Mathematical theory of non-equilibrium statistical mechanics

这个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，发现了所谓的波动关系（FR）和波动定理（FT）。FR是与时间反演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）的数值实验和Evans和Searles（1994）的物理理论工作。FT的第一个数学公式和数学证明是由Gallavotti和Cohen在1995年给出的紧黎曼流形的Anosov同构。FR对量子力学环境的扩展可以追溯到21世纪初，Kurchan，Tasaki和Tasaki-Matsui的工作，与以前研究的经典环境相比，涉及一些根本上新的想法。这些发现产生了一个巨大的身体物理理论，数值和实验工作，从根本上提高了我们的理解非平衡物理与应用延伸到化学和生物学。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。数学的进展一直受到阻碍，缺乏明确的数学结构背后的各种公式的FR在物理学文献和缺乏清晰的证明FT在一个给定的模型。 这个建议继续我的长期研究计划，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。在过去的六年里，这个研究项目已经发表了二十三篇科学论文，并形成了一个相互联系的研究网络。 子方向和项目，是不可能描述所有在分配的空间。在研究计划中，我将重点关注三个超越非平衡统计力学边界的问题，我预计这三个问题将对数学和数学物理产生最大的影响。 1.随机受迫Navier-Stokes系统拉格朗日轨线的遍历性、大偏差原理和涨落关系。 2.非平衡统计力学 重复的量子测量过程。 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.