Large deviation analysis of interacting systems of Brownian motions and random interlacements at positive temperature

正温度下布朗运动和随机交错相互作用系统的大偏差分析

• 批准号: 2273598
• 金额: --
• 依托单位: University of Warwick
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2019
• 资助国家: 英国
• 起止时间: 2019 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2273598
• 关键词: Large; deviation; analysis; interacting; systems

The overall theme is interacting particle systems and their critical phenomena. The novelty is to combine large deviation analysis for interacting Brownian motions with the recently developed theory of random interlacements. A second novelty is the specific scaling towards the scattering length, which describes interactions of Brownian motions and random interlacements. The aim to prove that condensation onto infinitely long cycles, which are given as random interlacements, is a signal of the so-called BoseEinstein condensation for interacting Bosons. This result will for the first time establish probabilistic condensation and its applications to mathematical physics. The main novelty is to prove that 'infinitely long' cycles appear as some random interlacements process. The aim is to find such a random process. It is primarily a probabilistic project using large deviation techniques, stochastic analysis, multi-scale analysis and interacting particle systems.Background. In this project, one of the most fascinating and challenging models is analysed, namely, interacting Bosons at positive temperature. Since the experiments on cold atoms in the late 1990s and two Nobel Prizes, mathematical research has started aiming to prove the so-called Bose-Einstein condensation critical phenomena like for example the superfluidity of liquid Helium at low temperatures. The project is using probabilistic methods for the quantum interacting systems - the so-called Feynman-Kac formula allows to transfer the quantum problem to a classical problem in probability theory.Main techniques to be used are variants of large deviation analysis, stochastic analysis and concentration inequalities. In particular, the project studies marked Poisson point processes with interaction, and the project involves the following steps:(1) Examination of the large N limit (number of Brownian motions) coupled with the large time limit for interacting Brownian motions in trap potential in the Gross-Pitaevskii scaling limit. The first step is to develop and employ a probabilistic version of the so-called scattering length to obtain a compelling description of the role of scaling of interaction terms.(2) The second step is to study the marked point process with cycle length distributions and interlacements distributions (both systems with no interaction terms). And relate the Bose-Einstein condensation phenomenon to the onset of positive probability weight on the so-called random interlacements, a random process of double-infinite (time horizon) paths (paths coming from infinity and disappearing to infinity).Random interlacements are a novel class of process and have recently attained a lot of research activity. We aim to showcase a useful application of this novel notion for the analysis of Bose-Einstein condensation-like phenomena.(3) Once step (2) proves the condensation, the major part will be to allow for interactions among finite cycles with or without the random interlacement processes. As a previous step, the project will show that the positive scattering length of step (1) can trigger condensation phenomena. Once the project finishes this analysis, the project studies the systems without the scaling of the interaction terms. The step is the most challenging of the whole project as it aims to demonstrate connections between spatial correlations and condensation onto interlacements. This result will establish a breakthrough in the field and is quite likely to have a lasting impact in the direction of many-particle systems and their condensation phenomena.(4) Once step 3 shows the novel condensation phenomena, the project will study the role of Gibbs measures for the coupled systems of Brownian motions and random interlacements.

总体主题是相互作用的粒子系统及其关键现象。新颖性是将大型偏差分析与最近开发的随机讲述理论相结合。第二个新颖性是朝向散射长度的特定缩放，它描述了布朗运动和随机讲述的相互作用。目的是证明将其缩合到无限长的循环上（作为随机的插座）是所谓的Boseeinstein凝结的信号，用于相互作用的玻色子。该结果将首次建立概率凝结及其在数学物理学上的应用。主要的新颖性是证明“无限长”的周期似乎是一些随机的中间过程。目的是找到这样的随机过程。它主要是使用大偏差技术，随机分析，多尺度分析和相互作用的粒子系统的概率项目。在这个项目中，分析了最引人入胜且最具挑战性的模型之一，即在正温度下相互作用的玻色子。自从1990年代后期进行冷原子的实验和两次诺贝尔奖以来，数学研究已经开始旨在证明所谓的玻色 - 因斯坦凝结关键现象，例如例如低温下液体氦气的超流体。该项目正在为量子相互作用系统使用概率方法 - 所谓的Feynman-KAC公式允许将量子问题转移到概率理论中的经典问题中。要使用的介质技术是大偏差分析，随机分析和浓度的变体不平等。特别是，该项目研究标志着泊松点过程与相互作用，并且该项目涉及以下步骤：（1）检查较大的N极限（Brownian Motions的数量），以及较大的时间限制，以使Brownian Motions在陷阱中相互作用的陷阱潜力相互作用。毛taevskii缩放限制。第一步是开发和采用所谓散射长度的概率版本，以获取相互作用术语缩放范围的作用的引人注目的描述。（2）第二步是研究具有周期长度分布和周期长度分布的标记点过程讲述分布（两个没有交互项的系统）。并将Bose-Einstein凝结现象与所谓的随机中间的阳性概率重量的发作联系起来，这是一个随机的双infinite（时间范围）路径（来自无穷大的路径，消失到无穷大）。新的过程类别，最近获得了大量的研究活动。我们的目的是展示这一新颖概念在分析类似于bose-内的凝结现象的有用应用。（3）一旦一步（2）证明了凝结，主要部分是允许在有或没有有限周期之间进行相互作用随机讲述过程。作为上一步，该项目将表明步骤（1）的正散射长度可以触发冷凝现象。一旦项目完成了此分析，该项目就会研究系统，而无需扩展交互项。该步骤是整个项目中最具挑战性的一步，因为它旨在证明空间相关性和凝结与中间隔离之间的联系。该结果将在该领域建立突破，并且很可能会在许多粒子系统及其凝结现象的方向上产生持久的影响。（4）一旦步骤3显示了新型的凝结现象，该项目将研究Gibbs测量了布朗尼运动和随机讲述的耦合系统。