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

Advanced Bogolyubov model of imperfect bose gas and superfluidity

N. N. Bogolyubov, Дмитрий Петрович Санкович

Published Jan 1, 2012• 0 citations
OpenAlex

Abstract

The equilibrium properties of a system of interacting bosons are studied from a microscopic point of view. We calculate the superfluid density in the Bogolyubov model of imperfect Bose gas. The model superstable Hamiltonian is considered. We examine the case of some pair potential and find the estimate for temperature and density in the λ-point.

Related papers

Superfluidity without a Bose condensate

S. I. Shevchenko

We have developed a microscopic theory of superfluidity in a two-dimensional, slightly nonideal Bose gas. The theory is rooted in the fact that a number of bosons are concentrated in the small momentum region at low temperatures when the interaction is weak, in spite of the fact that there is no Bose condensate in the two-dimensional case when T≠0. This fact lets us reduce the description of the system from one employing creation ak+ and annihilation ak operators for bosons to one based on combinations of these operators ρk and φk(ρk = S−1Σqaq+aq+k, φk· = −(i/ns) Σ(kq/k2 + l)aq+aq+k, where n is the boson density, and S is the area of the system). The description in terms of ρk and φk is simple because the hamiltonian is quadratic with respect to ρk and φk. The operators ρk and φk can also be used to express the occupation number Nk = ak+ak and current operators. We find the elementary excitation spectrum (which is identical to the Bogolyubov spectrum) and the momentum distribution of the bosons. We calculate the normal density due to thermal oscillations in the Bose gas and to interactions with impurity centers in the substrate.

Soviet Journal of Low Temperature Physics•Oct 1, 1983•1 citation

Statistical Mechanics for the Nonideal Bose Gas

A. E. Glassgold, Allan N. Kaufman, Kenneth Watson

The equilibrium and quasi-equilibrium properties of a system of interacting bosons are studied from a microscopic point of view. For equilibrium, the model of Bogolyubov is generalized to finite temperature by using the grand partition function. The thermodynamic properties and the pair-correlation function are calculated. The statistical mechanics for moving systems is then developed and applied to the problem of a rotating fluid. For quasi-equilibrium, general transport equations are derived from first principles, independent of statistics and model. For the Bogolyubov model, the familiar two-fluid hydrodynamics is then derived, leading to the phenomena of first and second sound.

Physical Review•Nov 1, 1960•52 citations

New theory of superfluidity. Method of equilibrium density matrix

Boris Volodumurovich Bondarev

The variational theory of equilibrium boson system state to have been previously developed by the author under the density matrix formalism is applicable for researching equilibrium states and thermodynamic properties of the quantum Bose gas which consists of zero-spin particles. Particle pulse distribution function is obtained and duly employed for calculation of chemical potential, internal energy and gas capacity temperature dependences. It is found that specific phase transition, which is similar to transition of liquid helium to its superfluid state, occurs at the temperature exceeding that of the Bose condensation.

arXiv:1412.6004•Dec 25, 2013•0 citations

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

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

Calculation of nonequilibrium thermodynamic potential of Bose system near the condensation point

K. M. Haponenko, Alexander I. Sokolovsky

Bose system of zero spin particles is considered in the presence of the Bose–Einstein condensate in the vicinity of the phase transition point. The system is investigated in the framework of the Bogolyubov model with the separated condensate. In this model an effective Hamiltonian of the system is introduced by replacing condensate creation and annihilation operators in system Hamiltonian by n01/2 where n0 is occupation number of the condensate state. According to Bogolyubov, the grand canonical thermodynamic potential related to the effective Hamiltonian is considered as nonequilibrium thermodynamic potential. In the present paper this potential is investigated as a function of the small variable n0. With the help of the thermodynamic perturbation theory it is shown that it is expanded in a series over integer powers of n0. This corresponds to the basic idea of the Landau theory of the phase transitions of the second kind. Coefficients at terms of the first and second orders in n0 in the expansion are calculated for Bose gas in the main approximation in small interaction. Calculation of the coefficients at terms of the third and fourth orders needs accounting contributions of the thermodynamic perturbation theory at least of the 4th order and will be done elsewhere. It is established that the results obtained for Bose gas do not fit into the Landau theory of phase transitions of the second kind. Some comments that discuss the situation are given.

