Rigorous Results for Self-Organizing Complex Systems

点击翻译按钮获取中文摘要

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