Topics in Statistical Mechanics and Related Graphical Models

[306167]蔡斯：研究人员将研究各种各样的系统，这些系统要么直接涉及处于平衡状态的经典自旋系统，要么以某种近似统计力学模型为基础。有六个单独的项目，他们都已经在某种程度上进行了分析。第一个项目是研究不连续相变的一般理论。对于通常的d维立方晶格上的一类最近邻吸引系统，他们可以证明，每当相关的平均场理论预测一个不连续相变时，实际系统也会经历一个不连续相变，每当维数d很大时。第二个项目是建议研究一个模型，该模型通过随机环境的确定性进化来模拟著名的SOC沙堆模型。第三，他们研究了在统计系综中相位共存点处的系统，确定了系统中少数相的过量。结果表明，在正确的尺度下，存在一个少数相体积的尖锐值，其中少数相的液滴会自发地出现。第四，他们将研究某些渗透系统，这些系统局部看起来与完全图上的渗透相似，但包含在通常的完全图系统中不存在的潜在几何结构。在第五个项目中，他们将研究各种相互作用的随机漫步，包括吸引和排斥（以及混合）相互作用。最后一个项目涉及多分散系统中自由能的热力学极限的正确定义，例如，具有连续粒子种类的系统。首席研究员将研究统计力学领域或密切相关领域的一些问题（提案中有六个问题，更多问题即将提出）。这些问题的本质涉及由简单成分组成的系统，这些成分以简单的方式相互作用。然而，由于有无数这样的成分，事实证明，这样的系统能够高度复杂的行为。特别令人感兴趣的是相变现象，在这种现象中，系统的整个集体特征经历了剧烈的变化。这个概念要么在所有被提议研究的系统的核心，要么就在幕后。一个项目涉及到“大多数”系统经历一种更灾难性的转变，有时被称为第一类相变。第二项研究涉及到水滴（如雨滴）的形成，结果证明，水滴的形成发生在被称为介观尺度的亚宏观尺度上。第三，正在调查的是“干扰”在一个系统中的渗透，从一个角度来看，这个系统模拟了一个社会网络。第四，将对上述基本成分具有不同（但密切相关）特征的系统进行调查。剩下的调查涉及所谓的渗透和随机游走系统。虽然这些模型起源于对化学聚合物的研究，但其应用涵盖了广泛的主题，其中之一，用现代术语来说，就是通信网络中的信息传播。

0306167Chayes The investigators will study a variety of systems all of which either directly concern classical spin systems in equilibrium or have their basis in some approximation to a statistical mechanics model. There are six individual projects, all of which they have already analyzed to some degree. The first project concerns an approach to a general theory of discontinuous phase transitions. For a certain class of nearest neighbor attractive systems on the usual d-dimensional cubic lattices they can show that whenever the associated mean field theory predicts a discontinuous phase transition, the actual system also undergoes a discontinuous transition whenever the dimension d is large. The second project is a proposal to study a model which emulates, through a deterministic evolution of a random environment, the celebrated SOC sandpile models. Third, they study systems at points of phase coexistence in a statistical ensemble fixing the excess of the minority phase in the system. It appears that, in a correct scaling, there is a sharp value of minority-phase volume where a droplet of the minority phase spontaneously emerges. Fourth, they will investigate certain percolation systems which locally appear similar to percolation on the complete graph but contain an underlying geometrical structure that is absent in the usual complete graph systems. In the fifth project, they will study a variety of interacting random walks with both attractive and repulsive (and mixed) interactions. The final project concerns the correct definition of thermodynamic limit of the free energy in polydisperse systems, e.g., systems with a continuum of particle species. The Principal Investigators will be examining a number of problems (six in the proposal with more on the horizon) in the area of Statistical Mechanics or closely related fields. The essence of these problems concerns systems consisting of simple constituents which interact in a simple manner. However, since there are a myriad of these constituents it turns out that such systems are capable of highly complex behavior. Of particular interest is the phenomena of phase transitions in which the entire collective character of the system undergoes a drastic change. This notion is either at the heart of or just behind the scenes in all of the systems that are proposed for study. One project concerns a demonstration that "most" systems undergo a more catastrophic sort of transition sometimes known as a phase change of the first kind. A second line of investigation concerns the formation of droplets (e.g. raindrops) which, as it turns out, takes place on a submacroscopic scale known as a mesoscopic scale. Third, under investigation is the infiltration of "disturbances" in a system which -- from one perspective -- models a social network. Fourth, there will be investigations of systems in which the above mentioned elementary constituents are all of different (but closely related) character. The remainder of the investigations concern the so called percolation and random walk systems. While these models have their origins in the study of chemical polymers, the applications cover a large range of topics, one of which, in modern terms, is the dissemination of information in communications networks.