Quantum Friction, Resonance Problems and New Methods in Geometric Flows

几何流中的量子摩擦、共振问题和新方法

基本信息

  • 批准号:
    1308985
  • 负责人:
  • 金额:
    $ 9.79万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Standard Grant
  • 财政年份:
    2013
  • 资助国家:
    美国
  • 起止时间:
    2013-08-15 至 2014-08-31
  • 项目状态:
    已结题

项目摘要

The problem of quantum friction concerns the motion of an invading classical particle in the Bose Einstein condensate. The PI considers the problem on a partial differential equation (PDE) model. Specifically the PI uses a nonlinear Schrodinger equation coupled to the trajectory of a classical particle, obtained after taking the mean field limit on the Bose Einstein condensate. The PI will prove the following three results: (1) if the initial speed of the classical particle is higher than the speed of sound in the Bose Einstein condensate, i.e. supersonic, then the particle will decelerate due to the friction until the speed reaches the speed of sound, (2) if the initial speed of the particle is subsonic, then the particle will travel ballistically as the time goes to infinity, (3) the whole system will converge to some inertial mode. Technically, the PI has to develop a better understanding of integro-differential equations, together with other techniques in nonlinear PDEs, for example Fermi Golden rules. The second problem is the resonance problem. The PI wants to consider the problem in the context of linear and nonlinear Schrodinger equations. Very often the presence of degeneracy, or resonance, in perturbation expansions makes the problem hard. The PI hopes to develop a better understanding of normal form transformations, and then to tackle some of the problems in Schrodinger equations. The third problem is to develop a new method for geometric flows, specifically mean curvature flow and Ricci flow. Different from the previous works, by Huisken for example, the PI mainly uses modulation equations and spectral analysis to perform almost precise estimates, instead of the maximum principle and entropy estimates. Hopefully the PI can solve some open problems here. For example, in the context of mean curvature flow, the evolving surface will collapse in finite time and form a cylinder around the collapsing point. However, whether the cylinder is unique is an open problem. The PI hopes to solve it, also a similar problem in Ricci flow.Quantum friction has many applications nowadays. One example is to test the speed of particles, for example neutrinos, by shooting the particle to some medium, for example argon. This phenomenon is known as Cerenkov radiation (Noble prize 1958). Despite its importance, the mathematical understanding of Cerenkov radiation is not satisfactory. In a broader context, the problem is in the class of non-equilibrium statistical mechanics and quantum fluid, which are popular at the moment. The second problem, the resonance problem, will deepen the understanding of normal form transformations, and help to tackle other problems, for example, in dynamical system (specifically KAM theory), and spin model in quantum mechanics. For the third problem, in the recent years, people have applied mean curvature flow to classify topological structures of different surfaces, and have estimated the amount of mass in general relativity. The PI's techniques provide more precise information on the evolution of the surfaces. Hopefully the PI's method will find applications there.
量子摩擦问题涉及玻色-爱因斯坦凝聚体中入侵的经典粒子的运动。PI在偏微分方程(PDE)模型上考虑该问题。具体来说,PI使用耦合到经典粒子轨迹的非线性薛定谔方程,在玻色爱因斯坦凝聚体上取平均场极限后获得。PI将证明以下三个结果:(1)如果经典粒子的初速度高于玻色爱因斯坦凝聚体中的声速,即超音速,则粒子将由于摩擦而减速,直到速度达到声速,(2)如果粒子的初速度是亚音速,则粒子将随着时间的推移而弹道地行进到无穷大,(3)整个系统会收敛到某个惯性模式。从技术上讲,PI必须更好地理解积分微分方程,以及非线性偏微分方程中的其他技术,例如费米黄金法则。第二个问题是共振问题。PI希望在线性和非线性薛定谔方程的背景下考虑这个问题。在微扰展开中,简并或共振的存在常常使问题变得困难。PI希望能更好地理解范式变换,然后解决薛定谔方程中的一些问题。第三个问题是发展一种新的几何流方法,特别是平均曲率流和里奇流。与Huisken等人以往的工作不同的是,PI主要利用调制方程和谱分析来进行几乎精确的估计,而不是利用极大值原理和熵估计。希望PI能解决一些悬而未决的问题。例如,在平均曲率流的上下文中,演化表面将在有限时间内塌陷,并在塌陷点周围形成圆柱。然而,圆柱体是否是唯一的是一个悬而未决的问题。PI希望解决这个问题,这也是Ricci流中的一个类似问题。一个例子是测试粒子的速度,例如中微子,通过将粒子射到某种介质中,例如氩气。这种现象被称为切伦科夫辐射(1958年诺贝尔奖)。尽管它的重要性,切伦科夫辐射的数学理解并不令人满意。在更广泛的背景下,这个问题属于非平衡统计力学和量子流体,这是目前流行的。第二个问题,共振问题,将加深对正规形变换的理解,并有助于解决其他问题,例如,在动力系统(特别是KAM理论)和量子力学中的自旋模型。对于第三个问题,近年来,人们应用平均曲率流对不同表面的拓扑结构进行分类,并在广义相对论中估计了质量的数量。PI的技术提供了关于表面演变的更精确的信息。希望PI的方法能在那里找到应用。

