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

Thermodynamics of a dilute Bose-Einstein gas with repulsive interactions

Yves Pomeau A, Sergio Rica

Published Jan 24, 2000• 6 citations
Publisher

Abstract

The thermodynamics of a system with N identical Bose particles is studied in the diluted gas limit. Not too close to the Bose-Einstein transition we find results differing from the ones in the literature. A relevant consequence of this work is that at low momenta the momentum distribution of quasiparticles is not in the usual form of the Bose factor.

DOI
10.1088/0305-4470/33/4/303

Related papers

Thermodynamics of a dilute non-perfect Bose-Einstein gas

Yves Pomeau A, Sergio Rica

We study the thermodynamics of a dilute Bose gas. Not too close to the Bose-Einstein transition we find results differing from the ones in the literature. As a consequence the low momentum part of the distribution of quasiparticules is not a Bose factor, and the effective spectrum is not of the phonon type. We also discuss the possibility of a Bose-Einstein transition in 2 D .

Europhysics Letters (EPL)•Jul 1, 2000•5 citations

Equilibrium and Non-Equilibrium Properties of Dilute Bose Gases

Oldenburg, Jakob Felix

Quantum systems consisting of many particles are both interesting from the physics and the mathematics perspective. One aspect that makes them fascinating from the physics point of view is that they can exhibit quantum effects on macroscopic scales such as Bose-Einstein condensation. This phenomenon, first predicted 100 years ago by Bose [19] and Einstein [44], occurs when a dilute gas of bosons is cooled down to very low temperatures. Below a certain critical temperature a condensate forms, meaning that a macroscopic fraction of the particles is in the same one-particle state, and thus, quantum phenomena become relevant on length scales much larger than the individual atom. The occurrence of Bose-Einstein condensation is also related to superfluidity as suggested in [88]. Bose-Einstein condensation has been achieved experimentally independently by the groups of Cornell and Wieman [3] and Ketterle [40] earning them a Nobel prize in 2001. While Bose and Einstein analyzed theoretically the non-interacting gas, an effective theory describing weakly interacting Bose gases was proposed by Bogoliubov in 1947 with the goal of explaining the occurrence of superfluidity in liquid helium [18]. Using assumptions such as that condensation indeed occurs, his theory yields predictions for the low-energy states of the system. Seen from the mathematics perspective, describing the Bose gas is an interesting challenge due to the strong correlations between particles introduced by the interaction making this case much harder to study than the non-interacting one. In particular, a proof that Bose-Einstein condensation occurs in the thermodynamic limit for continuous systems is still out of reach. However, there has been a lot of work regarding the rigorous analysis of bosonic systems in various relevant regimes. This thesis contains three new contributions establishing rigorous results on bosonic systems in different, physically relevant settings. It is based on the articles [37, 27, 31]. The first result concerns the time-evolution of initially trapped Bose gases in the ultra-dilute Gross-Pitaevskii regime. Under some assumptions on the initial data satisfied in particular by a class of states that are expected to approximate the ground state in a trap, we derive a quasi-free approximation that is close to the true solution in $L^2$-norm. In particular, we show that this approximate wave function is described in terms of a Bose-Einstein condensate in a one-particle state following the time-dependent Gross-Pitaevskii equation together with pair excitations with respect to this condensate. As a corollary, we establish a central limit theorem for bounded one-particle observables. The second result is a simple proof of Bose-Einstein condensation for interacting, ultra-dilute gases. More specifically, we consider scalings interpolating between the Gross-Pitaevskii regime and the thermodynamic limit. While condensation had been shown to hold in these regimes before, the advantage of the method used here is its simplicity. Lastly, the third result establishes an upper bound on the ground state energy of a dilute Bose gas in the thermodynamic limit. It captures rigorously the third order correction to the energy per unit volume for sufficiently low densities as predicted in the physics literature first by Wu and subsequently others. $\textit{Full-text embargoed until: 2026-01-01}$

•Dec 9, 2025•0 citations

Thermodynamics of a dilute Bose gas with condensate

Sergio Rica

The partition function of a dilute Bose gas with repulsive interaction, fixed number of particles and in the presence of a condensate is computed up to first order in the interactions. An equilibrium condition appears for the fraction of particles in the condensate and the chemical potential of the particles. We show that for a dilute gas the Bose–Einstein transition is not of second order. Moreover, the thermodynamical quantities obtained may look different from those in the literature, because the chemical potential enters in a nontrivial way in the quasi particle spectrum.

