Theory and Applications of Random Cellular Automata and Related Models

9971543Griffeath在理论方面，目前的建议将进一步发展研究者对确定性和概率元胞自动机（CA）的阈值增长和相关动力学的严格理论。增长模型理论在空间复杂系统分析中起着重要作用，其历史可以追溯到本世纪中叶。在目前的资助期间，Griffeath将通过关注五个主要研究项目来寻求进一步的理论：临界单调凝固的标度定律，阈值增长随机扰动的形状理论，spinodal聚类的原型，“小世界网络”的定量分析，以及von Koch型CA凝固的渐近性。这些研究主要是理论性的，他们的目标是得到严谨的数学结果。在实验方面，Griffeath已经开始研究更复杂的相互作用，这些相互作用表现出各种各样的有趣的自组织效应，这些效应在最简单的“线性”空间系统中是不存在的。目前初步的实证建模项目包括数字生物膜、社会互动的非线性空间建模、CA模式动力学以及交通拥堵的简单CA范式。概括地说，David Griffeath正在进行的研究在复杂空间系统的研究中结合了数学分析和计算机可视化。研究者利用这种相互作用进行确定性和随机细胞自动机的理论和实证研究，局部晶格动力学作为可激发介质中螺旋形成的原型，晶体形态，以及各种其他非线性过程，如成核，群集，宿主-寄生虫相互作用，枝晶生长和表面张力的相分离。对这些基本增长模型的聚集、形状、界面和液滴相互作用的分析为应用科学的许多领域提供了新的思路。研究者的研究为这些过程的数学基础做出了贡献。

9971543Griffeath On the theoretical side, the present proposal will further develop the investigator's rigorous theory of threshold growth, and related dynamics, for deterministic and probabilistic cellular automata (CA). The theory of growth models has played a key role in the analysis of spatial complex systems, dating back to the middle of the century. During the present funding period, Griffeath will seek to further that theory by focusing on five main research projects: scaling laws for critical monotone solidification, shape theory for random perturbations of threshold growth, a prototype of spinodal clustering, quantitative analysis of 'small-world networks', and asymptotics for von Koch - type CA solidification. These studies are primarily theoretical, their goal being rigorous mathematical results. On the experimental side, Griffeath has begun to examine more complex interactions which exhibit a wide variety of intriguing self-organizational effects not present in the simplest "linear" spatial systems. Preliminary empirical modeling projects at present include digital biofilms, nonlinear spatial modeling of social interaction, CA pattern dynamics, and a simple CA paradigm for traffic jams. In broad strokes, the ongoing research of David Griffeath combines mathematical analysis and computer visualization in the study of complex spatial systems. The investigator exploits this interplay for the theoretical and empirical study of deterministic and random cellular automata, local lattice dynamics which serve as prototypes for spiral formation in excitable media, the morphology of crystals, and various other nonlinear processes such as nucleation, flocking, host-parasite interactions, dendritic growth, and phase separation by surface tension. The analysis of aggregation, shape, interfaces, and droplet interaction for such basic growth models sheds new light on many areas of applied science. The investigator's research contributes to the mathematical foundations of such processes.