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(最早可以追溯到L.Boltzmann和J.W.Gibbs的基础性工作)在原则上有一个从微观动力学到宏观系统描述的精心制定的程序。执行这一程序可能非常困难；然而，大体轮廓是很容易理解的。大多数物理系统并不处于平衡状态，人们主要关注的是它们的长期行为。高斯分布(熟悉的钟形曲线)很重要，因为它具有普遍性；也就是说，它适用于各种各样看似无关的问题。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”；或者用更多的物理术语来说，是不相互作用的(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。非平衡统计物理的基础没有平衡统计物理那么好理解，特别是当过程涉及相互作用的粒子时。这个项目旨在理解强相互作用的过程；因此，没有令人满意的一般理论，因为存在非相互作用的情况。简单的模型在理解远离均衡的系统方面发挥了更大的作用。非对称简单排斥过程(ASEP)--晶格上粒子相互作用的模型--就是这样一个模型。ASEP是研究输运现象的最简单、非平凡的随机模型之一，因为它模拟远离平衡的过程。这个项目是在早期单点关联工作的基础上进行的，将分析一类高阶关联。第一高阶相关性是步骤初始条件的块概率。我们所说的L长度的块是指连续的L粒子块(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。该方案将这种块概率结果推广到阶跃伯努利初始条件的情况。这个相同的结果也是该项目的起点，在那里将分析类似的高阶关联。例如，第一个粒子与位置m处的粒子的相关性。所采用的方法既有组合方法，也有分析方法。特别是，新的组合恒等式推广了为区块概率找到的恒等式，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，该定律概括了一点波动的极限定律中出现的Tracy-Widom分布。该奖项反映了NSF的法定使命，并已通过使用基金会的智力优势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.