Interacting Particle Systems and Beyond

该项目主要研究统计力学中的晶格模型。这些模型中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对底层代数数据的新观点和解释。例如，这些系统可以用来研究晶体是如何融化的，神经元是如何在大脑中移动的，火线是如何前进的，癌症是如何扩散的，海洋中浮游生物是如何生长的。该研究项目旨在更好地理解随着系统规模的增长，一些重要模型的宏观行为。所考虑的模型通常被称为“可积的”或“精确可解的”。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是一个更大的模型族的代表。通过这种方式，除了本身有趣之外，完全可解系统是其推测的普适性类的范例，可用于建立直觉和做出预测。目的是获得研究通用性类的各种鲁棒方法。这项研究计划旨在更好地理解某些相互作用粒子系统背景下的一些“普遍性”特征。研究项目包括三个主要方向——对数气体研究（受外部电位限制的直线上粒子的集合，它们以对数方式相互排斥），卡达-帕里西-张方程（最初作为地表生长模型提出的非线性随机偏微分方程）的平稳测量研究，以及交通模型的研究。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.