero value of the total momentum P. For an ordinary gas the effect is trivial, the whole distribution being merely shifted in momentum space. However, in the case of a degenerate Bose gas whose single-particle energy spectrum has a sharp minimum (gap or cusp), only the excited part of the gas (normal fluid) participates in the imposed motion, the condensate (superfluid) remaining "frozen" in momentum space. This rigidity of the condensate in momentum space plays the same role as the rigidity of the superelectrons on imposition of a magnetic field in the London theory of superconductivity. P acts as an additional thermodynamic variable, states with P \ensuremath{\ne} 0 being macroscopically metastable and corresponding to the existence of a relative velocity between normal and superfluid. The basic hydrodynamic assumption of the two-fluid model is thus reduced to an assumption concerning the form of an effective single-particle energy spectrum, and the parallelism between the theories of superconductivity and superfluidity is clearly exhibited. The present theory permits, in particular, the introduction of the mathematical form of Landau's phonon and "roton" spectra within the framework of the Bose-Einstein condensation picture. The statistical-thermodynamic formulas are derived and are shown to lead to characteristic two-fluid equations derived previously from a variational principle.
We extend our recent work on the two-fluid hydrodynamics of a Bose-condensed gas by including collisions involving both condensate and noncondensate atoms. These collisions are essential for establishing a state of local thermodynamic equilibrium between the condensate and noncondensate. Our theory is more general than the usual Landau two-fluid theory, to which it reduces in the appropriate limit, in that it allows one to describe situations in which a state of complete local equilibrium between the two components has not been reached. The exchange of atoms between the condensate and noncondensate is associated with a new relaxational mode of the gas.
The equations of motion for the reduced density matrices of an interacting Bose system are treated using the concept of off-diagonal long-range order. The equations are decoupled on the one-particle level. With this approximation, the hydrodynamic equations for the densities and the velocities of the condensate and the depletion are derived. The interaction between the two fluid components depends on density variations and on the relative velocity. The latter interaction terms appear as kinetic pressure terms and a Magnus force term. Via the equation for the total momentum, a connection between the velocities of the condensate and the depletion, and the velocities in Landau's two-fluid model is derived. The excitation spectrum of the system is investigated in the phonon region. A comparison of the resulting velocities for the first and second sound with the experimentally determined ones shows that the functional dependence of the condensate density on temperature is similar to that of the superfluid component.
The dynamics of a confined fluid of Bose atoms is treated within the linear response regime, with a view to establishing a current-density functional formalism for an inhomogeneous superfluid state. After evaluating in full detail a simplified case of an external coupling to the density and phase of the condensate, the theory is extended to include the coupling to the total current density. The Kohn-Sham response functions of the condensate and all the exchange-correlation kernels for the superfluid are introduced from the microscopic equations of motion and are expressed in a physically transparent way through functional derivatives of correlation functions. A microscopic formula for the superfluid density is derived and used to introduce a generalized hydrodynamic approach for a weakly inhomogeneous two-fluid model in isothermal conditions. Local-density expressions are thereby derived for the velocities of first and second sound in the weakly inhomogeneous superfluid and for visco-elastic functions describing the transition from the hydrodynamic to the collisionless regime. Landau's hydrodynamic theory and known results in Green's functions language are recovered in the limiting case of a homogeneous superfluid.
The aim of this paper is to derive the equations of motion of quantum vortices in a weakly non-ideal Bose–Einstein condensate (BEC) at finite temperatures and in the presence of macroscopic superfluid and normal velocities. The motivation for this work is that the dynamics of quantum vortices in superfluid helium has been studied at the phenomenological level, and a number of issues require clarification, which can be addressed by studying the corresponding problem in BEC. In BEC, quantum vortices are located at points where the macroscopic wave function (MWF) ψ(r,T), which describes the BEC dynamics, vanishes ψ(r,T)=0. The intersection of the surfaces Reψ(r,T)=0 and Imψ(r,T)=0 forms a one-dimensional line, which is just the quantum vortex line. During the evolution of the MWF, the position of the zero surfaces changes, that is, the vortex lines move. Conversely, the motion of these vortex filaments determines the evolution of the MWF. Thus, there is a mutual correspondence, and the quantity ψ(r,T) can be considered as a functional of the entire vortex configuration, i.e., {s(ξ,t)}. This viewpoint allows us to derive the equations of motion for vortices from the Gross–Pitaevskii equation (GPE), which describes the dynamics of MWF. The key finding is the confirmation that the equation derived from the fundamental quantum description (the GPE) for a BEC has the same mathematical structure as the equation derived from macroscopic, phenomenological reasoning for superfluid helium.