Journal of Physics and Electronics•Dec 26, 2018•0 citations
The equilibrium properties of a system of interacting bosons are studied from a microscopic point of view. We calculate the superfluid density in the Bogolyubov model of imperfect Bose gas. The model superstable Hamiltonian is considered. We examine the case of some pair potential and find the estimate for temperature and density in the λ-point.
We have developed a microscopic theory of superfluidity in a two-dimensional, slightly nonideal Bose gas. The theory is rooted in the fact that a number of bosons are concentrated in the small momentum region at low temperatures when the interaction is weak, in spite of the fact that there is no Bose condensate in the two-dimensional case when T≠0. This fact lets us reduce the description of the system from one employing creation ak+ and annihilation ak operators for bosons to one based on combinations of these operators ρk and φk(ρk = S−1Σqaq+aq+k, φk· = −(i/ns) Σ(kq/k2 + l)aq+aq+k, where n is the boson density, and S is the area of the system). The description in terms of ρk and φk is simple because the hamiltonian is quadratic with respect to ρk and φk. The operators ρk and φk can also be used to express the occupation number Nk = ak+ak and current operators. We find the elementary excitation spectrum (which is identical to the Bogolyubov spectrum) and the momentum distribution of the bosons. We calculate the normal density due to thermal oscillations in the Bose gas and to interactions with impurity centers in the substrate.
The equilibrium and quasi-equilibrium properties of a system of interacting bosons are studied from a microscopic point of view. For equilibrium, the model of Bogolyubov is generalized to finite temperature by using the grand partition function. The thermodynamic properties and the pair-correlation function are calculated. The statistical mechanics for moving systems is then developed and applied to the problem of a rotating fluid. For quasi-equilibrium, general transport equations are derived from first principles, independent of statistics and model. For the Bogolyubov model, the familiar two-fluid hydrodynamics is then derived, leading to the phenomena of first and second sound.
The variational theory of equilibrium boson system state to have been previously developed by the author under the density matrix formalism is applicable for researching equilibrium states and thermodynamic properties of the quantum Bose gas which consists of zero-spin particles. Particle pulse distribution function is obtained and duly employed for calculation of chemical potential, internal energy and gas capacity temperature dependences. It is found that specific phase transition, which is similar to transition of liquid helium to its superfluid state, occurs at the temperature exceeding that of the Bose condensation.
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.
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
Bose system of zero spin particles is considered in the presence of the Bose–Einstein condensate in the vicinity of the phase transition point. The system is investigated in the framework of the Bogolyubov model with the separated condensate. In this model an effective Hamiltonian of the system is introduced by replacing condensate creation and annihilation operators in system Hamiltonian by n01/2 where n0 is occupation number of the condensate state. According to Bogolyubov, the grand canonical thermodynamic potential related to the effective Hamiltonian is considered as nonequilibrium thermodynamic potential. In the present paper this potential is investigated as a function of the small variable n0. With the help of the thermodynamic perturbation theory it is shown that it is expanded in a series over integer powers of n0. This corresponds to the basic idea of the Landau theory of the phase transitions of the second kind. Coefficients at terms of the first and second orders in n0 in the expansion are calculated for Bose gas in the main approximation in small interaction. Calculation of the coefficients at terms of the third and fourth orders needs accounting contributions of the thermodynamic perturbation theory at least of the 4th order and will be done elsewhere. It is established that the results obtained for Bose gas do not fit into the Landau theory of phase transitions of the second kind. Some comments that discuss the situation are given.