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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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Global Dynamics and Photon Loss in the Kompaneets Equation
The Kompaneets equation governs dynamics of the photon energy spectrum in certain high temperature (or low density) plasmas. We prove several results concerning the long time convergence of solutions to Bose–Einstein equilibria and the failure of photon conservation. In particular, we show the total photon number can decrease with time via an outflux of photons at the zero-energy boundary. The ensuing accumulation of photons at zero energy is analogous to Bose–Einstein condensation. We provide two conditions that guarantee that photon loss occurs, and show that once loss is initiated then it persists forever. We prove that as 𝑡→∞, solutions necessarily converge to equilibrium and we characterize the limit in terms of the total photon loss. Additionally, we provide a few results concerning the behavior of the solution near the zero-energy boundary, an Oleinik inequality, a comparison principle, and show that the solution operator is a contraction in 𝐿1. None of these results impose a boundary condition at the zero-energy boundary.
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- PAR ID:
- 10531431
- Publisher / Repository:
- SIAM
- Date Published:
- Journal Name:
- SIAM Journal on Mathematical Analysis
- Volume:
- 55
- Issue:
- 5
- ISSN:
- 0036-1410
- Page Range / eLocation ID:
- 5715 to 5750
- Subject(s) / Keyword(s):
- Kompaneets equation Sunyaev–Zeldovich effect Bose–Einstein condensate quantum entropy LaSalle invariance principle
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1137/22M1533207
