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

Vainchtein/Grigoriev 本研究计划的目标是构建一个一般的定量理论的长时间共振驱动的混沌运输和混合在近可积的自治和非自治体积保持流。微尺度流的具体例子将被用来说明一般的方法，并开发特定的工具，可以自然地推广到广泛的一类体积保持和哈密顿系统。共振过程的确定性理论将与随机游走理论和稳定岛理论相结合，以建立具有分离面和/或共振的系统中拉格朗日输运的统计长期描述。计划的方法的一个独特之处是，它将适用于当混沌平流是唯一的运输机制，以及当混沌平流与热或分子扩散竞争。这项工作包括集成的经常断开的方法和技术，用于描述共振相互作用和定期运输到一个一般的运输理论的近可积系统，和开发一种新的技术来量化混合速率，彻底性和不可压缩流体流动的均匀性。研究所的研究直接应用于科学和工程中的广泛问题，例如彗星和小行星在太阳系中的运输，凝聚态物质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，并应用于大气科学和磁约束聚变装置。其关键应用是微流体领域，通过其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。该研究计划将与本科和研究生阶段的教学紧密结合，并将包括旨在增加代表性不足的群体参与研究和将研究进展融入课程的活动。PI还将寻求与等离子体物理界扩大和建立微流体合作。

0900177/0900018 Vainchtein/Grigoriev The goal of this 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.