The Mathematics of Stochastic Systems with Many Interacting Constituents

一般而言，这一提议涉及具有多个自由度的系统的行为。在这里，人们寻找(A)整个系统的集体行为或(B)单个组件可能表现出的行为的统计描述。强调指出，(A)和(B)并不是相互脱节的标准：事实上，它们经常(但并非总是)重合。将用于促进这些研究的设置有四种类型；所有这些类型或多或少都是相互关联的，而且所有四种类型都源于对身体问题的描述。这些是(I)[经典自旋系统]格子模型，自旋或粒子在某些规则下相互作用，主要是局域规则，这些规则被称为经典平衡统计力学定律。(Ii)[量子自旋--系统]与(I)中类似的设置，但遵循量子平衡统计力学规则。(Iii)[相互作用粒子系统]模型，其中占据晶格位置的单个粒子或自旋根据随机规则动态地更新其位置或状态，这些随机规则本身受现有局部构型的影响。(4)[集群模型/图形表示法]图形模型，其中除本地规则外，还根据全局考虑或非本地规则随机确定配置。除了基于标准格子的大规模描述之外，一些工作还将涉及连续统模型--以及格子模型的连续统极限。始于20世纪60年代末的S的一种哲学是普适性哲学：一旦确定了组成及其相互作用的基本性质和前提，在具有多个自由度的系统中产生的大规模、长时间集体行为的结果(可能的模式)不依赖于模型的细致细节。因此，将研究各种系统，其中相互作用的成分的性质是最简单的类型。然而，预计将展出集体行为的全部画卷。因此，尽管这些类型的模型有物理上的起源，但至少在PI看来，这些类型的模型允许广泛的应用--甚至超越了物理科学。(在一项涉及犯罪行为研究和警察资源分配的相关提案的背景下，其中一些已经取得成果。)此外，也许更切合实际的是：这项提案及其前身包含了一系列广泛的问题，从多个方面提供了每个层面的研究机会，该项目的成功完成将需要从本科生到高级研究合作者的许多合作者的帮助。

In general terms this proposal concerns the behavior of systems with many degrees of freedom. Here one looks for (a) collective behavior of the system as a whole or (b) a statistical description of the behavior that an individual component might exhibit. It is emphasized that (a) and (b) are not disjoint criteria: indeed quite often (but not always) they coincide. The setups that will be used to promote these studies are of four types; all, to a greater or lesser extent, are interrelated and all four have their origins in the description of physical problems. These are (i) [Classical Spin--Systems] Lattice models of spins or particles interacting under certain rules, primarily local, which are known as the laws of classical equilibrium statistical mechanics. (ii) [Quantum Spin--Systems] A similar setup as in (i) but under the rules of quantum equilibrium statistical mechanics. (iii) [Interacting Particle Systems] Models where individual particles or spins occupying lattice positions update their position or state dynamically in time according to stochastic rules that are themselves effected by the existing local configuration. (iv) [Cluster Models/Graphical Representations] Graphical models where, in addition to local rules, configurations are determined, stochastically, according to global considerations or non-local rules. In addition to the standard lattice based large-scale descriptions, some of the work will concern continuum models --as well as continuum limits of lattice models.A philosophy which became prevalent beginning in the late 1960's is that of universality: Once the basic nature and premise of the constituents and their interactions are determined, the resultant (possible modes of) large?scale, long?time collective behaviors in systems with many degrees of freedom do not depend on the meticulous details of the model. Thus a variety of systems will be investigated where the nature of the interacting constituents is of the simplest type. However, it is anticipated that the full tapestry of collective behavior will be exhibited. Thus, notwithstanding the physical origins of these types of models, it is anticipated, at least by the PI, that these types of models allow for a broad range of applications--even beyond the physical sciences. (Already some of this has come to fruition in the context of a related proposal which concerns the study of criminal behavior and allocation of police resources.) Moreover and perhaps more pertinently: This proposal and its predecessor encompass a broad array of problems with multiple facets providing research opportunities at every level and the successful completion of the program will require the assistance of many collaborators ranging from the undergraduate to the senior research collaborator.