Integrable Structure of Interacting Particle Systems

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也可以是量子的。平衡统计力学（可以追溯到L.玻尔兹曼和J.W.吉布斯的基础工作）有一个从微观动力学到宏观系统描述的良好表述，在原则上。执行这一程序可能极其困难；然而，大致的大纲是很容易理解的。大多数物理系统都不处于平衡状态，主要关注的是它们的长期行为。高斯分布（我们熟悉的钟形曲线）之所以重要，是因为它具有普适性；也就是说，它适用于各种各样看似不相关的问题。所有这些问题的潜在共同主题是，研究对象具有某种程度的“独立性”；或者用更物理的术语来说，是非相互作用（或弱相互作用）的。这类问题在物理上和数学上都很容易理解。非平衡统计物理的基础不像平衡统计物理那样被很好地理解，特别是当过程涉及相互作用的粒子时。该项目旨在了解强烈相互作用的过程；因此，对于非相互作用的情况，没有令人满意的一般理论。在理解远离平衡的系统时，简单模型扮演着更重要的角色。不对称简单不相容过程（ASEP）是晶格上粒子相互作用的一个模型，就是这样一个模型。ASEP是研究输运现象的最简单的非平凡随机模型之一，因为它模拟了远离平衡态的过程。这个项目，建立在先前的一点相关工作的基础上，将分析一类高阶相关。第一个高阶相关性是阶跃初始条件下的块概率。我们所说的长度为L的块是指由L个粒子组成的连续块（比方说，从晶格上的m位置开始）。最近计算了这一事件发生的概率。项目将此块概率结果推广到步伯努利初始条件的情况。同样的结果也是项目的起点，在这里将分析类似的高阶相关性。例如，第一个粒子与位置m的粒子的相关性。所采用的方法既有组合方法，也有分析方法。特别是新的组合恒等式推广那些发现的块概率是项目的基本要素。该项目可能的长期影响是发现新的通用极限定律，推广出现在单点波动极限定律中的tracey - widom分布。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。

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.