The Geometry of Transport in Symplectic and Volume-Preserving Dynamics

任何一个把奶油倒进热咖啡里的人都观察到了两种液体混合过程中出现的复杂模式。也许令人惊讶的是，基本的过程还没有完全理解，特别是当流体运动是层流时，即，或者速度慢，或者粘度高，则难以实现均匀、有效的混合。这种层流过程对许多应用都很重要，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，甚至是海洋生物的营养物质扩散和产卵功效。层流会导致平流粒子的混沌运动，从而导致预测单个轨迹的准确性迅速下降，这些轨迹对环境中的微小变化非常敏感。研究人员和他的同事通过了解这种敏感性的原因，并通过开发方法来全局优化搅拌方案，来研究高效混合器的设计。这些模型的数学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，计算等离子体设备中的约束时间，了解高激发原子系统的光谱，以及预测小行星和航天器的轨迹。研究生将通过参与这个研究项目得到培训。亚音速流体是不可压缩的，产生的流动是体积保持的。虽然不可压缩流体中的混沌运动类似于哈密顿或辛体系中的混沌运动，但由于动量和坐标之间缺乏正则配对，因此存在深刻的几何差异。研究人员研究如何几何的动态变化时，它是“近”辛，导致新的椭圆形结构和发现违反指数准稳定性的近可积系统隐含的Nekhoroshev的理论。这些研究包括通过被破坏的不变环面和由规则岛屿和加速器模式产生的长时间相关性来理解运输的技术的发展。由于混沌平流的混合是由拉伸和折叠引起的，这促进了同宿缠结和结构如此精细，以至于扩散率即使在很小的时候也是有效的。研究者和他的学生将通过确定混合协议的有限序列的搅拌事件来对此进行建模。几何和能量约束下的优化技术被用来极端化索博列夫规范，给出混合的措施。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.