Comptes Rendus Physique•Jan 1, 2004•0 citations

Normal fluid/condensate interaction

Yves Pomeau A

In a dilute Bose gas the interaction between the normal gas and the condensate can be studied both at equilibrium and for its relevance to superfluid mechanics. In this article, one reviews first the thermodynamics of the interacting Bose gas in the low density limit and in the low temperature–finite density case. This is extended to thermodynamics with a velocity difference between the two components, normal and superfluid. It is argued that, if the initial value of the momentum related to this velocity difference is larger than the one given by thermodynamic stability, a new condensate should form in a finite time with a velocity close to the that of the normal gas. In the dilute case this occurrence of a new condensate is reminiscent of the condensation in the isotropic case at high enough density.

Comptes Rendus Physique•Jan 1, 2004•1 citation

Dilute Bose gas revised

Alexander Yu. Cherny, Arkady A Shanenko

The well-known results concerning a dilute Bose gas with the short-range repulsive interaction should be reconsidered due to a thermodynamic inconsistency of the method being basic to much of the present understanding of this subject. The aim of our paper is to propose another way of treating the dilute Bose gas with an arbitrary strong interaction. Using the reduced density matrix of the second order and a variational procedure, this way allows us to escape the inconsistency mentioned and operate with singular potentials of the Lennard-Jones type. The derived expansion of the condensate depletion in powers of the boson density n=N/V reproduces the familiar result, while the expansion for the mean energy per particle is of the form epsilon=2 pi planck(2)an/m[1+128/(15 square root of [pi]) square root of [na(3)](1-5b/8a)+...], where a is the scattering length and b> or =0 stands for one more characteristic length depending on the shape of the interaction potential (in particular, for the hard spheres a=b). All the consideration concerns the zero temperature.

arXiv:cond-mat/9903073•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Jul 1, 1999•7 citations

Thermodynamics of a dilute Bose gas: A path-integral Monte Carlo study

G. Spada, Sebastiano Pilati, S. Giorgini

We present precise path-integral Monte Carlo results for the thermodynamics of a homogeneous dilute Bose gas. Pressure and energy are calculated as a function of temperature both below and above the Bose-Einstein transition. Specifically, we address interaction effects, focusing on deviations from the ideal-gas law in the thermodynamic limit. We also calculate the isothermal compressibility and the contact parameter, which provide a clear signature of the role played by interactions. In particular, we obtain indications of a discontinuity of the compressibility at the transition point. To gain physical insight, numerical results are systematically compared with the predictions of first-order Hartree-Fock and second-order Popov theories, both giving an approximate description of the gas thermodynamics. The comparison shows the extension of the critical region around the transition point, where the inaccuracies of the perturbative expansions are more pronounced.

