CAREER: Halting Problems In Statistical Mechanics

职业：解决统计力学中的问题

• 批准号: 1455272
• 负责人: Lionel Levine
• 金额: $ 50万
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-06-01 至 2020-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1455272&HistoricalAwards=false
• 关键词: CAREER; Halting; Problems; Statistical; Mechanics

Landslides, earthquakes, avalanches, wildfires, and financial crises are examples of disaster events in complex systems. The aim of this research is to prove theorems about mathematical models of these systems, with three particular goals:(1) Quantify the dependence of large-scale observations on small details. Which details actually matter?(2) Develop efficient algorithms to determine whether a given system will stabilize and how long it will take.(3) Examine the threshold state that precedes a large disaster event, and what triggers the disaster.While simulation is an important tool for modeling complex systems, the focus of this research is on mathematical proof. A simulation reveals how a system behaves, but a proof reveals why.Many interacting particle systems have a phase transition between a state that eventually stabilizes and a state in which activity persists forever. These systems have a second level of dynamics, operating on a slower time scale and driving the system toward greater instability, leading ultimately to a threshold state in which activity persists forever. Until recently this threshold state remained beyond the realm of rigorous proof, but it is now understood in a particular case, the abelian sandpile in the limit as the initial condition tends to negative infinity. Dr. Levine will leverage this understanding to locate the activity phase transition in a wider class of models. This project also has an educational component, whose goal is to encourage and enable bright young people to pursue science and technology careers by providing them with the working knowledge of probability they will need to succeed in those careers.

滑坡、地震、雪崩、野火和金融危机都是复杂系统中的灾难事件。本研究的目的是证明这些系统的数学模型的定理，有三个特定的目标：（1）量化的小细节的大规模观测的依赖。哪些细节真正重要？(2)开发有效的算法来确定给定的系统是否会稳定以及需要多长时间。(3)研究大灾难事件发生前的临界状态以及引发灾难的原因。虽然仿真是建模复杂系统的重要工具，但本研究的重点是数学证明。模拟揭示了系统的行为方式，而证明揭示了原因。许多相互作用的粒子系统在最终稳定的状态和活动永远持续的状态之间存在相变。这些系统具有第二层次的动力学，在较慢的时间尺度上运行，并将系统推向更大的不稳定性，最终导致活动永远持续的阈值状态。直到最近，这个阈值状态仍然超出了严格证明的范围，但现在可以理解，在一个特定的情况下，阿贝尔沙堆在极限作为初始条件趋于负无穷大。Levine博士将利用这一理解在更广泛的模型中定位活动相变。该项目还有一个教育部分，其目标是通过向聪明的年轻人提供他们在这些职业中取得成功所需的概率方面的工作知识，鼓励他们并使他们能够从事科学和技术职业。