Interacting Particle Systems and Beyond

该项目的重点是研究统计力学中的格模型。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个新的观点和解释的基础代数数据。例如，这些系统可用于研究晶体如何融化，神经元如何在大脑中移动，火锋如何前进，癌症如何传播，浮游生物如何在海洋中生长。该研究项目旨在更好地理解一些重要模型随着系统规模增长的宏观行为。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是更大的模型族的代表。通过这种方式，除了本身有趣之外，精确可解系统是它们被证明的普适性类的范例，并且可以用来建立直觉和进行预测。目的是获得各种鲁棒的方法来研究普适类。这个研究项目旨在建立一个更好的理解某些相互作用的粒子系统的背景下的一些“普适性”的功能。该研究计划包括三个主要方向-对数气体的研究（系综的粒子线上限制的一个外部的潜力，相互排斥的热力学），研究定态措施的Kardar-Parisi-Zhang方程（一种非线性随机偏微分方程，最初被提出作为表面生长的模型），该奖项反映了NSF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.