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

arXiv:cond-mat/0106363 · Journal of Physics B Atomic Molecular and Optical Physics

Fluctuations of the condensate in ideal and interacting Bose gases

Hongwei Xiong, Shujuan Liu, Guoxiang Huang, Zhijun Xu, Cunyuan Zhang

Published Oct 31, 2001• 16 citations
arXivHTMLPDF

Abstract

We investigate the fluctuations of the condensate in the ideal and weakly interacting Bose gases confined in a box of volume V within a canonical ensemble. A canonical ensemble is developed to describe the behaviour of the fluctuations when different methods of approximation to the weakly interacting Bose gases are used. Research shows that the fluctuations of the condensate exhibit anomalous behaviour for the interacting Bose gas confined in a box.

DOI
10.1088/0953-4075/34/21/310

Related papers

Anomalous Fluctuations of the Condensate in Interacting Bose Gases

S. Giorgini, Лев П. Питаевский, S. Stringari

We find that the fluctuations of the condensate in a weakly interacting Bose gas confined in a box of volume $V$ follow the law $〈\ensuremath{\delta}{N}_{0}^{2}〉\ensuremath{\sim}{V}^{4/3}$. This anomalous behavior arises from the occurrence of infrared divergencies due to phonon excitations and holds also for strongly correlated Bose superfluids. The analysis is extended to an interacting Bose gas confined in a harmonic trap where the fluctuations are found to exhibit a similar anomaly.

arXiv:cond-mat/9711065•Physical Review Letters•Jun 8, 1998•125 citations

Grand-canonical condensate fluctuations in weakly interacting Bose-Einstein condensates of light

Christoph Weiß, Jacques Tempere

Grand-canonical fluctuations of Bose-Einstein condensates of light are accessible to state-of-the-art experiments [J. Schmitt et al., Phys. Rev. Lett. 112, 030401 (2014).PRLTAO0031-900710.1103/PhysRevLett.112.030401]. We phenomenologically describe these fluctuations by using the grand-canonical ensemble for a weakly interacting Bose gas at thermal equilibrium. For a two-dimensional harmonic trap, we use two models for which the canonical partition functions of the weakly interacting Bose gas are given by exact recurrence relations. We find that the grand-canonical condensate fluctuations for weakly interacting Bose gases vanish at zero temperature, thus behaving qualitatively similarly to an ideal gas in the canonical ensemble (or microcanonical ensemble) rather than the grand-canonical ensemble. For low but finite temperatures, the fluctuations remain considerably higher than for the canonical ensemble, as predicted by the ideal gas in the grand-canonical ensemble, thus clearly showing that we are not in a regime in which the ensembles are equivalent.

arXiv:1511.03064•Physical review. E•Oct 19, 2016•15 citations

Microcanonical fluctuations of the condensate in weakly interacting Bose gases

Zbigniew Idziaszek

We study fluctuations of the number of Bose condensed atoms in a weakly interacting homogeneous and trapped gases. For a homogeneous system we apply the particle-number-conserving formulation of the Bogoliubov theory and calculate the condensate fluctuations within the canonical and the microcanonical ensembles. We demonstrate that, at least in the low-temperature regime, predictions of the particle-number-conserving and traditional, nonconserving theory are identical, and lead to the anomalous scaling of fluctuations. Furthermore, the microcanonical fluctuations differ from the canonical ones by a quantity which scales normally in the number of particles, thus predictions of both ensembles are equivalent in the thermodynamic limit. We observe a similar behavior for a weakly interacting gas in a harmonic trap. This is in contrast to the trapped, ideal gas, where microcanonical and canonical fluctuations are different in the thermodynamic limit.

arXiv:cond-mat/0406052•Physical Review A•May 13, 2005•23 citations

Thermodynamic identities and particle number fluctuations in weakly interacting Bose-Einstein condensates

