Integrable Structure of Interacting Particle Systems

相互作用粒子系统的可积结构

基本信息

批准号：
1809311
负责人：
Craig Tracy
金额：
$ 32万
依托单位：
University of California-Davis
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-08-01 至 2023-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1809311&HistoricalAwards=false
关键词：
Integrable Structure Interacting Particle Systems

项目摘要

Statistical mechanics is that area of theoretical physics that predicts macroscopic behavior of physical systems given the microscopic dynamics. The dynamics can be either classical or quantum. Equilibrium statistical mechanics (the foundational work going back to L. Boltzmann and J.W. Gibbs) has a well-formulated, in principle, procedure to go from the microscopic dynamics to the description of the macroscopic system. Carrying out this procedure can be extremely difficult; nevertheless, the broad outlines are well understood. Most physical systems are not in equilibrium and the main interest is their long-time behavior. The Gaussian distribution (the familiar bell-shaped curve) is important because of its universality; that is, it applies to a wide variety of seemingly unrelated problems. The underlying common theme for all these problems is the fact that the objects under study have some degree of "independence"; or stated in more physical terms, are non-interacting (or weakly interacting). These types of problems are well-understood both physically and mathematically. The foundations of nonequilibrium statistical physics are not as well understood as equilibrium statistical physics, in particular when the processes involve interacting particles. This project is directed at understanding the processes which are strongly interacting; and as such, there is no satisfactory general theory as there exists for the non-interacting cases. Simple models occupy a bigger role in understanding systems far from equilibrium. The asymmetric simple exclusion process (ASEP), a model of interacting particles on a lattice, is one such model. ASEP is one of the simplest, nontrivial stochastic models in which to study transport phenomena as it models processes far from equilibrium. This project, building on earlier work of the one-point correlation, will analyze a class of higher-order correlations. The first higher-order correlations are block probabilities for step initial conditions. By a block of length L we mean a contiguous block of L particles (starting, say, at position m on the lattice). Recently the probability of this event was computed. The project will generalize this block probability result to the case of step Bernoulli initial conditions. This same result is also the starting point of the project where similar higher-order correlations will be analyzed. For example, the correlation of the first particle with the particle at position m. The methods employed are both combinatorial and analytical. In particular new combinatorial identities generalizing those found for block probabilities are an essential element of the project. The possible long term impact of this project is to discover new universal limit laws that generalize the Tracy-Widom distributions that appear in the limit laws for the one-point fluctuations.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.

统计力学是理论物理学的一个领域，它在给定微观动力学的情况下预测物理系统的宏观行为。动力学可以是经典的或量子的。平衡统计力学（可以追溯到 L. Boltzmann 和 J.W. Gibbs 的基础工作）原则上有一个完善的程序，可以从微观动力学到宏观系统的描述。执行此过程可能非常困难；尽管如此，总体轮廓还是很好理解的。大多数物理系统并不处于平衡状态，主要关注的是它们的长期行为。高斯分布（熟悉的钟形曲线）因其普遍性而很重要；也就是说，它适用于各种看似无关的问题。所有这些问题的根本共同主题是研究对象具有一定程度的“独立性”。或者用更物理的术语来说，是非相互作用的（或弱相互作用的）。这些类型的问题在物理和数学上都是很好理解的。非平衡统计物理学的基础不如平衡统计物理学那样容易理解，特别是当过程涉及相互作用的粒子时。该项目旨在理解强相互作用的过程；因此，对于非相互作用的情况，没有令人满意的一般理论。简单模型在理解远离平衡的系统方面发挥着更大的作用。非对称简单排除过程 (ASEP) 是一种晶格上相互作用的粒子模型，就是这样的模型之一。 ASEP 是研究传输现象的最简单、最重要的随机模型之一，因为它模拟远离平衡的过程。该项目以单点相关性的早期工作为基础，将分析一类高阶相关性。第一个高阶相关性是步骤初始条件的块概率。长度为 L 的块是指 L 个粒子的连续块（例如，从晶格上的位置 m 开始）。最近计算了该事件的概率。该项目将将此块概率结果推广到步骤伯努利初始条件的情况。同样的结果也是该项目的起点，将分析类似的高阶相关性。例如，第一个粒子与位置 m 处的粒子的相关性。所采用的方法包括组合方法和分析方法。特别是新的组合恒等式概括了块概率中发现的组合恒等式，是该项目的基本要素。该项目可能的长期影响是发现新的通用极限定律，概括出现在单点波动极限定律中的 Tracy-Widom 分布。该奖项反映了 NSF 的法定使命，并通过评估认为值得支持利用基金会的智力优势和更广泛的影响审查标准。