Probabilistic aspects of growth processes

生长过程的概率方面

• 批准号: 0805970
• 负责人: Janko Gravner
• 金额: $ 21万
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0805970&HistoricalAwards=false
• 关键词: Probabilistic; aspects; growth; processes

This project is devoted to studying deterministic and random cellular automata models of growth processes, with the emphasis on several new directions. The first field of study are long-range growth processes, specifically the behavior at the edge of the expanding wave by investigating, depending on the regime, mixing times and hydrodynamics limits.The investigator also aims to study nucleation and metastability properties of monotone growth models, and space-covering properties of nonmonotone ones. A new research direction is the utilization of three-dimensional mesoscopic modeling to understand growth of snow crystals. Computational and analytical methods are developed to connect the mesoscopic scale dynamics with the microscopic (particle) and macroscopic (continuum) versions. Finally, the investigator will devise combinatorial and probabilistic methods to understand the effects of random environments on growth processes in high-dimensional spaces important in genetics.In a broad scientific context, the aim of this project is to understand principles by which various physical systems propagate disturbances. For example, crystal growth has been a subject of intense research for more than a century, yet growth of objects as familiar as snowflakes remains mysterious.The investigator's research sheds light on how microscopic description of a growth process effects the resulting shape, its space-covering ability, behavior at the expanding front, and stability to perturbations in the environment and in the dynamics. An important feature is the interplay between rigorous mathematical analysis and large--scale computer experimentation.This component aims to develop connections to physics, engineering and biology for fundamentals of crystal growth and other expansion processes, to computer science for efficient simulation and visualization algorithms, and to theoretical genetics for realistic insights into the structure of genotype and phenotype spaces. Together with the advancement of mathematical and computational methods, such research has potential applications to crystal manufacturing in material sciences, understanding snowfall in atmospheric sciences, and genetic diversity in biology. Finally, communication with general public by means of attractive computer graphics, and ample opportunities for undergraduate and graduate students' involvement in all aspects of such studies, will popularize mathematics and its applications.

本计画致力于研究成长过程的决定性与随机性细胞自动机模型，并着重于几个新的方向。第一个研究领域是长程增长过程，特别是在膨胀波的边缘的行为，通过调查，取决于制度，混合时间和流体力学的限制。调查员还旨在研究单调增长模型的成核和亚稳性质，以及非单调增长模型的空间覆盖性质。一个新的研究方向是利用三维介观模拟来理解雪晶的生长。计算和分析方法的发展，连接的微观（粒子）和宏观（连续）版本的介观尺度动力学。最后，研究者将设计组合和概率方法，以了解随机环境对遗传学中重要的高维空间中生长过程的影响。在广泛的科学背景下，本项目的目的是了解各种物理系统传播干扰的原理。例如，晶体生长是一个世纪以来的热门研究课题，但雪花这样熟悉的物体的生长仍然是一个谜。研究人员的研究揭示了生长过程的微观描述如何影响最终的形状，其空间覆盖能力，在扩展前沿的行为，以及对环境和动力学扰动的稳定性。一个重要的特点是严格的数学分析和大规模的计算机实验之间的相互作用，这一部分的目的是发展连接到物理学，工程学和生物学的晶体生长和其他扩展过程的基础知识，计算机科学的有效模拟和可视化算法，和理论遗传学的基因型和表型空间的结构的现实见解。随着数学和计算方法的进步，这些研究在材料科学中的晶体制造，大气科学中的降雪和生物学中的遗传多样性方面具有潜在的应用。最后，通过有吸引力的计算机图形与公众沟通，以及本科生和研究生参与这些研究的各个方面的充分机会，将普及数学及其应用。