Mathematical theory of non-equilibrium statistical mechanics

非平衡统计力学数学理论

基本信息

批准号：
RGPIN-2019-04485
负责人：
Jaksic, Vojkan
金额：
$ 2.77万
依托单位：
McGill University
依托单位国家：
加拿大
项目类别：
Discovery Grants Program - Individual
财政年份：
2019
资助国家：
加拿大
起止时间：
2019-01-01 至 2020-12-31
项目状态：
已结题

来源：
https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=687151
关键词：
Mathematical theory non equilibrium statistical

项目摘要

***This proposal concerns mathematical theory of non-equilibrium statistical mechanics (classical and quantum) in dynamical systems setting. It is a continuation of a research program whose origin can be traced back to early 1990's and the discovery of the so called Fluctuation Relations (FR) and Fluctuation Theorems (FT). The FR is a universal property of the statistics of entropy production linked to a time-reversal and the probability of violation of the Second Law of Thermodynamics, while FT refers to the corresponding mathematical result in theory of Large Deviations. The discovery of FR goes back to numerical experiments of Evans-Cohen-Morris (1993) and physics-theoretical works of Evans and Searles (1994). The first mathematical formulation and a mathematical proof of FT was given by Gallavotti and Cohen in 1995 in context of Anosov diffeomorphisms of compact Riemannian manifolds. The extensions of FR to quantum mechanical setting goes back to early 2000's and works of Kurchan, Tasaki, and Tasaki-Matsui, involving some fundamentally new ideas compared to the previously studied classical setting. These discoveries generated an enormous body physics-theoretical, numerical, and experimental works which have fundamentally improved our understanding of non-equilibrium physics with applications extending to chemistry and biology. In spite of this spectacular progress, there are comparatively very few mathematically rigorous results on the derivations of FR and FT. The mathematical progress has been hampered by the lack of clear mathematical structures behind various formulations of the FR in the physics literature and a lack of clarity what constitutes the proof of FT in a given model.******This proposal continues with my long-term research program dealing with uncovering mathematical structures behind the FR and the FT, refining our general mathematical understanding of non-equilibrium statistical mechanics, and then using these results as a starting point in mathematical analysis of concrete physically relevant models in non-equilibrium statistical mechanics. In the last six years this research program has led to twenty three scientific publications, and has generated an interconnected web of research sub-directions and projects that are impossible to describe all in the allotted space. In the research proposal I will focus on three that go beyond boundaries of non-equilibrium statistical mechanics and which I expect will have the largest impact on mathematics and mathematical physics.******1. Ergodicity, Large Deviation Principle, and Fluctuation Relation for Lagrangian trajectories of randomly forced Navier-Stokes systems. ******2. Non-equilibrium statistical mechanics *of repeated quantum measurement processes. ******3. Vanishing noise limit for the entropy production for diffusion processes. ********

***该建议涉及动态系统设置中非平衡统计力学（经典和量子）的数学理论。这是一个研究计划的延续，其起源可以追溯到1990年代初，并且发现所谓的波动关系（FR）和波动定理（FT）。 FR是熵生产统计的普遍特性，该特性与时间反转以及违反第二种热力学定律的可能性有关，而FT是指大偏差理论的相应数学结果。 FR的发现可以追溯到Evans-Cohen-Morris（1993）和Evans and Searles（1994）的物理学理论作品的数值实验。 Gallavotti和Cohen在1995年就紧凑的Riemannian歧管的Anosov差异性提供了第一个数学公式和FT的数学证明。 FR向量子机械设置的扩展可以追溯到2000年代初期以及Kurchan，Tasaki和Tasaki-Matsui的作品，与先前研究的经典环境相比，涉及一些根本新的想法。这些发现产生了巨大的身体物理学理论，数值和实验性工作，从根本上讲，通过扩展到化学和生物学的应用，从根本上提高了我们对非平衡物理学的理解。尽管取得了壮观的进步，但在FR和FT的推导方面，数学上很少有严格的结果。由于物理学文献中FR的各种表述背后缺乏明确的数学结构的缺乏，并且缺乏清晰度，并且缺乏明确的数学结构，并且在给定模型中构成了FT的证明。*******非平衡统计力学中混凝土物理相关模型的数学分析。在过去的六年中，该研究计划导致了23个科学出版物，并产生了一个相互联系的研究子指导和项目网络，这些网络无法在分配的空间中描述所有内容。在研究建议中，我将重点关注三个超越非平衡统计力学的边界，我希望这将对数学和数学物理学产生最大的影响。****** 1。拉格朗日轨迹随机强迫Navier-Stokes Systems的拉格朗日轨迹的奇迹性，大偏差原理和波动关系。 ****** 2。重复量子测量过程的非平衡统计力学 *。 ****** 3。扩散过程的熵产生的噪声极限。 ********