Exploiting emergent scale invariance

非技术概述该奖项支持理论和计算研究和教育，旨在将强大的基本理论磨练成实用的方法，以更现实地描述各种现象。我们世界的重要部分在许多尺度上波动。云层稀疏，分形图凹凸不平，爆米花发出嘎吱嘎吱的声音。(地球--地震也是如此)。材料，如金属合金和生物细胞膜，当它们的成分设置在特定点附近时，可能会在许多尺度上波动。50年前，对这些波动的解释被发现是一种尺度不变性：这些系统放大后看起来几乎一样，例如大小云束看起来相似。这种标度不变性是由一种理论解释的，该理论描述了当系统放大时，支配这些系统的定律是如何变化的。膨化米饭中的几缕云和牛奶的规则随着规模的变化而变得相同。但这一优雅的理论还没有发挥出它的潜力，例如在预测天气方面，或者在描述骨头、混凝土和海壳等材料如何形成和破裂方面。科学可以解释湍流旋涡中的旋涡，但它需要将最大的旋涡与工程师设计飞机机翼的模型或气象学家的海岸和山脉相结合。这个项目将建立在PI最近开发的方法的基础上，以更全面和系统地解决这一理论。PI旨在将这一优雅的物理理论转化为实际工程工具的骨架。特别是，除了这些特殊的、比例不变的点之外，还有一些成熟的方法来计算材料和合金的性质。PI将开发工具，将这两种描述合并为一个集成的、强大的工具，用于描述复杂的材料和系统。技术总结该奖项支持理论和计算研究和教育，旨在扩大重整化小组的数量有效性，使其能够参与更现实的问题。理解临界现象和新出现的尺度不变性是解决我们许多科学和工程挑战的关键。目前该领域的表述分为对普遍标度定律的推崇和对精确可解系统(例如2D Ising模型)和特殊点附近系统(例如一维和四维Ising模型)的密集而难以理解的计算。这些后一种计算经常违反通常预期的天真的幂定律和Widom伸缩。PI最近利用分支理论的数学原理将这些异常的普适性类排列成普适性家族，表征了标度变量的对数、指数和不变组合，其中许多是第一次。这些新方法与将解析和奇异校正结合到标度中的已知技术密切相关，PI将新的校正临界点奇点与标度校正结合在一起，以显著扩展定量有效性的范围。该项目将使用2D非平衡随机场伊辛模型来说明新的预测标度形式和修正对标度的作用，生成一个量化理论，预测雪崩大小和磁化曲线在巨大的无序范围内，直到大小尺度变得像晶格结构一样小。该项目还将开发工具，为细胞膜实验提取整个相图和伴随的材料行为，使用微乳液和三临界伊辛现象的多参数膜模型的复杂相图来研磨和验证工具。

NONTECHNICAL SummaryThis award supports theoretical and computational research and education that aims to hone a powerful fundamental theory into a practical approach toward a more realistic description of a wide variety of phenomena. Important parts of our world fluctuate on many scales. Clouds are wispy, fractals are bumpy, and Rice Krispies crackle. (So does the Earth - earthquakes). Materials, like metal alloys and the membranes of biological cells, can fluctuate on many scales, when their compositions are set near special points. Fifty years ago, the explanation for these fluctuations was discovered to be a kind of scale invariance: these systems look nearly the same after they are magnified, for example big and small cloud wisps look similar. This scale invariance was explained by a theory, describing how the laws governing these systems change when the systems are magnified. The wisps of clouds and the milk in puffed rice have rules that become the same upon changing scale. But this elegant theory has not reached its potential, for example in predicting the weather, or in describing how materials like bone, concrete, and sea-shells form and break. Science can explain the whorls within whorls of turbulence, but it needs to mesh the biggest whorls into the engineer's models designing airplane wings, or the meteorologist's seashores and mountains. This project will build on the PI's recently developed methods for solving this theory more completely and systematically. The PI aims to turn this elegant theory of physics into the bones of a practical engineering tool. In particular, far from these special, scale-invariant points there are well-established methods to calculate the properties of materials and alloys. The PI will develop tools to merge these two descriptions into an integrated, powerful tool for describing complex materials and systems. TECHNICAL SUMMARYThis award supports theoretical and computational research and education that aims to extend the quantitative validity of the renormalization group and enable it to engage more realistic problems. Understanding critical phenomena and emergent scale invariance is a key to many of our scientific and engineering challenges. The current formulation of the field is split between an admiration of universal scaling laws and dense, inscrutable calculations for exactly solvable systems, for example the 2D Ising model, and systems near special points, for example the Ising model in 1D and 4D. These latter calculations often violate the naive power laws and Widom scaling usually expected. The PI has recently used the mathematics of bifurcation theory to arrange these anomalous universality classes into universality families, characterizing the logarithms, exponentials, and invariant combinations of scaling variables, many for the first time. These new methods are closely associated with known techniques for incorporating analytic and singular corrections to scaling, and the PI will combine the new correct critical-point singularities with corrections to scaling to dramatically extend the realm of quantitative validity. The project will use the 2D nonequilibrium random-field Ising model to illustrate both the new predicted scaling forms and the role of corrections to scaling, generating a quantitative theory predicting avalanche sizes and magnetization curves over an enormous range of disorder, until the size scale becomes as small as that of the lattice structure. The project will also develop tools to extract entire phase diagrams and concomitant materials behavior for experiments on cellular membranes, honing and validating the tools using the complex phase diagram of a multiparameter membrane model for microemulsions and tricritical Ising phenomena.