The Geometry of Transport in Symplectic and Volume-Preserving Dynamics

辛和保体积动力学中的输运几何

基本信息

批准号：
1812481
负责人：
James Meiss
金额：
$ 38.33万
依托单位：
University of Colorado at Boulder
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-09-01 至 2023-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1812481&HistoricalAwards=false
关键词：
Geometry Transport Symplectic Volume Preserving

项目摘要

Anyone who has poured cream into hot coffee has observed the complex patterns that occur during the mixing of two fluids. It is perhaps surprising that the underlying process is not fully understood, especially when the fluid motion is laminar, i.e., either it is slow or the viscosity is high; then uniform, efficient mixing is hard to achieve. Such laminar processes are important to many applications including the development of micrometer scale bioreactors, effective mixing of polymer and granular materials, spreading of pollutants in the atmosphere, and even nutrient dispersal and spawning efficacy for life in the sea. Laminar flows can cause chaotic motion of advected particles, yielding rapid loss of accuracy for prediction of individual trajectories that turn out to be extremely sensitive to the miniscule changes in environment. The investigator and his colleagues study the design of efficient mixers by developing an understanding of the causes of this sensitivity, and by developing methods to globally optimize stirring protocols. The mathematics of these models is closely related to that, ubiquitous for conservative motion in physics. Techniques the investigator develops are used to predict the lifetime of particles in accelerators, obtain rates for elemental chemical reactions, calculate confinement times in plasma devices, understand the spectra of highly excited atomic systems, and predict asteroid and spacecraft trajectories. Graduate students will be trained through participation in this research project.Subsonic fluids are incompressible and the resulting flows are volume-preserving. Though chaotic motion in incompressible fluids is similar to that in Hamiltonian or symplectic systems, there are profound geometrical differences due to the lack of a canonical pairing between momenta and coordinates. The investigator studies how the geometry of such dynamics changes when it is "nearly" symplectic, leading to novel elliptic structures and to a discovery of the violation of the exponential quasi-stability of nearly-integrable systems implied by Nekhoroshev's theory. The studies include the development of techniques for understanding transport through destroyed invariant tori and the long-time correlations engendered by regular islands and accelerator modes. Mixing due to chaotic advection is caused by stretching and folding, which promotes homoclinic tangles and structure so fine that diffusivity can be effective even when small. The investigator and his students will model this by a finite sequence of stirring events that determine a mixing protocol. Optimization techniques under geometric and energy constraints are used to extremize Sobolev norms that give measures of mixing.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.

将奶油倒入热咖啡中的任何人都观察到在混合两种液体过程中发生的复杂图案。不足为奇的是，基础过程尚未完全理解，尤其是当流体运动是层流时，即速度很慢或粘度高；那么很难实现均匀，有效的混合。这样的层流过程对于许多应用至关重要，包括生物反应器的开发，聚合物和颗粒状材料的有效混合，大气中污染物的传播，甚至营养的分散和产卵对海中生命的效力。层流流可以引起前流颗粒的混乱运动，从而迅速丧失了对单个轨迹的预测，这些轨迹对环境的微小变化非常敏感。研究者及其同事通过对这种敏感性的原因进行理解以及开发全球优化搅拌协议的方法来研究有效混合器的设计。这些模型的数学与该模型的数学密切相关，无处不在，对于物理学的保守运动。研究者开发的技术用于预测加速器中颗粒的寿命，获得元素化学反应的速率，计算血浆设备中的限制时间，了解高度激发的原子系统的光谱，并预测小行星和飞船轨迹。研究生将通过参与该研究项目进行培训。Subsonic流体是不可压缩的，并且由此产生的流量很容易。尽管不可压缩流体中的混乱运动与哈密顿量或符号系统的混乱相似，但由于Momenta和坐标之间缺乏规范配对，因此存在很大的几何差异。研究者研究了这种动力学的几何形状在“几乎”符号符合性时如何变化，从而导致新型的椭圆结构，并发现Nekhoroshev理论所暗示的几乎可综合系统的指数准稳定性的发现。研究包括开发通过破坏的托里（Tori）理解运输的技术，以及常规岛屿和加速器模式产生的长期相关性。由于伸展和折叠而引起的混合对流引起的混合，这促进了同型缠结和结构，因此即使在较小的情况下，扩散率也可以有效。研究人员和他的学生将通过确定混合协议的搅拌事件的有限序列对此进行建模。在几何和能量限制下的优化技术用于极大化sobolev规范，以衡量混合的措施。该奖项反映了NSF的法定任务，并且使用基金会的知识分子优点和更广泛的影响审查标准，被认为值得通过评估。