Mathematical theory of non-equilibrium statistical mechanics

*这一建议涉及动力系统背景下的非平衡统计力学(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20世纪90年代初S和所谓涨落关系(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有关的熵产生统计的普遍性质，而FT指的是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。在紧致黎曼流形的Anosov微分同胚的背景下，Gallavotti和Cohen于1995年给出了FT的第一个数学公式和一个数学证明。FR对量子力学的扩展可以追溯到2000年初的S以及库昌、玉崎和松井龙崎的著作，与以前研究的经典背景相比，涉及到一些根本上的新想法。这些发现产生了巨大的身体物理学--理论、数值和实验工作，这些工作从根本上改善了我们对非平衡物理的理解，并将其应用到化学和生物学。尽管取得了这一惊人的进展，但关于FR和FT的派生的数学严谨结果相对很少。由于物理文献中FR各种公式背后缺乏明确的数学结构，以及在给定模型中缺乏构成FT证明的清晰的数学结构，数学进步一直受到阻碍。*这个建议继续我的长期研究计划，旨在揭示FR和FT背后的数学结构，提炼我们对非平衡统计力学的一般数学理解，然后将这些结果用作非平衡统计力学中具体物理相关模型的数学分析的起点。在过去的六年里，这一研究计划导致了23种科学出版物的发表，并产生了一个相互关联的研究子方向和项目的网络，这些研究方向和项目不可能在分配的空间内全部描述。在研究提案中，我将重点讨论三个超越非平衡统计力学界限，并预计将对数学和数学物理产生最大影响的因素。*1.随机强迫N-S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.扩散过程的熵产生的消噪极限。********

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