Energy-driven systems: Geometry of energy landscapes and applications

能源驱动系统：能源景观和应用的几何形状

• 批准号: 0908415
• 负责人: Dejan Slepcev
• 金额: $ 11.23万
• 依托单位: Carnegie-Mellon University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2012-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0908415&HistoricalAwards=false
• 关键词: Energy; driven; systems; Geometry; energy

SlepcevDMS-0908415 Energy-driven systems are characterized by two elements: the energy being dissipated and the mechanism by which the energy is dissipated. The dissipation mechanism endows the configuration space with a geometric structure. The investigator studies the geometry of configuration spaces and relevant energy landscapes, and their applications to the dynamics of gradient flows. Gradient flows in Wasserstein metric provide a prime example, given the number of the systems that can be described as such flows and the wealth of results obtained over the last decade. The investigator continues studying gradient flows in the Wasserstein metric, but also examines gradient flows with respect to other, less studied, metrics that appear naturally in physical and biological systems. Understanding the geometry of energy landscapes enables one to make conclusions and predictions about dynamics of complex nonlinear systems. In such systems the detailed dynamics are often practically intractable, but many system-averaged quantities, such as energy, follow precise scaling laws. The investigator studies scaling laws in several energy-driven systems that display coarsening behavior. The investigator studies complex nonlinear systems relevant to applications in materials science, physics, and population biology. Being able to accurately predict the large-scale features of these systems is important for designing better man-made materials, controlling processes involving phase separation, achieving controlled self-assembly of nano-particles, and understanding (and potentially controlling) the behavior of large-scale animal groups (such as insect swarms). Additionally, he works on geometrical approaches to understanding collections of microscopy images in biology and medicine, in particular, on developing novel tools for image classification and registration, which can lead to a variety of applications to automated processing of biological and medical images.

SLEPCEVDMS-0908415能量驱动的系统的特征是两个要素：耗散的能量和能量消散的机制。 耗散机制将配置空间赋予了几何结构。 研究者研究了构型空间和相关能量景观的几何形状，及其应用于梯度流动的动力学。 考虑到可以描述为这种流量的系统的数量以及在过去十年中获得的大量结果，Wasserstein Metric中的梯度流提供了一个主要的例子。 研究者继续研究Wasserstein指标中的梯度流，但还研究了与其他在物理和生物系统中自然显然研究的，较少研究的指标相对于梯度流。 了解能量景观的几何形状使人们能够就复杂非线性系统的动态得出结论和预测。 在这样的系统中，详细的动态通常在实际上是棘手的，但是许多系统平均数量（例如能量）遵循精确的缩放定律。 研究者研究了显示出粗略行为的几个能量驱动系统的扩展定律。 研究人员研究了与材料科学，物理和人群生物学应用有关的复杂非线性系统。 能够准确预测这些系统的大规模特征对于设计更好的人为材料，控制涉及相位分离，实现纳米粒子的自由组装以及理解（以及可能控制）大规模动物群体（例如昆虫群）的行为的过程很重要。 此外，他致力于了解生物学和医学中显微镜图像的收集的几何方法，特别是开发用于图像分类和注册的新型工具，这可以导致各种应用于生物学和医学图像的自动处理。