Studies in Interacting Random Systems

这个项目研究的是模拟复杂相互作用行为的随机过程，比如相互作用的粒子系统和界面模型。目标是通过数量级指数、极限分布和大偏差率函数来获得描述大空间和时间尺度上的行为的数学结果，以及界面的粗糙度或粒子流的波动。重点是发现新的现象，以及捕捉不同类别的微观规则和相关的大规模行为之间联系的整体图景。在非对称系统的背景下，一个特别有趣的问题是电流在描述系统宏观演化的偏微分方程特征上的波动。这项研究的目的是对某些宏观现象的精确数学理解，这些现象是由大量微观成分之间的相互作用造成的。该项目研究的数学模型包含了各个组成部分行为的随机性。这些模型是对许多复杂系统常见的组织和行为基本原则的抽象。目标是描述模型的典型大规模行为，并量化偏离典型行为的机会。复杂的相互作用发生在我们周围的自然和人为过程中。例如高速公路上的汽车、排队的顾客、通过通信网络的数据包或管道中的流体颗粒。因此，理解复杂相互作用的数学对科学和工程具有潜在的广泛影响。数学家、自然科学家、社会科学家和工程师同时研究了提案中描述的类型模型以及与之相关的其他模型。

0402231Seppalainen This project studies stochastic processes that model complex, interacting behavior, such as interacting particle systems and interface models. The goal is to obtain mathematical results that describe behavior over large space and time scales in terms of limit theorems, and roughness of interfaces or fluctuations in particle flow by order-of-magnitude exponents, limit distributions, and large deviation rate functions. The emphasis is on finding new phenomena, and on capturing the overall picture of connections between different classes of microscopic rules and the associated large-scale behavior. One particular problem of interest in the context of asymmetric systems is the fluctuations of the current across a characteristic of the partial differential equation that describes the macroscopic evolution of the system. The objective of this research is precise mathematical understanding of certain kinds of macroscopic phenomena that result from interactions among a large number of microscopic components. The project studies mathematical models that include randomness in the behavior of the individual components. These models are abstractions of basic principles of organization and behavior common to many types of complex systems. The goal is to describe the typical large scale behavior of the model, and also to quantify the chances of deviations from the typical behavior. Complex interactions occur all around us in natural and man-made processes. Examples include cars on a freeway, customers in a queue, packets making their way through a communication network, or fluid particles in a tube. Consequently understanding the mathematics of complex interactions has potentially wide implications for science and engineering. Models of the types described in the proposal and others related to them are concurrently studied by mathematicians, natural scientists, social scientists, and engineers.