arXiv:2110.00283•Physical Review A•Jan 31, 2022•11 citations
The thermodynamics of a system with N identical Bose particles is studied in the diluted gas limit. Not too close to the Bose-Einstein transition we find results differing from the ones in the literature. A relevant consequence of this work is that at low momenta the momentum distribution of quasiparticles is not in the usual form of the Bose factor.
Quantum systems consisting of many particles are both interesting from the physics and the mathematics perspective. One aspect that makes them fascinating from the physics point of view is that they can exhibit quantum effects on macroscopic scales such as Bose-Einstein condensation. This phenomenon, first predicted 100 years ago by Bose [19] and Einstein [44], occurs when a dilute gas of bosons is cooled down to very low temperatures. Below a certain critical temperature a condensate forms, meaning that a macroscopic fraction of the particles is in the same one-particle state, and thus, quantum phenomena become relevant on length scales much larger than the individual atom. The occurrence of Bose-Einstein condensation is also related to superfluidity as suggested in [88]. Bose-Einstein condensation has been achieved experimentally independently by the groups of Cornell and Wieman [3] and Ketterle [40] earning them a Nobel prize in 2001. While Bose and Einstein analyzed theoretically the non-interacting gas, an effective theory describing weakly interacting Bose gases was proposed by Bogoliubov in 1947 with the goal of explaining the occurrence of superfluidity in liquid helium [18]. Using assumptions such as that condensation indeed occurs, his theory yields predictions for the low-energy states of the system. Seen from the mathematics perspective, describing the Bose gas is an interesting challenge due to the strong correlations between particles introduced by the interaction making this case much harder to study than the non-interacting one. In particular, a proof that Bose-Einstein condensation occurs in the thermodynamic limit for continuous systems is still out of reach. However, there has been a lot of work regarding the rigorous analysis of bosonic systems in various relevant regimes. This thesis contains three new contributions establishing rigorous results on bosonic systems in different, physically relevant settings. It is based on the articles [37, 27, 31]. The first result concerns the time-evolution of initially trapped Bose gases in the ultra-dilute Gross-Pitaevskii regime. Under some assumptions on the initial data satisfied in particular by a class of states that are expected to approximate the ground state in a trap, we derive a quasi-free approximation that is close to the true solution in L2L^2L2-norm. In particular, we show that this approximate wave function is described in terms of a Bose-Einstein condensate in a one-particle state following the time-dependent Gross-Pitaevskii equation together with pair excitations with respect to this condensate. As a corollary, we establish a central limit theorem for bounded one-particle observables. The second result is a simple proof of Bose-Einstein condensation for interacting, ultra-dilute gases. More specifically, we consider scalings interpolating between the Gross-Pitaevskii regime and the thermodynamic limit. While condensation had been shown to hold in these regimes before, the advantage of the method used here is its simplicity. Lastly, the third result establishes an upper bound on the ground state energy of a dilute Bose gas in the thermodynamic limit. It captures rigorously the third order correction to the energy per unit volume for sufficiently low densities as predicted in the physics literature first by Wu and subsequently others. Full-text embargoed until: 2026-01-01\textit{Full-text embargoed until: 2026-01-01}Full-text embargoed until: 2026-01-01
The partition function of a dilute Bose gas with repulsive interaction, fixed number of particles and in the presence of a condensate is computed up to first order in the interactions. An equilibrium condition appears for the fraction of particles in the condensate and the chemical potential of the particles. We show that for a dilute gas the Bose–Einstein transition is not of second order. Moreover, the thermodynamical quantities obtained may look different from those in the literature, because the chemical potential enters in a nontrivial way in the quasi particle spectrum.
In a dilute Bose gas the interaction between the normal gas and the condensate can be studied both at equilibrium and for its relevance to superfluid mechanics. In this article, one reviews first the thermodynamics of the interacting Bose gas in the low density limit and in the low temperature–finite density case. This is extended to thermodynamics with a velocity difference between the two components, normal and superfluid. It is argued that, if the initial value of the momentum related to this velocity difference is larger than the one given by thermodynamic stability, a new condensate should form in a finite time with a velocity close to the that of the normal gas. In the dilute case this occurrence of a new condensate is reminiscent of the condensation in the isotropic case at high enough density.
The well-known results concerning a dilute Bose gas with the short-range repulsive interaction should be reconsidered due to a thermodynamic inconsistency of the method being basic to much of the present understanding of this subject. The aim of our paper is to propose another way of treating the dilute Bose gas with an arbitrary strong interaction. Using the reduced density matrix of the second order and a variational procedure, this way allows us to escape the inconsistency mentioned and operate with singular potentials of the Lennard-Jones type. The derived expansion of the condensate depletion in powers of the boson density n=N/V reproduces the familiar result, while the expansion for the mean energy per particle is of the form epsilon=2 pi planck(2)an/m[1+128/(15 square root of [pi]) square root of [na(3)](1-5b/8a)+...], where a is the scattering length and b> or =0 stands for one more characteristic length depending on the shape of the interaction potential (in particular, for the hard spheres a=b). All the consideration concerns the zero temperature.
We present precise path-integral Monte Carlo results for the thermodynamics of a homogeneous dilute Bose gas. Pressure and energy are calculated as a function of temperature both below and above the Bose-Einstein transition. Specifically, we address interaction effects, focusing on deviations from the ideal-gas law in the thermodynamic limit. We also calculate the isothermal compressibility and the contact parameter, which provide a clear signature of the role played by interactions. In particular, we obtain indications of a discontinuity of the compressibility at the transition point. To gain physical insight, numerical results are systematically compared with the predictions of first-order Hartree-Fock and second-order Popov theories, both giving an approximate description of the gas thermodynamics. The comparison shows the extension of the critical region around the transition point, where the inaccuracies of the perturbative expansions are more pronounced.