The Kardar-Parisi-Zhang (KPZ) Universality of Random Growing Interfaces

随机增长界面的 Kardar-Parisi-Zhang (KPZ) 普遍性

基本信息

批准号：
2321493
负责人：
Kanstantsin Matetski
金额：
$ 14.9万
依托单位：
Michigan State University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2023
资助国家：
美国
起止时间：
2023-02-15 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2321493&HistoricalAwards=false
关键词：
Kardar Parisi Zhang KPZ Universality

项目摘要

The project concerns universal large-scale behavior of random growing interfaces, which are mathematical models that describe real physical processes such as propagating fire fronts, growing liquid crystals, bacterial colonies and coffee stains. These processes are typically modeled using the statistical mechanics approach, i.e. by considering large systems of interacting particles, where each particle corresponds to a molecule or an individual bacterium. Imprecision of our measurements and dependence of physical processes on many factors are typically described by random perturbations of particles. Large-scale behavior of such models is observed when one looks at the systems from a sufficiently large distance, after a sufficiently long period of time. Universality in this case refers to the fact that such growing interfaces exhibit similar large-scale behavior. Description of this universal behavior gives a better understanding of the real physical processes in nature. Recent breakthroughs in probability theory, particularly in stochastic partial differential equations and integrable probability, have provided the tools to study such mathematical models.The Kardar-Parisi-Zhang (KPZ) universality arises in non-equilibrium statistical mechanics, when studying limiting behavior of random growing interfaces. On the mathematical level, growing interfaces appear in free energies of directed random polymers, random growth models, interacting particle systems, stochastic Burgers and Hamilton-Jacobi-Bellman equations, and stochastically perturbed reaction-diffusion equations. Depending on the characteristics of models, two universal objects are conjectured to govern such interfaces: the KPZ equation and the KPZ fixed point. Recent developments in the area of stochastic PDEs allow to prove scaling limits of various discrete systems to singular stochastic PDEs. Moreover, methods of integrable probability (exactly solvable models) provided a complete characterization of the KPZ fixed point. The goal of this project is to study universal scaling limits of such random growing interfaces using these new results.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.

该项目涉及随机生长界面的通用大规模行为，这些行为是数学模型，它们描述了真实的物理过程，例如传播火锋，不断增长的液晶，细菌菌落和咖啡污渍。这些过程通常是使用统计力学方法进行建模的，即通过考虑大型相互作用颗粒的系统，其中每个粒子对应于分子或单个细菌。不精确地测量和物理过程对许多因素的依赖性通常是通过颗粒的随机扰动来描述的。当人们从足够长的时间后从足够长的距离看系统时，就会观察到这种模型的大规模行为。在这种情况下，普遍性是指这样的界面表现出相似的大规模行为。对这种普遍行为的描述可以更好地理解自然界的真实物理过程。概率理论的最新突破，尤其是在随机部分微分方程和可集成的概率中，已经提供了研究此类数学模型的工具。在研究非平衡统计机制中，Kardar-Parisi-Zhang（KPZ）普遍性在研究中会出现。增长的界面。在数学层面上，生长的接口出现在定向的随机聚合物，随机生长模型，相互作用的粒子系统，随机汉堡和汉密尔顿 - 雅各布利 - 贝尔曼方程以及随机扰动的反应 - 扩散方程中。根据模型的特征，两个通用对象被猜想以控制此类接口：KPZ方程和KPZ固定点。随机PDE区域的最新发展允许证明各种离散系统的缩放限制到奇异的随机PDE。此外，可集成概率的方法（确切的可解决模型）提供了KPZ固定点的完整表征。该项目的目的是使用这些新结果研究这种随机增长界面的通用缩放限制。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛的影响评估标准通过评估来支持的。