Slow Dynamics in Random Media

本项目探讨的主题是无序介质的概率论和统计物理之间的接口。它提出对无序系统的随机动力学的减速或捕获建立一个系统的理解。我们的理解是，对于这些系统中的一些，减速是由于系统在构型空间中花费大部分时间的局部区域。一个一般的范例（陷阱模型）已经在物理文献中被介绍，最近由PI和各种合作者研究。该项目建议做三件事：加深对这种现象学模型的理解，表明它在大范围时间尺度下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），并在渗透集群上随机漫步的难题中引入陷阱模型的观点。PI将与研究生和已建立的国际合作者网络一起研究第一个和第二个主题。在他在Courant研究所任职期间（大约两年），Co-PI将与PI一起研究第三个主题（超临界簇上的有偏随机漫步）的两个主题之一。第二个主题将由PI， Co-PI和外部合作者调查。现代科学的许多分支都面临着一项复杂任务的挑战，即对复杂系统，或者更具体地说，对由许多组成部分和具有复杂局部规则的大系统的时间演化有一个清晰的认识。特别是，上个世纪关于统计平衡特性的经验教训似乎对大多数这些复杂系统没有什么价值，因为它们往往在通常情况下，总是离平衡很远，在某种意义上，总是处于瞬态状态。它们非常缓慢地达到平衡，并在此过程中表现出迷人的特性；其中之一就是“衰老”。这些非常缓慢的进化是本提案的主题，旨在解决精心选择的范例，这些范例已知是复杂的和“普遍的”，即代表了此类现象的广泛类别。

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.