Interacting Particle Systems and Beyond

该项目主要研究统计力学中的晶格模型。这种模型固有的代数结构允许许多精确的计算，而它们的概率性质提供了一种新的观点和对潜在代数数据的解释。例如，这些系统可以用来研究晶体如何融化，神经元如何在大脑中移动，火线如何推进，癌症如何扩散，浮游生物群体如何在海洋中生长。该研究项目的目的是更好地了解一些重要模型随着系统规模的增长而产生的宏观行为。所考虑的模型通常被称为“可积”或“精确可解”。虽然精确可解系统是非常特殊的，但它们的渐近性质被认为对更大的模型族具有代表性。通过这种方式，除了有趣之外，精确可解的系统也是它们猜想的普适性类的范例，可以用来建立直觉和做出预测。其目的是获得各种稳健的方法来研究普适类。这项研究计划试图在某些相互作用的粒子系统的背景下建立对一些“普遍性”特征的更好的理解。该研究计划包括三个主要方向--对数气体(受外部势能以对数方式相互排斥的线上的粒子集合)的研究，Kardar-Parisi-Zhang方程(最初作为地表增长模型提出的非线性随机偏微分方程式)的平稳度量的研究，以及交通模型的研究。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The project focuses on the study of lattice models in statistical mechanics. The algebraic structure inherent in such models allows for many exact computations, while their probabilistic nature provides a new point of view and interpretation of the underlying algebraic data. For instance, these systems can be used to study how crystals melt, how neurons move through the brain, how a fire front advances, how a cancer spreads, how a plankton colony grows in the ocean. The research project aims at achieving a better understanding of the macroscopic behavior of some important models as the size of the system grows.The models under consideration are usually referred to as “integrable” or “exactly solvable.” Though exactly solvable systems are very special, their asymptotic properties are believed to be representative for a larger family of models. In this way, besides being interesting themselves, exactly solvable systems are exemplars of their conjectured universality classes and can be used to build intuition and make predictions. The aim is to obtain a variety of robust methods to study the universality classes. This research program seeks to establish a better understanding of some “universality” features in the context of certain interacting particle systems. The research program consists of three main directions— the study of log-gases (ensemble of particles on the line confined by an external potential that repel each other logarithmically), the study of the stationary measures for the Kardar-Parisi-Zhang equation (a non-linear stochastic partial differential equation that was originally proposed as a model of surface growth), and the study of traffic models.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.