The Mathematics of Stochastic Systems with Many Interacting Constituents

一般来说，这个建议涉及具有多个自由度的系统的行为。在这里，人们寻找(a)作为一个整体的系统的集体行为，或(b)对单个组件可能表现出的行为的统计描述。需要强调的是，(a)和(b)并不是互不相干的标准：事实上，它们经常（但并非总是）是一致的。将用于促进这些研究的设置有四种类型；所有这些或多或少都是相互关联的，这四种都源于对身体问题的描述。这些是(i)[经典自旋-系统]自旋或粒子在某些规则下相互作用的晶格模型，主要是局部的，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)类似的设置，但在量子平衡统计力学的规则下。（iii）[相互作用粒子系统]模型，其中单个粒子或自旋占据晶格位置，根据随机规则及时动态地更新其位置或状态，这些规则本身受现有局部配置的影响。（iv）[聚类模型/图形表示]除局部规则外，根据全局考虑或非局部规则随机确定配置的图形模型。除了基于标准晶格的大规模描述之外，一些工作将涉及连续统模型-以及晶格模型的连续统极限。从20世纪60年代末开始流行的一种哲学是普遍性哲学：一旦组成部分及其相互作用的基本性质和前提被确定，结果（可能的模式）就会很大？规模,很长时间吗?多自由度系统中的时间集体行为不依赖于模型的精细细节。因此，将研究各种系统，其中相互作用成分的性质是最简单的类型。然而，预计将展示集体行为的完整挂毯。因此，尽管这些类型的模型的物理起源，它是预期的，至少由PI，这些类型的模型允许广泛的应用-甚至超出了物理科学。（在一项有关研究犯罪行为和警察资源分配的相关提案中，其中一些已经取得成果。）此外，也许更有针对性的是：本提案及其前身包含了广泛的问题，在各个层面提供了研究机会，成功完成该计划将需要许多合作者的帮助，从本科生到高级研究合作者。

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.