Fractal Geometry for Dynamics on Random Media

随机介质动力学的分形几何

• 批准号: 1855550
• 负责人: Alan Hammond
• 金额: $ 21万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-01 至 2022-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1855550&HistoricalAwards=false
• 关键词: Fractal; Geometry; Dynamics; Random; Media

When a laser pulse is fired into a liquid crystal, causing an unstable region to grow from where the pulse makes contact, or when the stain of an ink blot spreads on paper, interfaces grow in the presence of a locally disordered environment. A central task of the rigorous theory of statistical mechanics is to analyze which features of physical systems are universal, shared by many models with apparently very different microscopic specifications. Understanding of the Kardar-Parisi-Zhang universality class, which concerns many surfaces growing with local randomness, has been greatly advanced by many mathematicians and physicists over the last forty years. Often these advances have been aided by the use of integrable techniques, in which sometimes rather special models are analyzed via algebraic tools or exact formulas. The Principal Investigator will analyze certain important universality classes, including the Kardar-Parisi-Zhang class, by using probabilistic proof methods in unison with limited but crucial integrable inputs. The structure of certain random fractal geometries arising in scaling limits of dynamics on random media will thus be explicated. The proposed probabilistic investigation of fractal geometries has three principal components. The first concerns anomalous transport in disordered random media. Namely, when a particle with a uniform tendency to move in a preferred direction does so in a milieu that is random and strongly disordered, trapping is a vital phenomenon that interrupts and sometimes eliminates linear progress. The phenomenon will be elucidated by a detailed inquiry into biased motion on the infinite cluster of supercritical percolation in the integer lattice of dimensions at least two. The second direction concerns scaled last passage percolation. Rich aspects of Kardar-Parisi-Zhang universality will be explored by studying the scaled random geometry of extremal paths moving through random fields using probabilistic modes of proof such as resampling of random energy profiles and surgery on paths. The final principal component will address noise sensitivity and last passage percolation. The stability and sensitivity of last passage percolation paths under perturbation by noise of the random environment will be studied, strengthening the bridge between two key theories: Kardar-Parisi-Zhang universality and discrete harmonic analysis.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

当激光脉冲射入液晶，导致不稳定区域从脉冲接触的地方生长，或者当墨水印迹的污渍在纸上扩散时，界面在局部无序环境的存在下生长。严格统计力学理论的一个中心任务是分析物理系统的哪些特征是普遍的，被许多具有明显差异的微观规范的模型所共有。Kardar-Parisi-Zhang普适性类涉及许多具有局部随机性增长的曲面，在过去的40年里，许多数学家和物理学家已经极大地推进了对它的理解。这些进展往往是通过使用可积技术，其中有时相当特殊的模型分析通过代数工具或精确的公式。主要研究者将分析某些重要的普适类，包括Kardar-Parisi-Zhang类，通过使用概率证明方法与有限但关键的可积输入相结合。因此，在随机介质上的动力学的标度极限中产生的某些随机分形几何的结构将被解释。分形几何的概率研究有三个主要组成部分。第一个是无序随机介质中的反常输运。也就是说，当一个粒子在一个随机的、高度无序的环境中以一种均匀的趋势向一个优选的方向移动时，捕获是一种重要的现象，它会中断甚至消除线性进程。这一现象将阐明了详细的调查偏向运动的无限集团的超临界渗流在整数晶格的尺寸至少为2。第二个方向涉及缩放的最后一个通道渗流。Kardar-Parisi-Zhang普适性的丰富方面将通过研究通过随机场移动的极端路径的缩放随机几何来探索，使用概率模式的证明，例如随机能量分布的恢复和路径上的手术。最后的主成分将解决噪声敏感性和最后一次通过渗透。将研究随机环境噪声扰动下最后一段渗流路径的稳定性和敏感性，加强两个关键理论之间的桥梁：Kardar-Parisi-Zhang普适性和离散谐波分析。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。