Mathematical theory of non-equilibrium statistical mechanics

这一建议涉及非平衡统计力学（经典和量子）在动力系统设置的数学理论。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，并发现了所谓的波动关系（FR）和波动定理（FT）。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统计的普遍性质，而FT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）的数值实验和Evans和Searles（1994）的物理理论著作。1995年，Gallavotti和Cohen在紧黎曼流形的Anosov微分同态的背景下给出了傅里叶变换的第一个数学公式和数学证明。FR对量子力学环境的扩展可以追溯到2000年初，Kurchan， Tasaki和Tasaki- matsui的作品，与先前研究的经典环境相比，涉及一些根本性的新思想。这些发现产生了大量的身体物理学理论、数值和实验工作，从根本上提高了我们对非平衡物理学的理解，并将其应用于化学和生物学。尽管取得了这一惊人的进展，但相对而言，在FR和FT的推导上，数学上严格的结果很少。由于物理文献中FR的各种公式背后缺乏清晰的数学结构，以及在给定模型中缺乏FT证明的清晰性，数学上的进展受到了阻碍。这个提案继续了我的长期研究计划，即揭示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，然后将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。在过去的六年里，这个研究项目已经发表了23篇科学论文，并形成了一个由研究子方向和项目组成的相互关联的网络，这些研究子方向和项目不可能在规定的篇幅内全部描述清楚。在研究计划中，我将重点关注超越非平衡统计力学边界的三个问题，我预计它们将对数学和数学物理产生最大的影响。1. 随机强迫Navier-Stokes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. 扩散过程熵产生的消失噪声极限。

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.