项目成果

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Gang Zhou其他文献

Security enhanced and cost-effective user authentication scheme for wireless sensor networks
无线传感器网络安全增强且经济有效的用户认证方案
  • DOI:
    10.5755/j01.itc.47.2.16397
  • 发表时间:
    2018
  • 期刊:
  • 影响因子:
    1.1
  • 作者:
    Wenfen Liu;Gang Zhou;Jianghong Wei;Xuexian Hu;Saru Kumari
  • 通讯作者:
    Saru Kumari
Integrated design of a lightweight metastructure for broadband vibration isolation
宽带隔振轻质元结构的集成设计
  • DOI:
    10.1016/j.ijmecsci.2022.108069
  • 发表时间:
    2022-12
  • 期刊:
  • 影响因子:
    7.3
  • 作者:
    Jianlei Zhao;Gang Zhou;Duzhou Zhang;Ivana Kovacic;Rui Zhu;Haiyan Hu
  • 通讯作者:
    Haiyan Hu
Curriculum Knowledge Representation and Manipulation in Knowledge-Based Tutoring Systems
基于知识的辅导系统中的课程知识表示和操作
  • DOI:
    10.1109/69.542023
  • 发表时间:
    1996
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Gang Zhou;J. Wang;P. Ng
  • 通讯作者:
    P. Ng
The Quenching Problem in the Nonlinear Heat Equations
非线性热方程中的淬火问题
  • DOI:
  • 发表时间:
    2006
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Gang Zhou
  • 通讯作者:
    Gang Zhou
Effect of Deoxycholic Acid on the Performance of Dye-Sensitized Solar Cell Bbased on the Black Dye
脱氧胆酸对黑色染料敏化太阳能电池B性能的影响
  • DOI:
  • 发表时间:
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Quanyou Feng;Hong Wang;Gang Zhou;Zhongsheng Wang
  • 通讯作者:
    Zhongsheng Wang

Gang Zhou的其他文献

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{{ truncateString('Gang Zhou', 18)}}的其他基金

CSR: EAGER: A Wearable Body Motion Sensing Platform Using Conductive Stretchable Fabric
CSR:EAGER:使用导电可拉伸织物的可穿戴身体运动传感平台
  • 批准号:
    1841129
  • 财政年份:
    2018
  • 资助金额:
    $ 9.79万
  • 项目类别:
    Standard Grant
Quantum Friction, Resonance Problems and New Methods in Geometric Flows
几何流中的量子摩擦、共振问题和新方法
  • 批准号:
    1801387
  • 财政年份:
    2017
  • 资助金额:
    $ 9.79万
  • 项目类别:
    Standard Grant
TWC: Small: Collaborative: Towards Energy-Efficient Privacy-Preserving Active Authentication of Smartphone Users
TWC:小型:协作:实现智能手机用户的节能隐私保护主动身份验证
  • 批准号:
    1618300
  • 财政年份:
    2016
  • 资助金额:
    $ 9.79万
  • 项目类别:
    Standard Grant
Quantum Friction, Resonance Problems and New Methods in Geometric Flows
几何流中的量子摩擦、共振问题和新方法
  • 批准号:
    1443225
  • 财政年份:
    2013
  • 资助金额:
    $ 9.79万
  • 项目类别:
    Standard Grant
CAREER: Exploiting Sensing Diversity and Conquering Communication Reality to Meet User Requirements in Performance-Critical Wireless Sensor Networks
职业:利用传感多样性并征服通信现实,以满足性能关键的无线传感器网络中的用户需求
  • 批准号:
    1253506
  • 财政年份:
    2013
  • 资助金额:
    $ 9.79万
  • 项目类别:
    Standard Grant
CSR: EAGER: Network Traffic Aware Smartphone Energy Savings
CSR:EAGER:网络流量感知智能手机节能
  • 批准号:
    1250180
  • 财政年份:
    2012
  • 资助金额:
    $ 9.79万
  • 项目类别:
    Standard Grant
NeTS:Small:Collaborative Research:Holistic Transparent Performance Assurance within the Crowded Spectrum
NetS:小型:协作研究:拥挤范围内的整体透明性能保证
  • 批准号:
    0916994
  • 财政年份:
    2009
  • 资助金额:
    $ 9.79万
  • 项目类别:
    Standard Grant
Collaborative Research: Multi-Scale QoS for Body Sensor Networks
协作研究:身体传感器网络的多尺度 QoS
  • 批准号:
    0901437
  • 财政年份:
    2009
  • 资助金额:
    $ 9.79万
  • 项目类别:
    Standard Grant

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通过 X 射线透射成像实现异种材料搅拌摩擦焊中的三维材料流动可视化
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博士后奖学金:EAR-PF:滚动、流动或断裂 - 这就是问题:研究摩擦和断层稳定性背后的机制
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    2305630
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    2024
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