Rigorous Results for Self-Organizing Complex Systems

0204018 Griffeath Griffeath将继续他正在进行的研究计划，最初重点是交通元胞自动机（TCA）的定理，以及冯·科赫二维凝固规则的渐进密度的严格推导。交通项目的基础在最近与明尼苏达大学的劳伦斯·格雷联合发表在《统计物理学杂志》上的一篇论文中进行了描述。受凯内格尔等人的经验工作的启发，作者给出了令人信服的证据，即随着汽车密度的增加，TCA表现出几个相变，并在一定范围内的密度集群成两个自组织的极端遍历状态的混合物，称为自由流和同步堵塞。本研究计画将进一步分析此模型，以建立保守性群集的新现象，并揭示其内在机制。在一个单独的项目中，Griffeath将研究Von Koch晶体，这是二维整数晶格上非常简单的确定性最近邻元胞自动机生长规则，其中许多是精确可解但非周期性的。将开发一种新的计算机器，以严格描述自组织的，往往是分形结构的这些晶体通过广义的形式语言，交互式可视化，非线性递归，和计算机辅助证明。大卫格里菲思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。在他的职业生涯中，研究人员利用这种相互作用对各种各样的动力学进行理论和实证研究，这些动力学作为跨科学现象的原型：可激发介质中的螺旋形成，晶体生长和各种其他非线性过程，如成核，群集，宿主-寄生虫相互作用，树枝状生长，表面张力相分离，生态竞争和生物膜的生长。Griffeath目前正在开展一项主要研究，即模拟交通堵塞出现的“交通元胞自动机”模型。该项目的大部分初始工作都是基于模拟的。既然已经开发出有效的软件工具来可视化和定量测量这些微妙的自组织系统，Griffeath打算推导出一个数学上严格的理论，

0204018Griffeath Griffeath will continue his ongoing research program, focusing initially on theorems for the Traffic Cellular Automaton (TCA), and on rigorous derivation of asymptotic densities for von Koch two-dimensional solidification rules. Foundations of the traffic project are described in a recent paper, joint with Lawrence Gray of the University of Minnesota, in the Journal of Statistical Physics. Motivated by empirical work of Kai Nagel and others, the authors give convincing evidence that the TCA exhibits several phase transitions as the density of cars increases, and for a certain range of densities clusters into a mixture of two self-organized extreme ergodic states called free flow and synchronized jam. The research program will further analyze this model, intended to establish the new phenomenon of conservative clustering, and also to shed light on the underlying mechanism. In a separate project, Griffeath will study Von Koch crystals, very simple deterministic nearest-neighbor Cellular Automaton growth rules on the two-dimensional integer lattice, many of which are exactly solvable but aperiodic. A novel computational machinery will be developed in order to rigorously describe the self-organized, often fractal structure of these crystals by means of a generalized formal language, interactive visualization, nonlinear recursion, and computer-aided proof. The research of David Griffeath combines mathematical analysis and computer visualization in the study of complex spatial systems. Over his career, the investigator has exploited this interplay for the theoretical and empirical study of a wide variety of dynamics which serve as prototypes for phenomena across the sciences: spiral formation in excitable media, crystal growth, and various other nonlinear processes such as nucleation, flocking, host-parasite interactions, dendritic growth, phase separation by surface tension, ecological competition, and growth of biofilms. Griffeath is currently launching a major study of "Traffic Cellular Automata" models which emulate the emergence of traffic jams. Most of the initial effort on this project has been simulation-based. Now that effective software tools have been developed for visualization and quantitative measurement of these subtle, self-organizing systems, Griffeath intends to derive a mathematically rigorous theory of the observed