Fabrizio Illuminati, Patrick Navez, Martin Wilkens

We derive exact thermodynamic identities relating the average number of condensed atoms and the root-mean-square fluctuations determined in different statistical ensembles for the weakly interacting Bose gas confined in a box. This is achieved by introducing the concept of auxiliary partition functions for model Hamiltonians that do conserve the total number of particles. Exploiting such thermodynamic identities, we provide the first, completely analytical prediction of the microcanonical particle number fluctuations in the weakly interacting Bose gas. Such fluctuations, as a function of the volume V of the box are found to behave normally, in contrast with the anomalous scaling behaviour V 4/3 of the fluctuations in the ideal Bose gas.

arXiv:cond-mat/9909157•Journal of Physics B Atomic Molecular and Optical Physics•Aug 3, 1999•27 citations

Fluctuations of a weakly interacting Bose-Einstein condensate

Zbigniew Idziaszek, Łukasz Zawitkowski, Mariusz Gajda +1 more

Fluctuations of the number of condensed atoms in a finite-size, weakly interacting Bose gas confined in a box potential are investigated for temperatures up to the critical region. The canonical partition functions are evaluated using a recursive scheme for smaller systems, and a saddle-point approximation for larger samples that allows to treat realistic size systems containing up to N ∼10 5 particles. We point out the importance of particle-number constraint and interactions between out of condensate atoms for the statistics near the critical region. For sufficiently large systems, the crossover from the anomalous to normal scaling of the fluctuations is observed. The excitations are described in a self-consistent way within the Bogoliubov-Popov approximation, and the interactions between thermal atoms are described by means of the Hartree-Fock method.

arXiv:0708.0092•Europhysics Letters (EPL)•Apr 1, 2009•16 citations

Condensation ofNbosons. III. Analytical results for all higher moments of condensate fluctuations in interacting and ideal dilute Bose gases via the canonical ensemble quasiparticle formulation

Vitaly V. Kocharovsky, Vl. V. Kocharovsky, Marlan O. Scully

We consider equilibrium canonical-ensemble fluctuations of the Bose-Einstein condensate (BEC) in a trap using the particle-number conserving operator formalism of Girardeau and Arnowitt [Phys. Rev. 113, 755 (1959)] and the concept of the canonical-ensemble quasiparticles. We find explicit expressions for the characteristic function and all cumulants of the ground-state occupation statistics both for the dilute weakly interacting and ideal Bose gases. We present analytical formulas for the moments of the ground-state occupation fluctuations in the weakly interacting and ideal Bose gases in a box (``homogeneous gas'') as well as in the ideal Bose gas in an arbitrary power-law trap, in particular, in an arbitrary harmonic trap. We calculate the effect of Bogoliubov coupling between quasiparticles on suppression of the ground-state occupation fluctuations at moderate temperatures and their enhancement at very low temperatures. We clarify a crossover between ideal-gas and weakly interacting gas statistics, which is governed by a pair-correlation, squeezing mechanism. The important conclusion is that in most cases the ground-state occupation fluctuations are anomalously large and are not Gaussian even in the thermodynamic limit.

