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
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DOI：
10.1007/s10955-023-03082-x
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发表时间：
2023-04
影响因子：
1.6
通讯作者：
Thomas Chen;M. Hott
中科院分区：
文献类型：
--
作者：
Thomas Chen;M. Hott
In this work, we study the quantum fluctuation dynamics in a Bose gas on a torusthat exhibits Bose–Einstein condensation, beyond the leading order Hartree–Fock–Bogoliubov (HFB) theory. Given a Bose–Einstein condensate (BEC) with densitysurrounded by thermal fluctuations with density 1, we assume that the system dynamics is generated by a Hamiltonian with mean-field scaling. We derive a quantum Boltzmann type dynamics from a second-order Duhamel expansion upon subtracting both the BEC dynamics and the HFB dynamics, with rigorous error control. Given a quasifree initial state, we determine the time evolution of the centered correlation functions,,at mesoscopic time scales, whereis the coupling constant determining the HFB interaction, anda,denote annihilation and creation operators. While the BEC and the HFB fluctuations both evolve at a microscopic time scale, the Boltzmann dynamics is much slower, by a factor. For large but finiteN, we consider both the case of fixed system size, and the case. In the case, we show that the Boltzmann collision operator contains subleading terms that can become dominant, depending on time-dependent coefficients assuming particular values in; 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. In either of those cases, we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \sim \Big (\frac{\log \log N}{\log N}\Big )^{\alpha }$$\end{document}, for different values of.
In this work, we study the quantum fluctuation dynamics in a Bose gas on a torusthat exhibits Bose–Einstein condensation, beyond the leading order Hartree–Fock–Bogoliubov (HFB) theory. Given a Bose–Einstein condensate (BEC) with densitysurrounded by thermal fluctuations with density 1, we assume that the system dynamics is generated by a Hamiltonian with mean-field scaling. We derive a quantum Boltzmann type dynamics from a second-order Duhamel expansion upon subtracting both the BEC dynamics and the HFB dynamics, with rigorous error control. Given a quasifree initial state, we determine the time evolution of the centered correlation functions,,at mesoscopic time scales, whereis the coupling constant determining the HFB interaction, anda,denote annihilation and creation operators. While the BEC and the HFB fluctuations both evolve at a microscopic time scale, the Boltzmann dynamics is much slower, by a factor. For large but finiteN, we consider both the case of fixed system size, and the case. In the case, we show that the Boltzmann collision operator contains subleading terms that can become dominant, depending on time-dependent coefficients assuming particular values in; 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. In either of those cases, we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \sim \Big (\frac{\log \log N}{\log N}\Big )^{\alpha }$$\end{document}, for different values of.