Slow Dynamics in Random Media

这个项目的主题是在无序介质的概率论和统计物理之间的接口。它建议建立一个系统的理解，减缓或捕获无序系统的随机动力学。我们的理解是，对于这些系统中的一些系统，减速是由于配置空间中的局部区域，系统花费了大部分时间。一个通用的范例（陷阱模型）已经在物理学文献中引入，并由PI和各种合作者最近研究。该项目建议做三件事：加深对这种唯象模型的理解，表明它在广泛的时间尺度上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），并在渗流集群上的随机行走的硬问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调查第一个主题和第二个主题。Co-PI将调查第三个主题的两个主题之一（超临界集群上的有偏随机游走）与PI在柯朗研究所任职期间（约两年）。第二个主题将由PI、Co-PI和外部合作者进行研究。现代科学的许多分支都面临着一项复杂的任务，即要清楚地了解复杂系统的时间演化，或者更具体地说，要了解由许多组件组成的具有复杂局部规则的大型系统。特别是，上个世纪在统计平衡性质方面所吸取的教训似乎对这些复杂系统中的大多数没有什么价值，因为在通常条件下，它们往往远离平衡，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沿着迷人的特性，其中之一就是“老化”。这些非常缓慢的演变是本提案的主题，该提案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”，即代表了广泛的一类此类现象。

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.