Variational Theories for Defects and Patterns

缺陷和模式的变分理论

• 批准号: 0808059
• 负责人: Nicholas Ercolani
• 金额: $ 21.5万
• 依托单位: University of Arizona
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-09-15 至 2012-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0808059&HistoricalAwards=false
• 关键词: Variational; Theories; Defects; Patterns

The principal physical impetus for the projects studied in this proposal is to gain an analytical understanding of pattern formation. In particular the PI seeks to characterize the classes of defects that appear when one is far above the parameter threshold at which patterns emerge. These issues are studied through a non-convex variational problem known as the regularized Cross-Newell (RCN) model. Unlike similar models that have been studied recently, RCN incorporates features of non-trivial twist which enable the existence of a richer ?taxonomy? of defects (in particular, concave and convex disclinations). Such features are indeed seen in experiments and numerical simulations of the underlying microscopic physical equations. The issue has been to determine whether or not the variational model can capture these defects. In a recent work the PI has demonstrated that the minimizers of a variational model with twist must necessarily differ from those without twist. The main projects of this proposal are centered on establishing a precise analytical characterization of the competition between microstructure formation and overall topologically induced energetic stress in versions of the RCN model having some geometric symmetry. The study of pattern forming systems is a fundamental area of scientific investigation in which the tools of modern mathematical analysis can be brought to bear on the modeling of physical systems especially near a critical transition in the behavior of the system. For the projects studied in this proposal one is principally interested in patterns that arise when a continuous translational symmetry is reduced, at a critical threshold, to a discrete periodic symmetry resulting in what is often referred to as a "striped" pattern. Such patterns are for instance generic in Rayleigh-Benard convection (RBC) which is a principal theoretical model for the formation of weather patterns. In RBC the striped pattern corresponds to the formation, at a critical temperature, of periodic "convection rolls" of a uniform characteristic width. A major goal of this research is to study not just the patterns that form at a critical threshold but to characterize the types of defects that arise in these patterns when one is far from threshold. This work will make definite predictions on defect formation that can be tested in the laboratory. This will have relevance for modeling defect structure in liquid crystals, in animal coat patterns (including fingerprints) and in the evolution of plant patterns.

该提案中研究的项目的主要物理动力是对模式形成的分析理解。特别是，PI试图表征当一个出现模式出现的参数阈值时出现的缺陷类别。这些问题是通过称为正则化跨纽维尔（RCN）模型的非凸变量问题研究的。与最近研究的类似模型不同，RCN结合了非平凡扭曲的特征，从而可以存在更丰富的分类法？缺陷（尤其是凹形和凸脱节）。确实可以看到这种特征在实验和基础微观物理方程的数值模拟中。问题是确定变分模型是否可以捕获这些缺陷。在最近的一项工作中，PI表明，具有扭曲的变异模型的最小化必须与没有扭曲的模型有所不同。该提案的主要项目集中在建立微观结构形成与总体拓扑诱导的能量应力之间的竞争的精确分析表征，该版本具有一些几何对称性。对模式成型系统的研究是科学研究的一个基本领域，在该领域中，可以将现代数学分析的工具依靠物理系统的建模进行，尤其是在系统行为的关键过渡附近。对于本提案中研究的项目，一个主要对在临界阈值下降到离散的周期性对称性的持续翻译对称性时产生的模式感兴趣，从而导致通常称为“条纹”模式。这样的模式是例如雷利 - 贝纳德对流（RBC）中的通用模式，它是天气模式形成的主要理论模型。在RBC中，条纹图案对应于均匀特征宽度的周期性“对流”的临界温度的形成。这项研究的一个主要目的是不仅研究形成临界阈值的模式，而且要表征当一个模式远离阈值时出现的缺陷类型。这项工作将对可以在实验室中进行测试的缺陷形成做出明确的预测。这将与在液晶，动物外套图案（包括指纹）和植物模式的演化中建模缺陷结构相关。