Integrable Structure of Interacting Particle Systems

统计力学是理论物理学的一个领域，它预测给定微观动力学的物理系统的宏观行为。 动力学可以是经典的，也可以是量子的。平衡态统计力学（可以追溯到L。玻尔兹曼和J.W.吉布斯）有一个很好的公式，原则上，程序从微观动力学的描述宏观系统。 执行这一程序可能极其困难;尽管如此，大致轮廓是很好理解的。 大多数物理系统并不处于平衡状态，主要的兴趣是它们的长期行为。 高斯分布（我们熟悉的钟形曲线）之所以重要，是因为它具有普遍性;也就是说，它适用于各种看似无关的问题。所有这些问题的基本共同主题是，所研究的对象具有某种程度的“独立性”;或者用更物理的术语来表示，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很容易理解。非平衡态统计物理学的基础不像平衡态统计物理学那样被理解，特别是当过程涉及相互作用的粒子时。这个项目的目的是了解强相互作用的过程，因此，没有令人满意的一般理论，因为存在的非相互作用的情况下。 简单的模型在理解远离平衡的系统方面发挥了更大的作用。非对称简单排斥过程（ASEP），一种晶格上相互作用粒子的模型，就是这样一个模型。 ASEP是一个最简单的，非平凡的随机模型，在其中研究运输现象，因为它的模型过程远离平衡。 这个项目，建立在一点相关的早期工作，将分析一类高阶相关。 第一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（比如说，从晶格上的位置m开始）。 最近计算了这一事件的概率。 该项目将推广这个块概率结果的情况下，步骤伯努利初始条件。同样的结果也是该项目的出发点，在那里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。 所采用的方法是组合和分析。 特别是新的组合身份推广那些发现块概率是一个基本要素的项目。 该项目可能的长期影响是发现新的通用极限定律，该定律概括了出现在一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。

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.