Mathematical theory of non-equilibrium statistical mechanics

***该提案涉及动力系统设置中非平衡统计力学（经典和量子）的数学理论。它是一项研究计划的延续，该计划的起源可以追溯到 20 世纪 90 年代初，即所谓的涨落关系 (FR) 和涨落定理 (FT) 的发现。 FR是与时间反转和违反热力学第二定律的概率相关的熵产生统计的普遍性质，而FT是指大偏差理论中相应的数学结果。 FR 的发现可以追溯到 Evans-Cohen-Morris (1993) 的数值实验以及 Evans 和 Searles (1994) 的物理理论著作。 FT 的第一个数学公式和数学证明是由 Gallavotti 和 Cohen 于 1995 年在紧致黎曼流形的阿诺索夫微分同胚的背景下给出的。 FR 对量子力学设置的扩展可以追溯到 2000 年代初，Kurchan、Tasaki 和 Tasaki-Matsui 的作品，与之前研究的经典设置相比，涉及一些全新的想法。这些发现产生了大量的身体物理学理论、数值和实验成果，从根本上提高了我们对非平衡物理学的理解，并将其应用扩展到化学和生物学。尽管取得了如此惊人的进展，但关于 FR 和 FT 的推导，数学上严格的结果相对较少。由于物理文献中各种 FR 公式背后缺乏清晰的数学结构，以及在给定模型中构成 FT 证明的内容缺乏明确性，数学进展受到了阻碍。********该提案继续我的长期研究计划，涉及揭示 FR 和 FT 背后的数学结构，完善我们对非平衡统计力学的一般数学理解，然后使用这些结果作为非平衡统计力学中具体物理相关模型的数学分析的起点。在过去六年中，该研究计划已发表了二十三篇科学出版物，并产生了一个相互关联的研究子方向和项目网络，这些子方向和项目不可能在所分配的空间中描述所有内容。在研究计划中，我将重点关注三个超越非平衡统计力学界限的领域，我预计这将对数学和数学物理产生最大的影响。*****1。随机强迫纳维-斯托克斯系统拉格朗日轨迹的遍历性、大偏差原理和涨落关系。 ******2.重复量子测量过程的非平衡统计力学*。 ******3.扩散过程熵产生的噪声极限消失。 ******

***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. ********