Mathematical Sciences: Percolative Models

数学科学：渗透模型

• 批准号: 9618128
• 负责人: Yu Zhang
• 金额: $ 6.19万
• 依托单位: University of Colorado at Colorado Springs
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-15 至 2000-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9618128&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Percolative; Models

9618128 Zhang The project concentrates on percolation theory, a mathematical theory used to describe transitions of physical systems. Percolation theory has a variety of applications to solid physics, biology, computer science, and geology. A percolation process typically depends on one or more parameters. A dramatic change in physical properties may occur as a critical parameter value is passed. The research will focus on the behavior of percolative models in the following three areas: percolation near, above and below the critical thresholds. In particular, the project will investigate mathematically rigorous exact solutions for the percolative models. The research makes use of probability theory (the moment estimations, the ergodic theory, correlation and martingale inequalities and stochastic ordering), graph theory (duality, the fractal dimension), combinatorics (partition lattices, distributive lattices), and function analysis (the real analyticity). The project will use these mathematical tools to advance our rigorous understanding of critical phenomena. The project concentrates on percolation, a mathematical model used to describe transitions of physical systems. Percolation theory has a variety of applications to solid physics, biology, computer science, and geology. A percolation process typically depends on one or more parameters. A dramatic change in physical properties may occur as a critical parameter value is passed. For example, suppose we immerse a large porous solid in a bucket water. Clearly, how water penetrates the solid depends on the size of the pores of the solid. A simple mathematical model of such a process is defined by taking the pores to be distributed in some regular manner, and to be open or closed with probabilities p or 1-p. There is a critical threshold, for the probability p at which the behavior changes abruptly. For values of p below the critical value the water penetration is only superficial and above it the penetration is arbitrarily deep. The specific behavior near the critical threshold is more complicated. One of the most challenging problems is to give a mathematical description of deep penetration near the critical threshold. The research will focus on three areas: the behaviors of percolation near the critical threshold, above the critical threshold and below the critical threshold. In particular, the project will investigate mathematically rigorous exact solutions for the percolation process. The research makes use of probability theory, graph theory, combinatorics and function analysis. The project will use these mathematical tools to advance our rigorous understanding of critical phenomena.

9618128 张 该项目专注于渗透理论，一种用于描述物理系统转变的数学理论。渗流理论在固体物理学、生物学、计算机科学和地质学中有多种应用。渗滤过程通常取决于一个或多个参数。当超过临界参数值时，物理特性可能会发生巨大变化。该研究将重点关注渗透模型在以下三个领域的行为：接近、高于和低于临界阈值的渗透。特别是，该项目将研究渗透模型的数学上严格的精确解决方案。该研究利用了概率论（矩估计、遍历理论、相关性和鞅不等式以及随机排序）、图论（对偶性、分形维数）、组合学（划分格子、分配格子）和函数分析（实解析性）。该项目将使用这些数学工具来推进我们对关键现象的严格理解。 该项目的重点是渗透，一种用于描述物理系统转变的数学模型。渗流理论在固体物理学、生物学、计算机科学和地质学中有多种应用。渗滤过程通常取决于一个或多个参数。当超过临界参数值时，物理特性可能会发生巨大变化。例如，假设我们将一个大的多孔固体浸入一桶水中。显然，水如何渗透固体取决于固体孔隙的大小。这种过程的简单数学模型是通过将孔隙以某种规则方式分布并以 p 或 1-p 的概率打开或关闭来定义的。对于行为突然改变的概率 p，存在一个临界阈值。对于低于临界值的 p 值，水渗透只是表面的，而高于临界值时，渗透是任意深度的。临界阈值附近的具体行为更加复杂。最具挑战性的问题之一是对临界阈值附近的深度穿透进行数学描述。研究将集中在三个领域：接近临界阈值、高于临界阈值和低于临界阈值的渗流行为。特别是，该项目将研究渗滤过程的数学上严格的精确解决方案。该研究利用了概率论、图论、组合学和函数分析。该项目将使用这些数学工具来推进我们对关键现象的严格理解。