Physical Review A•Apr 13, 2000•68 citations
We investigate the fluctuations of the condensate in the ideal and weakly interacting Bose gases confined in a box of volume V within a canonical ensemble. A canonical ensemble is developed to describe the behaviour of the fluctuations when different methods of approximation to the weakly interacting Bose gases are used. Research shows that the fluctuations of the condensate exhibit anomalous behaviour for the interacting Bose gas confined in a box.
We find that the fluctuations of the condensate in a weakly interacting Bose gas confined in a box of volume VVV follow the law $〈\ensuremath{\delta}{N}_{0}^{2}〉\ensuremath{\sim}{V}^{4/3}$. This anomalous behavior arises from the occurrence of infrared divergencies due to phonon excitations and holds also for strongly correlated Bose superfluids. The analysis is extended to an interacting Bose gas confined in a harmonic trap where the fluctuations are found to exhibit a similar anomaly.
Grand-canonical fluctuations of Bose-Einstein condensates of light are accessible to state-of-the-art experiments [J. Schmitt et al., Phys. Rev. Lett. 112, 030401 (2014).PRLTAO0031-900710.1103/PhysRevLett.112.030401]. We phenomenologically describe these fluctuations by using the grand-canonical ensemble for a weakly interacting Bose gas at thermal equilibrium. For a two-dimensional harmonic trap, we use two models for which the canonical partition functions of the weakly interacting Bose gas are given by exact recurrence relations. We find that the grand-canonical condensate fluctuations for weakly interacting Bose gases vanish at zero temperature, thus behaving qualitatively similarly to an ideal gas in the canonical ensemble (or microcanonical ensemble) rather than the grand-canonical ensemble. For low but finite temperatures, the fluctuations remain considerably higher than for the canonical ensemble, as predicted by the ideal gas in the grand-canonical ensemble, thus clearly showing that we are not in a regime in which the ensembles are equivalent.
We study fluctuations of the number of Bose condensed atoms in a weakly interacting homogeneous and trapped gases. For a homogeneous system we apply the particle-number-conserving formulation of the Bogoliubov theory and calculate the condensate fluctuations within the canonical and the microcanonical ensembles. We demonstrate that, at least in the low-temperature regime, predictions of the particle-number-conserving and traditional, nonconserving theory are identical, and lead to the anomalous scaling of fluctuations. Furthermore, the microcanonical fluctuations differ from the canonical ones by a quantity which scales normally in the number of particles, thus predictions of both ensembles are equivalent in the thermodynamic limit. We observe a similar behavior for a weakly interacting gas in a harmonic trap. This is in contrast to the trapped, ideal gas, where microcanonical and canonical fluctuations are different in the thermodynamic limit.
We derive exact thermodynamic identities relating the average number of condensed atoms and the root-mean-square fluctuations determined in different statistical ensembles for the weakly interacting Bose gas confined in a box. This is achieved by introducing the concept of auxiliary partition functions for model Hamiltonians that do conserve the total number of particles. Exploiting such thermodynamic identities, we provide the first, completely analytical prediction of the microcanonical particle number fluctuations in the weakly interacting Bose gas. Such fluctuations, as a function of the volume V of the box are found to behave normally, in contrast with the anomalous scaling behaviour V 4/3 of the fluctuations in the ideal Bose gas.
Fluctuations of the number of condensed atoms in a finite-size, weakly interacting Bose gas confined in a box potential are investigated for temperatures up to the critical region. The canonical partition functions are evaluated using a recursive scheme for smaller systems, and a saddle-point approximation for larger samples that allows to treat realistic size systems containing up to N ∼10 5 particles. We point out the importance of particle-number constraint and interactions between out of condensate atoms for the statistics near the critical region. For sufficiently large systems, the crossover from the anomalous to normal scaling of the fluctuations is observed. The excitations are described in a self-consistent way within the Bogoliubov-Popov approximation, and the interactions between thermal atoms are described by means of the Hartree-Fock method.
We consider equilibrium canonical-ensemble fluctuations of the Bose-Einstein condensate (BEC) in a trap using the particle-number conserving operator formalism of Girardeau and Arnowitt [Phys. Rev. 113, 755 (1959)] and the concept of the canonical-ensemble quasiparticles. We find explicit expressions for the characteristic function and all cumulants of the ground-state occupation statistics both for the dilute weakly interacting and ideal Bose gases. We present analytical formulas for the moments of the ground-state occupation fluctuations in the weakly interacting and ideal Bose gases in a box (``homogeneous gas'') as well as in the ideal Bose gas in an arbitrary power-law trap, in particular, in an arbitrary harmonic trap. We calculate the effect of Bogoliubov coupling between quasiparticles on suppression of the ground-state occupation fluctuations at moderate temperatures and their enhancement at very low temperatures. We clarify a crossover between ideal-gas and weakly interacting gas statistics, which is governed by a pair-correlation, squeezing mechanism. The important conclusion is that in most cases the ground-state occupation fluctuations are anomalously large and are not Gaussian even in the thermodynamic limit.