Mathematical Analysis of Dispersion and Transport in Quantum Dynamics

量子动力学中色散和输运的数学分析

• 批准号: 2009800
• 负责人: Thomas Chen
• 金额: $ 50.54万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-08-01 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2009800&HistoricalAwards=false
• 关键词: Mathematical; Analysis; Dispersion; Transport; Quantum

This research addresses problems emerging from the physics of large systems of interacting quantum mechanical particles (of boson or fermion type), analyzed with rigorous mathematical methods. The PI will study the effects of impelling a heavy quantum tracer particle into a gas of bosons, and investigate the dynamical behavior of quantum fluctuations when the phenomenon of Bose-Einstein condensation occurs. In addition, the PI will study the motion of electrons (which are fermions) in disordered media, describing systems such as semiconductors. In another line of research, the PI and his collaborators have recently introduced an approach to study equations describing classical gases and fluids with methods in quantum mechanics, using a mathematical device known as the Wigner transform; this method will be further developed in this project. The projects includes training opportunities for graduate students and postdoctoral researchers. The PI will study the fluctuation dynamics around a Bose-Einstein condensate on the level of the Hartree-Fock-Bogoliubov approximation and beyond, using a combination of methods from nonlinear partial differential equations and renormalization in quantum field theory. Moreover, the PI will study the derivation of effective equations for systems of fermions propagating in a random medium, in the context of the weakly disordered manybody Anderson model with mean field interaction. The analysis of classical Boltzmann equations via Wigner transform and methods of dispersive nonlinear partial differential equations will be further developed. An important outcome of the analysis of these problems is the development of new mathematical techniques merging methods from nonlinear partial differential equations, harmonic analysis, and quantum field theory.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

这项研究解决了相互作用的量子力学粒子（玻色子或费米子类型）大系统物理学中出现的问题，并用严格的数学方法进行分析。PI将研究将重量子示踪粒子推进玻色子气体的影响，并研究玻色-爱因斯坦凝聚现象发生时量子涨落的动力学行为。此外，PI将研究电子（费米子）在无序介质中的运动，描述半导体等系统。在另一项研究中，PI和他的合作者最近引入了一种方法，使用称为Wigner变换的数学设备，用量子力学方法研究描述经典气体和流体的方程;该方法将在该项目中进一步发展。这些项目包括为研究生和博士后研究人员提供培训机会。PI将在Hartree-Fock-Bogoliubov近似及更高的水平上研究玻色-爱因斯坦凝聚体周围的波动动力学，使用非线性偏微分方程和量子场论中的重整化方法的组合。此外，PI将研究在随机介质中传播的费米子系统的有效方程的推导，在弱无序多体安德森模型与平均场相互作用的背景下。通过维格纳变换和色散非线性偏微分方程的方法的经典玻尔兹曼方程的分析将进一步发展。这些问题的分析的一个重要成果是新的数学技术的发展融合方法从非线性偏微分方程，谐波分析，量子场论。这个奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。

项目成果

On the Emergence of Quantum Boltzmann Fluctuation Dynamics near a Bose–Einstein Condensate

• DOI: 10.1007/s10955-023-03082-x
• 发表时间: 2023-04
• 期刊: Journal of Statistical Physics
• 影响因子: 1.6
• 作者: Thomas Chen;M. Hott

Small Data Global Well-Posedness for a Boltzmann Equation via Bilinear Spacetime Estimates

通过双线性时空估计的玻尔兹曼方程的小数据全局适定性

• DOI: 10.1007/s00205-021-01613-y
• 发表时间: 2021
• 期刊: Archive for Rational Mechanics and Analysis
• 影响因子: 2.5
• 作者: Chen, Thomas;Denlinger, Ryan;Pavlović, Nataša
• 通讯作者: Pavlović, Nataša

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

玻色子的时变 Hartree→Fock→Bogoliubov 方程

• DOI: 10.1007/s00028-022-00799-2
• 发表时间: 2022
• 期刊: Journal of Evolution Equations
• 影响因子: 1.4
• 作者: Bach, Volker;Breteaux, Sebastien;Chen, Thomas;Fröhlich, Jürg;Sigal, Israel Michael
• 通讯作者: Sigal, Israel Michael