Emergence of condensation in stochastic systems

一个复杂的系统经历凝聚，如果一个观察到的量的正分数集中在一个单一的状态，渐近时间（或其他一些模型参数）走向无穷大。凝聚是许多复杂系统的特征，包括宏观经济系统中的财富凝聚，分子的凝胶化，玻色-爱因斯坦凝聚的物理现象，以及交通堵塞的形成。在这个项目中，我们研究如何冷凝可以建立，或出现，在复杂的系统。这将通过研究相对简单的数学玩具模型来完成。在这些模型中，我们放大到一个小的凝聚态附近在有限的时间，并重新调整该网站周围的可观察到的分布。如果标度分布随着时间的无穷大和标度的零而收敛，我们称这种极限为凝聚波。换句话说，凝聚波描述了可观测量在坍缩形成凝聚体之前的分布形状。研究者和他的合作者最近的一项试点研究表明，对于来自非常不同背景的几个模型，凝聚波的形状是通用的，即，它不依赖于具体的模型细节。在所有的例子中，形状都是一个著名的概率分布，即伽马分布，从这些观察中产生的紧迫问题是：（1）以这种形式呈现凝聚的复杂系统的普适性类有多大？(2)是否有其他具有不同波形的普适类？(3)我们如何用可及参数来描述不同普适类的系统？(4)除了凝聚波的形状之外，还有什么特征是给定类的特征？该项目将针对不同复杂性的各种模型解决这些问题。

A complex system undergoes condensation if a positive fraction of an observed quantity concentrates in a single state, asymptotically as time (or some other model parameter) goes to infinity. Condensation is a characteristic feature of many complex systems including wealth condensation in macroeconomic systems, gelation of molecules, the physical phenomenon of Bose-Einstein condensation, and the formation of traffic jams. In this project we investigate how condensation can build up, or emerge, in complex systems. This will be done by investigating relatively simple mathematical toy models. In these models we zoom into a small neighbourhood of the condensation state at a finite time and rescale the distribution of the observable around this site. If the scaled distribution converges as time goes to infinity and scale goes to zero we call the limit a condensing wave. In other words, the condensing wave describes the shape of the distribution of the observable before it collapses to form the condensate. A recent pilot study by the investigator and his collaborators has shown that for several models from very different contexts the shape of the condensing wave is universal, i.e., it does not depend on the specific model details. In all the examples the shape was that of a well-known probability distribution, the gamma distribution.The pressing questions resulting from these observations are:(1) How large is the universality class of complex systems exhibiting condensation in this form?(2) Are there other universality classes with different wave shapes?(3) How can we characterize systems in different universality classes in terms of accessible parameters?(4) What features other than the shape of condensing wave are characteristic of a given class?The project will address these questions for a variety of models of different complexity.