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.

整个主题是相互作用的粒子系统和它们的临界现象。新颖之处在于将相互作用布朗运动的大偏差分析与最近发展的随机交织理论结合起来。第二个新颖之处是对散射长度的特定缩放，它描述了布朗运动和随机交织的相互作用。其目的是证明无限长循环上的凝聚（以随机穿插的形式给出）是相互作用玻色子的所谓玻色斯坦凝聚的一个信号。这一结果将首次确立概率凝聚及其在数学物理中的应用。主要的新颖之处在于证明了“无限长”周期表现为一些随机的交错过程。我们的目标是找到这样一个随机过程。它主要是一个使用大偏差技术、随机分析、多尺度分析和相互作用粒子系统的概率项目。在这个项目中，分析了最迷人和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20世纪90年代末的冷原子实验和两次诺贝尔奖以来，数学研究开始致力于证明所谓的玻色-爱因斯坦凝聚临界现象，例如液态氦在低温下的超流动性。该项目使用概率方法研究量子相互作用系统——所谓的费曼-卡茨公式允许将量子问题转化为概率论中的经典问题。主要使用的技术有大偏差分析、随机分析和浓度不等式。该项目特别研究了具有相互作用的泊松点过程，该项目包括以下步骤：(1)在Gross-Pitaevskii标度极限中，检查陷阱势中布朗运动的大N极限（布朗运动的数量）和相互作用布朗运动的大时间限制。第一步是开发和采用所谓散射长度的概率版本，以获得对相互作用项缩放作用的令人信服的描述。(2)第二步是研究周期长度分布和间隔分布（两个系统都没有相互作用项）的标记点过程。并将玻色-爱因斯坦凝聚现象与所谓的随机交错上的正概率权重的开始联系起来，这是一个双无限（时间视界）路径的随机过程（从无穷远开始并消失到无穷远）。随机交错是一类新的过程，近年来获得了大量的研究活动。我们的目标是展示这个新概念在分析玻色-爱因斯坦凝聚现象中的有用应用。(3)一旦步骤(2)证明了凝聚，主要部分将是考虑有或没有随机交错过程的有限循环之间的相互作用。作为前一步，该项目将证明步骤(1)的正散射长度可以触发冷凝现象。一旦项目完成了这个分析，项目就会在不缩放交互项的情况下研究系统。这一步是整个项目中最具挑战性的，因为它旨在展示空间相关性和相互交织之间的联系。这一结果将在该领域建立一个突破，并很可能在多粒子系统及其凝聚现象的方向上产生持久的影响。(4)一旦步骤3显示出新的凝聚现象，该项目将研究吉布斯测度在布朗运动和随机交织耦合系统中的作用。