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arXiv:1711.00610 · Annales Henri Poincaré

Dynamical Hartree–Fock–Bogoliubov Approximation of Interacting Bosons

Jacky J. Chong, Zehua Zhao

Published Oct 20, 2021• 13 citations
arXivHTMLPDF
DOI
10.1007/s00023-021-01100-w

Related papers

APPLICATION OF DISPERSIVE PDE TECHNIQUES TO THE STUDIES OF THE TIME-DEPENDENT HARTREE-FOCK-BOGOLIUBOV SYSTEM FOR BOSONS

Jacky J. Chong

The thesis provides rigorous quantitative analyses for studying quantum fluctuations of a non-relativistic Bose gas about a Bose-Einstein condensate. In recent years, the dynamics of the condensate and the excitations of a Bose gas was shown to be well approximated by the quasifree dynamics with governing equations given by a system of coupled nonlinear dispersive PDEs called the time-dependent Hartree-Fock-Bogoliubov (HFB) system (c.f. Grillakis and Machedon, J. Fix. Point Theory A., (2013), Grillakis and Machedon, Comm. PDEs, (2017), Bach et al. in arXiv:1602.05171v2, (2018), Benedikter et al., Ann. Henri Poincare, (2018)). However, both the quantitative and qualitative analysis of the time-dependent HFB system are still in their early stages of development. Thus, the primary purposes of the thesis are to further the development of some analytic tools necessary for studying the time-dependent HFB system and use these effective equations to provide quantitative estimates for the true dynamics of the Bose gas at absolute zero temperature in Fock space norm. The thesis comprises the entirety of the author's current and past projects on the time-dependent HFB system. Each project falls into one or more of two categories: studying the local or global well-posedness of the time-dependent HFB system, or obtaining global-in-time Fock space estimates for the error terms of the quasifree approximation to the dynamics of a system of interacting bosons. In the former category, the author employs techniques of dispersive PDE theory developed in the past three decades along with classical methods of harmonic analysis to study lower regularity solutions to the time-dependent HFB system. The lower regularity well-posedness of solutions for the time-dependent HFB system is necessary for studying norm approximation of the dynamics of a dilute Bose gas with strong interactions. In the latter category, we prove global a-priori estimates for the solutions to the HFB system and use them to obtain estimates for the error terms of the Fock space approximation.

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The time-dependent Hartree–Fock–Bogoliubov equations for Bosons

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Abstract We introduce the map of dynamics of quantum Bose gases into dynamics of quasifree states, which we call the “nonlinear quasifree approximation”. We use this map to derive the time-dependent Hartree–Fock–Bogoliubov (HFB) equations describing the dynamics of quantum fluctuations around a Bose–Einstein condensate. We prove global well-posedness of the HFB equations for pair potentials satisfying suitable regularity conditions, and we establish important conservation laws. We show that the space of solutions of the HFB equations has a symplectic structure reminiscent of a Hamiltonian system. This is then used to relate the HFB equations to the HFB eigenvalue equations discussed in the physics literature. We also construct Gibbs equilibrium states at positive temperature associated with the HFB equations, and we establish criteria for the appearance of Bose–Einstein condensation.

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Generalised Hartree-Fock method for bosons with attractive interactions

K. K. Singh

Physica•Jun 1, 1974•1 citation

Self-consistent Hartree-Fock-Bogoliubov approach for bosons: self-eliminating divergence and pure pair condensate

M. S. Bulakhov, A. S. Peletminskii

We investigate the thermodynamic properties of an interacting Bose gas with a condensate within the energy-functional formulation of the Hartree-Fock-Bogoliubov (HFB) approach. For a contact interaction, we derive a self-consistent solution to the HFB equations that intrinsically eliminates divergence. This solution characterizes the equilibrium state featuring a condensate of correlated pairs of particles. We analyze the temperature dependence of key thermodynamic quantities such as condensate density, chemical potential, entropy, pressure, specific heat capacity at constant volume, and isothermal compressibility and compare them with predictions from the Popov approximation (PA). We predict that the transition temperature shifts to higher values due interactions, with the HFB approach yielding a larger shift than the PA. Analysis of the compressibility indicates that a pure pair condensate is unstable, and the stable equilibrium corresponds to only a mixture of single-particle and pair condensates.

arXiv:2506.13413•Jun 16, 2025•0 citations
The thesis provides rigorous quantitative analyses for studying quantum fluctuations of a non-relativistic Bose gas about a Bose-Einstein condensate. In recent years, the dynamics of the condensate and the excitations of a Bose gas was shown to be well approximated by the quasifree dynamics with governing equations given by a system of coupled nonlinear dispersive PDEs called the time-dependent Hartree-Fock-Bogoliubov (HFB) system (c.f. Grillakis and Machedon, J. Fix. Point Theory A., (2013), Grillakis and Machedon, Comm. PDEs, (2017), Bach et al. in arXiv:1602.05171v2, (2018), Benedikter et al., Ann. Henri Poincare, (2018)). However, both the quantitative and qualitative analysis of the time-dependent HFB system are still in their early stages of development. Thus, the primary purposes of the thesis are to further the development of some analytic tools necessary for studying the time-dependent HFB system and use these effective equations to provide quantitative estimates for the true dynamics of the Bose gas at absolute zero temperature in Fock space norm. The thesis comprises the entirety of the author's current and past projects on the time-dependent HFB system. Each project falls into one or more of two categories: studying the local or global well-posedness of the time-dependent HFB system, or obtaining global-in-time Fock space estimates for the error terms of the quasifree approximation to the dynamics of a system of interacting bosons. In the former category, the author employs techniques of dispersive PDE theory developed in the past three decades along with classical methods of harmonic analysis to study lower regularity solutions to the time-dependent HFB system. The lower regularity well-posedness of solutions for the time-dependent HFB system is necessary for studying norm approximation of the dynamics of a dilute Bose gas with strong interactions. In the latter category, we prove global a-priori estimates for the solutions to the HFB system and use them to obtain estimates for the error terms of the Fock space approximation.
Abstract We introduce the map of dynamics of quantum Bose gases into dynamics of quasifree states, which we call the “nonlinear quasifree approximation”. We use this map to derive the time-dependent Hartree–Fock–Bogoliubov (HFB) equations describing the dynamics of quantum fluctuations around a Bose–Einstein condensate. We prove global well-posedness of the HFB equations for pair potentials satisfying suitable regularity conditions, and we establish important conservation laws. We show that the space of solutions of the HFB equations has a symplectic structure reminiscent of a Hamiltonian system. This is then used to relate the HFB equations to the HFB eigenvalue equations discussed in the physics literature. We also construct Gibbs equilibrium states at positive temperature associated with the HFB equations, and we establish criteria for the appearance of Bose–Einstein condensation.
We investigate the thermodynamic properties of an interacting Bose gas with a condensate within the energy-functional formulation of the Hartree-Fock-Bogoliubov (HFB) approach. For a contact interaction, we derive a self-consistent solution to the HFB equations that intrinsically eliminates divergence. This solution characterizes the equilibrium state featuring a condensate of correlated pairs of particles. We analyze the temperature dependence of key thermodynamic quantities such as condensate density, chemical potential, entropy, pressure, specific heat capacity at constant volume, and isothermal compressibility and compare them with predictions from the Popov approximation (PA). We predict that the transition temperature shifts to higher values due interactions, with the HFB approach yielding a larger shift than the PA. Analysis of the compressibility indicates that a pure pair condensate is unstable, and the stable equilibrium corresponds to only a mixture of single-particle and pair condensates.