The Mathematics of Stochastic Systems with Many Interacting Constituents

一般来说，这个建议涉及的行为的系统与许多自由度。 在这里，我们寻找的是（a）系统作为一个整体的集体行为，或者（B）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）并不是互不相关的标准：实际上，它们经常（但并不总是）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，并且所有四种类型都起源于对物理问题的描述。 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]类似的设置（i），但根据量子平衡统计力学的规则。 (iii)[交互粒子系统]占据晶格位置的单个粒子或自旋根据自身受现有局部配置影响的随机规则及时动态更新其位置或状态的模型。 (iv)[集群模型/图形表示]图形模型，其中除了局部规则之外，还根据全局考虑或非局部规则随机确定配置。 除了标准的晶格为基础的大规模的描述，一些工作将涉及连续模型-以及连续的限制晶格models.A哲学，成为流行的开始在20世纪60年代后期是普遍性：一旦基本性质和前提的成分和它们的相互作用是确定的，所产生的（可能的模式）大？规模，长吗？具有多个自由度的系统中的时间集体行为不依赖于模型的细致细节。 因此，将研究各种系统，其中相互作用组分的性质是最简单的类型。 然而，预计集体行为的全部挂毯将被展示。因此，尽管这些类型的模型的物理起源，但至少PI预计，这些类型的模型允许广泛的应用-甚至超越物理科学。 （其中一些已经在一项有关研究犯罪行为和警察资源分配的建议中取得了成果。）此外，也许更有针对性：该提案及其前身涵盖了一系列广泛的问题，这些问题具有多个方面，在各个层面提供研究机会，并且该计划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协助。

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.