On the Quantum Boltzmann Equation near Maxwellian and Vacuum

Ouyang, Zhimeng; Wu, Lei

doi:10.48550/arxiv.2102.00657

Cited by 3 publications

(8 citation statements)

References 35 publications

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“…Remark 2.5. Notice that Q d;T ´S,λ in Theorem 2.1, respectively Q c;T ´S,λ in Theorem 2.2, correspond to a mollification of the collision operator Q of the quantum Boltzmann equation typically found in the literature, see, e.g., [70,88,176,187,194,195], usually presented in the form Qpf, g, hqrJs " π ż dp 3 δ `Epp 1 q `Epp 2 q ´Epp 3 q ˘δpp 1 `p2 ´p3 q pvpp 1 q `vpp 2 qq 2 `Jpp 1 q `Jpp 2 q ´Jpp 3 q ȓ f pp 1 qr gpp 2 qhpp 3 q ´f pp 1 qgpp 2 q r hpp 3 q ˘.…”

Section: 94)mentioning

confidence: 90%

On the emergence of quantum Boltzmann fluctuation dynamics near a Bose-Einstein Condensate

Chen¹,

Hott²

2021

Preprint

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In this paper, we study the quantum fluctuation dynamics in a Bose gas on a torus Λ Ă R 3 that exhibits Bose-Einstein condensation, beyond the leading order Hartree-Fock-Bogoliubov (HFB) fluctuations. Given a mean-field Hamiltonian and Bose-Einstein condensate (BEC) with density N , we extract a quantum Boltzmann type dynamics from a second-order Duhamel expansion upon subtracting both the HFB dynamics and the BEC dynamics. Using a Fock-space approach, we provide explicit error bounds. Given an approximately quasi-free initial state, we determine the time evolution of the centered correlation functions xay, xaay ´xay 2 , xa `ay ´|xay| 2 for mesoscopic time scales. For large but finite N , we consider both the case of fixed system size |Λ| " 1, and the case |Λ| " N 6 53 ´. In the case |Λ| " 1, we show that the Boltzmann collision operator contains subleading terms that can become dominant, depending on time-dependent coefficients assuming particular values in Q; this phenomenon is reminiscent of the Talbot effect. For the case´, we prove that the collision operator is well approximated by the expression predicted in the literature. THOMAS CHEN AND MICHAEL HOTT 7.1. Discrete case 78 7.2. Approximation by continuous limit 81 Appendix A. Introductory observations 83 Appendix B. Proof of convergence to mean field equations 85 B.1. Error bounds due to HFB evolution 89 B.2. Various error bounds 90 References 95

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Section: 94)mentioning

confidence: 90%

On the emergence of quantum Boltzmann fluctuation dynamics near a Bose-Einstein Condensate

Chen¹,

Hott²

2021

Preprint

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“…In terms of mathematical methods, this article is closer to [5], [44] and [27] since all of these works fall into the close-to-equilibrium framework well established for the classical Boltzmann equation. There are many great works that contribute to this mathematically satisfactory theory for global well-posedness of the classical Boltzmann equation.…”

Section: Introductionmentioning

confidence: 98%

“…Note that all the above results on BBE equations are obtained in the homogeneous case. In the inhomogeneous case, there are at least two recent results: Bae-Jang-Yun [5] and Ouyang-Wu [44]. These two works have different focuses and will be revisited later.…”

Section: Introductionmentioning

confidence: 99%

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Global well-posedness of the quantum Boltzmann equation for bosons interacting via inverse power law potentials

Zhou¹

2022

Preprint

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We consider the spatially inhomogeneous quantum Boltzmann equation for bosons with a singular collision kernel, the weak-coupling limit of a large system of Bose-Einstein particles interacting through inverse power law. Global well-posedness of the corresponding Cauchy problem is proved in a periodic box near equilibrium for initial data satisfying high temperature condition. Contents 1. Introduction 1 2. Linear operator estimate 12 3. Bilinear operator estimate 20 4. Trilinear operator estimate 31 5. Commutator estimate and weighted energy estimate 37 6. Local well-posedness 47 7. A priori estimate and global well-posedness 53 8. Appendix 60 References 69

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“…For the spatially homogeneous case, we mention [23,26]. For the inhomogeneous case, Lu [24,25] obtained the global existence and weak stability of weak solutions for Fermi-Dirac particles with very soft potentials, see also [28,35,36]. Jiang-Xiong-Zhou [19] and Jiang-Zhou [20] studied the incompressible Navier-Stokes-Fourier limit and the compressible Euler and acoustic limits from the quantum Boltzmann equation, respectively.…”

Section: Introductionmentioning

confidence: 99%

Existence and uniqueness of solutions to the quantum Boltzmann equation for soft potentials

Li¹

2021

Preprint

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In this paper we consider a modified quantum Boltzmann equation with the quantum effect measured by a continuous parameter δ that can decrease from δ = 1 for the Fermi-Dirac particles to δ = 0 for the classical particles. In case of soft potentials, for the corresponding Cauchy problem in the whole space or in the torus, we establish the global existence and uniqueness of non-negative mild solutions in the function spacev with small defect mass, energy and entropy but allowed to have large amplitude up to the possibly maximum upper boundThe key point is that the obtained estimates are uniform in the quantum parameter 0 < δ ≤ 1. In particular, as δ → 0 we can recover the results on the classical Boltzmann equation around global Maxwellians for which solutions may have arbitrarily large oscillations.

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On the Quantum Boltzmann Equation near Maxwellian and Vacuum

Cited by 3 publications

References 35 publications

On the emergence of quantum Boltzmann fluctuation dynamics near a Bose-Einstein Condensate

On the emergence of quantum Boltzmann fluctuation dynamics near a Bose-Einstein Condensate

Global well-posedness of the quantum Boltzmann equation for bosons interacting via inverse power law potentials

Existence and uniqueness of solutions to the quantum Boltzmann equation for soft potentials

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