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Growth and Motion in a Random Medium

随机介质中的生长和运动

基本信息

批准号：
2152362
负责人：
Timo Seppalainen
金额：
$ 33.16万
依托单位：
University of Wisconsin-Madison
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-06-01 至 2025-05-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2152362&HistoricalAwards=false
关键词：
Growth Motion Random Medium

项目摘要

This project is fundamental research on mathematical models that describe complex interactions, growth, and motion in an irregular environment with stochastic unpredictability. The goal is to discover general mathematical laws that govern such systems. These systems appear quite different at small scales and large scales. So it is important to understand how different rules for small-scale evolution lead to different large-scale, system-wide behavior. Real-world phenomena that such mathematical studies can illuminate include the motion of vehicles, packets in communication networks, fluid particles in a tube, fluid spreading in a porous medium, epidemics advancing in a population, and the fluctuations of a polymer chain in a fluid. Laboratory experiments have demonstrated that these mathematical models capture essential features of physical reality. Over the long term, understanding complex interactions has profound implications for science and engineering and thereby for society. Mathematical systems of the kind described in the proposal are intensely and concurrently studied by mathematicians, natural scientists, social scientists, and engineers. This project provides research training opportunities for graduate students. This project investigates mathematical models of growth and motion in random media. Examples include first-passage percolation, the corner growth model, random walk in random environment, directed polymer models, and stochastic partial differential equations. The objectives of this work are mathematically rigorous descriptions of the behavior of these models and the development of robust tools for their analysis. Specific goals include regularity of limit shapes, properties of optimal paths such as their length, fluctuations and geometric features, ergodic properties, descriptions of large scale limits in terms of variational formulas and entropy, and descriptions of probability distributions of complicated random geometric objects such as trees of geodesics and competition interfaces. The methods employed in this work are those of rigorous mathematical research, aided by experimental computer simulation.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.

该项目是对数学模型的基础研究，该模型描述了具有随机不可预测性的不规则环境中的复杂相互作用，生长和运动。其目标是发现支配这些系统的一般数学定律。这些系统在小尺度和大尺度上表现出很大的不同。因此，了解小规模进化的不同规则如何导致不同的大规模系统行为是很重要的。这种数学研究可以阐明的现实世界现象包括车辆的运动、通信网络中的数据包、管道中的流体粒子、多孔介质中的流体传播、人口中的流行病传播以及流体中聚合物链的波动。实验室实验表明，这些数学模型捕捉物理现实的基本特征。从长远来看，理解复杂的相互作用对科学和工程，从而对社会具有深远的影响。数学家、自然科学家、社会科学家和工程师都在同时深入地研究提案中所描述的数学系统。该项目为研究生提供了研究培训机会。本项目研究随机介质中生长和运动的数学模型。例子包括第一次通过渗流，角增长模型，随机环境中的随机行走，定向聚合物模型和随机偏微分方程。这项工作的目标是数学上严格的描述这些模型的行为和强大的工具，他们的分析发展。具体目标包括规则性的限制形状，最佳路径的属性，如它们的长度，波动和几何特征，遍历属性，描述大规模的限制在变分公式和熵，和描述的概率分布的复杂随机几何对象，如树的测地线和竞争接口。在这项工作中采用的方法是严格的数学研究，辅助实验计算机模拟。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。