Exploiting emergent scale invariance

该奖项支持理论和计算研究和教育，旨在将强大的基础理论磨练成一种实用的方法，以更真实地描述各种现象。 我们世界的重要部分在许多尺度上波动。云是纤细的，分形是凹凸不平的，脆米饼是噼啪作响的。(So地震（Earthquakes）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，可以在许多尺度上波动。50年前，人们发现这些波动的原因是一种尺度不变性：这些系统在放大后看起来几乎一样，例如，大云和小云看起来很相似。这种尺度不变性可以用一种理论来解释，它描述了当系统被放大时，控制这些系统的定律是如何变化的。云的缕缕和爆米花中的牛奶有规则，在改变尺度时变得相同。但是，这个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、混凝土和贝壳等材料如何形成和破裂方面。科学可以解释漩涡中的漩涡，但它需要将最大的漩涡与设计飞机机翼的工程师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最近开发的更完整和系统地解决这个理论的方法。PI的目标是将这个优雅的物理学理论转化为实用工程工具的核心。特别是，远离这些特殊的，标度不变的点，有完善的方法来计算材料和合金的属性。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。该奖项支持理论和计算研究和教育，旨在扩展重整化群的定量有效性，使其能够参与更现实的问题。理解临界现象和涌现尺度不变性是我们许多科学和工程挑战的关键。目前的场的表述分为两种，一种是对普适标度律的崇拜，另一种是对精确可解系统（例如二维伊辛模型）和特殊点附近系统（例如一维和四维伊辛模型）的密集、难以理解的计算。这些后一种计算经常违反通常预期的朴素幂律和Widom缩放。PI最近使用分叉理论的数学将这些异常普适性类安排到普适性家族中，表征了尺度变量的非线性、指数和不变组合，其中许多是第一次。这些新的方法与已知的技术密切相关，用于将分析和奇异校正结合到缩放中，PI将联合收割机将新的正确的临界点奇异性与缩放校正相结合，以显著地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预测的缩放形式和缩放校正的作用，产生一个定量理论，预测雪崩大小和磁化曲线在巨大的无序范围内，直到尺寸尺度变得像晶格结构一样小。该项目还将开发工具来提取整个相图和伴随的材料行为，用于细胞膜实验，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磨练和验证工具。

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.