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.

任何将奶油倒入热咖啡中的人都观察到两种液体混合过程中发生的复杂模式。也许令人惊讶的是，潜在的过程尚未完全被理解，特别是当流体运动是层流时，即要么缓慢，要么粘度高；那么均匀、高效的混合就很难实现。这种层流过程对于许多应用都很重要，包括微米级生物反应器的开发、聚合物和颗粒材料的有效混合、污染物在大气中的扩散，甚至海洋生物的营养物扩散和产卵功效。层流会导致平流粒子的混沌运动，从而导致对环境微小变化极其敏感的单个轨迹预测的准确性迅速下降。研究人员和他的同事通过了解这种敏感性的原因并开发全局优化搅拌方案的方法来研究高效混合器的设计。这些模型的数学原理与物理学中普遍存在的保守运动密切相关。研究人员开发的技术用于预测加速器中粒子的寿命，获得元素化学反应的速率，计算等离子体装置中的限制时间，了解高度激发的原子系统的光谱，以及预测小行星和航天器的轨迹。研究生将通过参与该研究项目接受培训。亚音速流体是不可压缩的，并且产生的流动是体积保持的。尽管不可压缩流体中的混沌运动与哈密顿系统或辛系统中的混沌运动相似，但由于动量和坐标之间缺乏规范配对，因此存在深刻的几何差异。研究人员研究了这种动力学的几何形状在“接近”辛时如何变化，从而产生新颖的椭圆结构，并发现违反了涅霍罗舍夫理论所暗示的近可积系统的指数准稳定性。这些研究包括开发通过被破坏的不变环面理解输运的技术以及由常规岛屿和加速器模式产生的长期相关性。混沌平流引起的混合是由拉伸和折叠引起的，这促进了同宿缠结和结构的精细化，使得扩散率即使在很小的时候也能有效。研究人员和他的学生将通过确定混合方案的有限搅拌事件序列对此进行建模。几何和能量约束下的优化技术用于极端化提供混合测量的 Sobolev 规范。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力优点和更广泛的影响审查标准进行评估，被认为值得支持。