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: 1003651
• 负责人: Timo Seppalainen
• 金额: $ 27.81万
• 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: 2010
• 资助国家: 美国
• 起止时间: 2010-07-15 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1003651&HistoricalAwards=false
• 关键词: Stochastic; systems; complex; interactions; random

This project studies several classes of stochastic processes that possess complicated interactions and in some cases also inhomogeneous or random environments: interacting particle systems, polymer models, and random motion in a random medium.The goal is to describe typical large scale behavior and to quantify deviations from the typical behavior.In polymer models the goal is to settle long-standing open problems on scaling exponents that describe the order of magnitude of the fluctuations of the molecule chain and the free energy.For a class of asymmetric zero range processes this project proposes to prove scaling properties of space-time correlations and current fluctuations that confirm KPZ-type behavior.For a single particle moving in a random environment this project studies large deviations, especially the question of when the rate functions for quenched and averaged large deviations coincide.For a large collection of particles moving in a dynamically evolving environment the goal is to prove distributional limit theorems for the current.This project investigates mathematical models that describe complex interactions and motion of particles in an irregularenvironment. These mathematical systems incorporate randomnessto model irregularity and unpredictability.The goal is to discover general mathematical principlesthat govern such systems. A key point is that thesesystems appear quite different at microscopic and macroscopic scales. So it is important to understand how different rules for small-scale interactions and motions lead to different large-scale systemwide behavior.Real-world phenomena that such mathematical studies can illuminate include the motion of vehicles on a freeway, packets making their way through a communication network, fluid particles in a tube, wetting transitions where fluid spreads in a porous medium, epidemics advancing among individuals in a population, or the fluctuations of a polymer chain in a fluid.Over the longer term understanding these 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.

该项目研究几类具有复杂相互作用的随机过程，在某些情况下也是非均匀或随机环境： 相互作用的粒子系统，聚合物模型， 本项目的目标是描述典型的大尺度行为，并量化偏离典型行为的情况。在聚合物模型中，目标是解决描述分子链和自由能涨落量级的标度指数的长期未决问题。对于一类非对称零程过程，本项目提出证明空间标度指数的标度性质。时间相关性和电流波动证实了KPZ型行为。对于在随机环境中运动的单个粒子，该项目研究了大偏差，特别是淬灭和平均大偏差的速率函数何时重合的问题。对于在动态演化环境中运动的大量粒子，目标是证明电流的分布极限定理。本项目研究数学模型，描述复杂的相互作用和运动的粒子在不规则的。 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，目标是发现支配这些系统的一般数学原理。 一个关键点是，这些系统在微观和宏观尺度上看起来非常不同。 因此，了解小尺度相互作用和运动的不同规则如何导致大尺度系统范围内不同的行为是很重要的。这种数学研究可以阐明的现实世界现象包括高速公路上车辆的运动，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过渡，人口中个体之间的流行病传播，或流体中聚合物链的波动。从长远来看，理解这些复杂的相互作用对科学和工程，从而对社会具有深远的意义。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。