Stochastic Systems with Complex Interactions and Random Environments

具有复杂相互作用和随机环境的随机系统

• 批准号: 1306777
• 负责人: Timo Seppalainen
• 金额: $ 31.37万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1306777&HistoricalAwards=false
• 关键词: Stochastic; Systems; Complex; Interactions; Random

This project studies mathematical models of random paths in random environments and percolation models that describe random growth. The goal is to describe typical large scale behavior and to quantify deviations from the typical behavior. The emphasis is on finding universal principles that apply to classes of models that share fundamental characteristics. There are two main directions of research. (i) The study of the combinatorics, probability distributions and large deviations of 1+1 dimensional exactly solvable models in the Kardar-Parisi-Zhang universality class. Outcomes of this research include rigorous verification of fluctuation exponents conjectured in the physics literature, new connections between combinatorics and models from probability and statistical physics, and better understanding of the integrable structures underlying exact solvability. (ii) The study of more general models of random paths and growing clusters with probabilistic tools. This work will lead to variational descriptions of limiting free energies and limit shapes, and the identification of structures in these models that enable us to study their universal fluctuation properties. The grand goal is to establish some universal properties for models beyond the narrow set of explicitly solvable cases. This project investigates mathematical models that describe complex interactions, growth, and motion of particles in an irregular environment. These mathematical systems incorporate randomness to model irregularity and unpredictability. The goal is to discover general mathematical laws that govern such systems. These systems appear quite different at microscopic and macroscopic scales. So it is important to understand how different rules for small-scale evolution lead to different large-scale systemwide behavior. Real-world phenomena that such mathematical studies can illuminate include the motion of vehicles, packets in a communication network, fluid particles in a tube, wetting transitions where fluid spreads in a porous medium, epidemics advancing in a population, or the fluctuations of a polymer chain in a fluid. Over the long term understanding complex interactions has profound implications for science and engineering and thereby for society. Models of the kind described in the proposal are intensely and concurrently studied by mathematicians, natural scientists, social scientists, and engineers.

本计画研究随机环境中随机路径的数学模型，以及描述随机成长的逾渗模型。 目标是描述典型的大规模行为，并量化与典型行为的偏差。 重点是找到适用于共享基本特征的模型类的普遍原则。 有两个主要的研究方向。 (i)研究Kardar-Parisi-Zhang普适类中1+1维精确可解模型的组合学、概率分布和大偏差。 这项研究的成果包括严格验证的波动指数在物理学文献中，组合数学和模型之间的新的连接，从概率和统计物理，以及更好地理解的可积结构的基础上精确的可解性。 (ii)用概率工具研究随机路径和增长集群的更一般模型。这项工作将导致变分描述的限制自由能和限制形状，并在这些模型中，使我们能够研究其普遍的波动特性的结构识别。其宏伟目标是建立一些普遍的性质，超越了狭隘的明确可解的情况下模型。该项目研究描述粒子在不规则环境中的复杂相互作用，生长和运动的数学模型。 这些数学系统结合了随机性来模拟不规则性和不可预测性。 其目标是发现支配这些系统的一般数学定律。 这些系统在微观和宏观尺度上看起来完全不同。 因此，了解不同的小规模演化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，管道中的流体颗粒，流体在多孔介质中传播的润湿转变，人口中的流行病或流体中聚合物链的波动。 从长远来看，理解复杂的相互作用对科学和工程，从而对社会有着深远的影响。数学家、自然科学家、社会科学家和工程师都在同时深入研究提案中所描述的模型。