Studies in Interacting Particle Systems and Statistical Mechanics

9703814肖曼首席研究员将致力于相互作用粒子系统、统计力学、渗流和概率论其他相关学科领域的各种项目。主要的研究对象是几个相互作用的粒子系统的遍历行为和松弛模式，包括Ising模型和其他晶格自旋系统、任意图上的接触过程、生长模型和元胞自动机。所提出的几个问题的动机来自于在自然界中观察到的现象，例如，在相变区附近的系统的亚稳态。首席研究员将研究相互作用粒子系统中的问题。这些模型适用于包含大量相似组件的系统，其中每个组件都以一种简单的方式与其他几个组件交互。例如，人们可以想到晶体中的原子在足够近的时候相互作用，或者大量人口中的个人可能会受到或不会受到邻居之间发生的污染的感染。局部相互作用可能涉及大量的随机性，但由于系统作为一个整体是大的，人们预计并确实在全球水平上观察到更可预测的行为。对这种系统的极大兴趣源于这样一个事实，即整个系统的定性行为可能以非平凡的方式依赖于构建在局部交互中的参数，但这些参数仅以平稳的方式影响那些参数(例如，上面第二个例子中相邻个体之间的感染率)。因此，人们感兴趣的一般问题是理解这种类型的随机动力学可以达到哪种类型的平衡，如何达到平衡状态，以及影响当地规则的参数可能如何影响这些问题的答案。

9703814 Schonmann The Principal Investigator will work on various projects in the areas of interacting particle systems, statistical mechanics, percolation, and other related subjects in Probability Theory. The main objects of interest in this proposal are the ergodic behavior and the relaxation patterns of several interacting particle systems, including Ising models and other lattice spin systems, contact processes on arbitrary graphs, growth models and cellular automata. Several of the proposed problems have their motivation in phenomena observed in nature as, for example, metastable states of systems in the vicinity of phase transition regions. The Principal Investigator will study problems in interacting particle systems. These are models for systems which contain a large number of similar components and in which each component interacts in a simple way with a few other ones. For instance one can think of the atoms in a crystal interacting when they are close enough, or individuals in a large population which may suffer or not from some infection with contamination occurring between neighbors. The local interaction may involve a substantial amount of randomness, but since the system as a whole is large, one expects and indeed observes a much more predictable behavior at the global level. A great deal of interest in such systems stems from the fact that the qualitative behavior of the whole system may depend in non-trivial ways on parameters which are built into the local interactions, but which only affect those in a smooth way (for example, the rate of infection between neighboring individuals in the second example above). The general issues of interest are then the understanding of which sort of equilibria can be reached by such types of stochastic dynamics, how an equilibrium state is reached, and how the parameters which affect the local rules may affect the answers to these questions.