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

Journal of Physics A Mathematical and General

Exact solution of the Bogoliubov Hamiltonian for weakly imperfect Bose gas

J.-B. Bru, Valentin Anatol'evich Zagrebnov

Published Nov 27, 1998• 26 citations
Publisher

Abstract

We show that the pressure of the Bogoliubov weakly imperfect Bose gas (WIBG) can be calculated exactly in the thermodynamic limit. We point out the sufficient and necessary conditions for it not to equate with the pressure of the ideal Bose gas (IBG). We prove that they differ only in that part of the phase diagram where the WIBG has a Bose condensate. We show that in contast to the conventional Bose condensate (e.g. in the IBG) the condensate in the WIBG is due to an effective attraction between bosons in the zero-mode.

DOI
10.1088/0305-4470/31/47/002

Related papers

The Bogoliubov weakly imperfect Bose-gas

Valentin Anatol'evich Zagrebnov

It is shown that the condensation in the Bogoliubov Weakly Imperfect Bose{Gas (WIBG) may appear in two stages.If the interaction is such that the pressure of the WIBG does not coincide with the pressure of the Perfect Bose{Gas (PBG), then the WIBG may manifest two kinds of condensations: a non{conventional condensation in the zero{mode due to the interaction (the rst stage) and a conventional (generalized of type I) Bose{Einstein condensation in modes next to the zero{mode due to the particle density saturation (the second stage).

Journal of Physical Studies•Jan 1, 1999•1 citation

The condensate equation for some Bose systems

E. Buffet, Ph. De Smedt, Joseph V. Pulé

The exactness of the Bogoliubov approximation is discussed from the point of view of the condensation properties of the Bose gas. For the imperfect Bose gas, the authors find that the approximation yields the correct condensate density when the thermodynamic limit is taken by isotropic dilation; this is not the case in general for other ways of going to the infinite volume limit. They also prove the existence of Bose-Einstein condensation for a class of Bose gases with weak interactions and an energy gap.

Journal of Physics A Mathematical and General•Dec 21, 1983•20 citations

The influence of weak interaction on thermodynamic properties and the stability of imperfect Bose gas

Yuan Du-Qi, 宝鸡文理学院物理系，宝鸡 721007

Using the free energy and the equation of state for an imperfect Bose gas, the analytical expression between the critical temperature and the length of dispersion wave and the pressure for the imperfect Bose gas is derived. The unusual properties of the specific heat at constant pressure and the expansion coefficient at constant pressure and isothermal compressibility by the attractive force for the imperfect Bose gas are revealed. The instability arising from the attractive force is pointed out, and the temperature criterion and particle density criterion of instability is given for the imperfect Bose gas under the action of attraction.

Acta Physica Sinica•Jan 1, 2006•4 citations

Note on a Non Linear Perturbation of the Ideal Bose Gas

M. Corgini, R. Tabilo

In this work we show that the introduction of a U(1) symmetry breaking field in the energy operator of the boson-free gas, is equivalent, in the thermodynamic limit, to the inclusion, in the Hamiltonian of the ideal gas, of a non-linear function of the number operator associated with the zero mode. In other words, the limit pressures coincide. Moreover, both models undergo non conventional Bose-Einstein condensation (BEC) for strictly negative values of the chemical potential $μ.$ Finally, the proof of equivalence of limit pressures is extended to a class of full-diagonal models.

arXiv:1812.06380•Dec 16, 2018•0 citations

Thermodynamics of Weakly Interacting Bose Gas, BCS Theory

Boris V. Svistunov, Egor S. Babaev, Nikolay V. Prokof'ev

This chapter deals for the most part with a systematic microscopic description of equilibrium weakly interacting Bose gas (WIBG) in one, two, and three dimensions. Our analysis is based on Beliaev’s diagrammatic technique [1] for Matsubara Green’s functions, regularized with small but finite Bogoliubov’s symmetrybreaking terms. This approach was discussed in detail in Reference [2]; the analytic part of this chapter is based on that paper. At the end of the chapter, we will present the mean-field theory for weakly interacting Fermi gas, or the Bardeen-Cooper-Schrieffer (BCS) solution.

