Collaborative Research: Long-Term Chaotic Transport in Volume-Preserving Flows

0900177/0900018 Vainchtein/Grigoriev这项合作研究计划的目标是构建近可积自治和非自治保容流动中长时间共振驱动的混沌输运和混合的一般量化理论。微尺度流动的具体例子将被用来说明一般的方法和开发特定的工具，这些工具可以自然地推广到一大类体积守恒和哈密顿系统。共振过程的确定性理论将与随机游动理论和稳定岛理论相结合，发展具有分离线和/或共振的系统中拉格朗日输运的统计长期描述。计划方法的一个独特特点是，它既适用于混沌平流是唯一传输机制的情况，也适用于混沌平流与热扩散或分子扩散竞争的情况。这项工作包括将用于描述共振相互作用和规则输运的经常断开的方法和技术集成到近可积系统的一般输运理论中，以及开发一种新的技术来量化不可压缩流体流动的混合率、彻底性和均匀性。PIS的研究直接应用于科学和工程中的广泛问题，如彗星和小行星在太阳系中的传输，凝聚态物质中激发模式之间的能量交换，以及带电粒子在电磁场中的运动，以及在大气科学和磁约束聚变设备中的应用。关键的应用是在微流控领域，它通过对化学处理和传感器技术的影响，有望在药物发现、医疗诊断和国家安全方面取得重大进展。研究方案将与本科生和研究生的教与学紧密结合，并将包括旨在增加未被充分代表的群体参与研究和将研究进展纳入课程的活动。PIS还将寻求扩大和建立与等离子体物理界的微流体合作。

0900177/0900018Vainchtein/Grigoriev The goal of this collaborative research program is to construct a general quantitative theory of long-time resonance-driven chaotic transport and mixing in near-integrable autonomous and non-autonomous volume-preserving flows. Specific examples of microscale flows will be used to illustrate the general approach and develop specific tools that can be naturally generalized to a wide class of volume-preserving and Hamiltonian systems. The deterministic theory of resonance processes will be combined with the theory of random walks and theory of stability islands to develop a statistical long-term description of the Lagrangian transport in systems with separatrices and/or resonances. A unique feature of the planned approach is that it will apply both when chaotic advection is the only transport mechanism as well as when chaotic advection competes with thermal or molecular diffusion. This work includes integration of often disconnected methods and techniques used to describe resonant interactions and regular transport into a general transport theory for near-integrable systems, and development of a novel technique to quantify mixing rate, thoroughness, and uniformity for incompressible fluid flows. The PIs' research has direct applications to a wide range of problems in science and engineering, such as the transport of comets and asteroids through the solar system, energy exchange between excitation modes in condensed matter, and motion of charged particles in electromagnetic fields with applications to atmospheric science and magnetic confinement fusion devices. The key application is in the field of microfluidics which promises major advances in drug discovery, medical diagnostics, and national security through its impact on chemical processing and sensor technology. The research program will be tightly integrated with teaching and learning at the undergraduate and graduate levels and will include activities aimed at increased participation of underrepresented groups in research and integration of research advances into the curriculum. The PIs will also seek to extend and establish microfluidics collaborations with the plasma physics community.