CAREER: Halting Problems In Statistical Mechanics

山体滑坡、地震、雪崩、野火和金融危机都是复杂系统中灾难事件的例子。这项研究的目的是证明关于这些系统的数学模型的定理，有三个特别的目标：(1)量化大范围观测对小细节的依赖。哪些细节真正重要？(2)开发有效的算法来确定给定系统是否会稳定以及需要多长时间。(3)检查大型灾难事件之前的阈值状态，以及是什么触发了灾难。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，但证据揭示了为什么。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。这些系统有第二层次的动态，以较慢的时间尺度运行，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，这一阈值状态仍然超出了严格证明的范围，但现在它被理解为在一个特殊情况下，阿贝尔沙堆在初始条件趋于负无穷时处于极限。莱文博士将利用这一理解，在更广泛的模型类别中定位活动阶段转变。该项目还有一个教育部分，其目标是鼓励和使聪明的年轻人追求科学和技术事业，向他们提供他们在这些职业中取得成功所需的概率的实用知识。

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.