CRC Press eBooks•Apr 15, 2015•0 citations

Casimir energy of a BEC: from moderate interactions to the ideal gas

Jürgen Schiefele, Carsten Henkel

Considering the Casimir effect due to phononic excitations of a weakly interacting dilute Bose–Einstein condensate (BEC), we derive a renormalized expression for the zero-temperature Casimir energy of a BEC confined to a parallel plate geometry with periodic boundary conditions. Our expression is formally equivalent to the free energy of a bosonic field at finite temperature, with a nontrivial density of modes that we compute analytically. As a function of the interaction strength, smoothly describes the transition from the weakly interacting Bogoliubov regime to the non-interacting ideal BEC. For the weakly interacting case, reduces to leading order to the Casimir energy due to zero-point fluctuations of massless phonon modes. In the limit of an ideal Bose gas, our result correctly describes the Casimir energy going to zero.

arXiv:0809.1816•Journal of Physics A Mathematical and Theoretical•Dec 17, 2008•27 citations
We show that the pressure of the Bogoliubov weakly imperfect Bose gas (WIBG) can be calculated exactly in the thermodynamic limit. We point out the sufficient and necessary conditions for it not to equate with the pressure of the ideal Bose gas (IBG). We prove that they differ only in that part of the phase diagram where the WIBG has a Bose condensate. We show that in contast to the conventional Bose condensate (e.g. in the IBG) the condensate in the WIBG is due to an effective attraction between bosons in the zero-mode.
It is shown that the condensation in the Bogoliubov Weakly Imperfect Bose{Gas (WIBG) may appear in two stages.If the interaction is such that the pressure of the WIBG does not coincide with the pressure of the Perfect Bose{Gas (PBG), then the WIBG may manifest two kinds of condensations: a non{conventional condensation in the zero{mode due to the interaction (the rst stage) and a conventional (generalized of type I) Bose{Einstein condensation in modes next to the zero{mode due to the particle density saturation (the second stage).
The exactness of the Bogoliubov approximation is discussed from the point of view of the condensation properties of the Bose gas. For the imperfect Bose gas, the authors find that the approximation yields the correct condensate density when the thermodynamic limit is taken by isotropic dilation; this is not the case in general for other ways of going to the infinite volume limit. They also prove the existence of Bose-Einstein condensation for a class of Bose gases with weak interactions and an energy gap.
Using the free energy and the equation of state for an imperfect Bose gas, the analytical expression between the critical temperature and the length of dispersion wave and the pressure for the imperfect Bose gas is derived. The unusual properties of the specific heat at constant pressure and the expansion coefficient at constant pressure and isothermal compressibility by the attractive force for the imperfect Bose gas are revealed. The instability arising from the attractive force is pointed out, and the temperature criterion and particle density criterion of instability is given for the imperfect Bose gas under the action of attraction.
In this work we show that the introduction of a U(1) symmetry breaking field in the energy operator of the boson-free gas, is equivalent, in the thermodynamic limit, to the inclusion, in the Hamiltonian of the ideal gas, of a non-linear function of the number operator associated with the zero mode. In other words, the limit pressures coincide. Moreover, both models undergo non conventional Bose-Einstein condensation (BEC) for strictly negative values of the chemical potential μ.μ.μ. Finally, the proof of equivalence of limit pressures is extended to a class of full-diagonal models.
This chapter deals for the most part with a systematic microscopic description of equilibrium weakly interacting Bose gas (WIBG) in one, two, and three dimensions. Our analysis is based on Beliaev’s diagrammatic technique [1] for Matsubara Green’s functions, regularized with small but finite Bogoliubov’s symmetrybreaking terms. This approach was discussed in detail in Reference [2]; the analytic part of this chapter is based on that paper. At the end of the chapter, we will present the mean-field theory for weakly interacting Fermi gas, or the Bardeen-Cooper-Schrieffer (BCS) solution.
Considering the Casimir effect due to phononic excitations of a weakly interacting dilute Bose–Einstein condensate (BEC), we derive a renormalized expression for the zero-temperature Casimir energy of a BEC confined to a parallel plate geometry with periodic boundary conditions. Our expression is formally equivalent to the free energy of a bosonic field at finite temperature, with a nontrivial density of modes that we compute analytically. As a function of the interaction strength, smoothly describes the transition from the weakly interacting Bogoliubov regime to the non-interacting ideal BEC. For the weakly interacting case, reduces to leading order to the Casimir energy due to zero-point fluctuations of massless phonon modes. In the limit of an ideal Bose gas, our result correctly describes the Casimir energy going to zero.