Slow Dynamics in Random Media

这个项目的主题是无序介质的概率理论和统计物理之间的交界点。它建议对无序系统的随机动力学的减速或陷阱建立一个系统的理解。据了解，对于这些系统中的一些系统，速度变慢是由于配置空间中的局部区域，系统大部分时间都花在这些区域上。一种通用的范式(陷阱模型)已经在物理文献中被引入，并在最近被PI和各种合作者研究。这个项目建议做三件事：加深对这个唯象模型的理解，证明它在广泛的时间范围内与平均场自旋玻璃动力学(和老化)相关(建立在PI和合作者最近的工作基础上)，并介绍陷阱模型在渗流团簇上随机行走的困难问题中的观点。PI将与研究生和已建立的国际合作者网络一起调查第一个和第二个主题。第三个专题(在超临界星系团上有偏见的随机行走)的两个主题之一将在他被任命为库兰特研究所期间(大约两年)与国际和平联合会一起进行调查。第二个主题将由PI、Co-PI和外部合作者进行调查。现代科学的许多分支都面临着一项复杂的任务，即清楚地了解复杂系统的时间演化，更具体地说，是由许多组件组成的具有复杂局部规则的大系统的时间演化。特别是，上个世纪关于统计平衡性质的经验教训对这些复杂系统中的大多数似乎没有什么价值，因为在通常情况下，它们往往离平衡很远，在某种意义上，总是处于过渡状态。它们达到平衡的速度非常慢，并在此过程中表现出迷人的特性；其中之一就是“老化”。这些非常缓慢的演变是这项建议的主题，目的是处理精选的范例，这些范例被认为是复杂和“普遍的”，即代表了一大类此类现象。

This project addresses topics at the interface between probability theory and statistical physics of disordered media. It proposes to build a systematic understanding of the slowing down or trapping of stochastic dynamics of disordered systems. The understanding is that for some of these systems the slowing down is due to localized regions in configuration space where the system spends most of its time. A general paradigm (the Trap Model) has been introduced in physics literature and studied recently by the PI and various collaborators. This project proposes to do three things: sharpen this understanding of this phenomenological model, show that it is relevant for mean-field Spin Glass dynamics (and aging) in a wide range of time scale (building on recent work of the PI and collaborators), and introduce the point of view of trap models in hard problems of Random Walks on percolation clusters. The PI will investigate the first topic and second topic with graduate students and an established international network of collaborators. The Co-PI will investigate one of the two themes of the third topic (Biased random walks on supercritical clusters) with the PI for the duration of his appointment at Courant Institute (approximately two years). The second topic will be investigated by the PI, the Co-PI and external collaborators. Many branches of modern science are challenged with the complicated task of attaining a clear understanding of the time evolution of complex systems, or more specifically, of large systems comprised of many components and with complex local rules. In particular, the lessons learned in the last century in regards to the properties of statistical equilibrium seem of little value for most of these complex systems as they tend to be, under usual conditions, always very far from equilibrium and, in a sense, always in a transient regime. They achieve their equilibrium very slowly and exhibit fascinating properties along the way; one of them being "aging". These very slow evolutions are the subject of this proposal for addressing well chosen paradigms which are known to be complex and "universal", i.e. representative of a wide class of such phenomena.