Mathematical Theory of Non-Equilibrium Statistical Mechanics

非平衡统计力学数学理论

• 批准号: RGPIN-2014-05965
• 负责人: Jaksic, Vojkan
• 金额: $ 2.04万
• 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: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=654456
• 关键词: Mathematical; Theory; Non; Equilibrium; Statistical

The research proposal concerns continuation of the research program on which I have worked for over a decade. The specific goals are the following.**(I) Completion of the Entropic Fluctuation Program. This massive research program has been my main focus over the last five years and has already led to nearly 500 journal pages in print. The completion of the so-called "quantum Evans-Searles" part of the program (fluctuation theory with respect to the reference state) requires an additional year of work and completion of two major papers ("Non-equilibrium statistical mechanics of Pauli-Fierz systems" (estimated around 100 pages) and "Entropic fluctuations in statistical mechanics II. Quantum dynamical systems" (estimated around 400 pages)) and a completion of a research monograph "Non-equilibrium statistical mechanics of locally interacting fermionic systems" (estimated around 400 pages). *(II) Thermodynamics of non-equilibrium steady states. This research project is a natural continuation of the Entropic Fluctuation program. It concerns the problematic concept of "entropy" for physical systems far from equilibrium. I believe that in various special situations (like open quantum systems) a satisfactory result with possibly far reaching physical and mathematical implications can be obtained by combining the geometric ideas of Ruelle concerning "entropic connection and curvature" with the ideas of geometric parameter estimation theory (Efron).*(III) Rare events and fluctuation symmetries in the theory of stochastic PDE's. This project is devoted to study of large-time asymptotics (and in particular large deviation theory) for some stochastic PDE's arising in mathematical physics. The principal motivation is non-equilibrium statistical mechanics and the ultimate goal is mathematically rigorous understanding of the Gallavotti-Cohen Fluctuation Relation for physical systems described by stochastic PDE's. The motivating example are Navier-Stokes equations describing the motion of an incompressible viscous fluid. I also plan to study the complex Ginzburg-Landau equation and damped-driven dispersive PDE's.*(IV) Localization for interacting Fermi gases on a lattice. The Anderson localization for random Schrodinger operators describing the motion of an electron moving under the influence of a random external potential is very well understood in the large disorder regime. In contrast, virtually nothing is known about the Anderson localization in the physically important case where the interaction between electrons is not neglected. The traditional approach based on the spectral theory appears unsuitable and new ideas are needed. I plan to study this problem using the ideas and techniques that has recently emerged in mathematically rigorous literature on non-equilibrium quantum statistical mechanics. The main idea is to link the localization theory of a disordered sample of interacting fermions to the absence of the Landauer-Buttiker non-equilibrium steady state transport when thermal reservoirs are attached to the sample.*(V) Open XY spin chains and spectral theory of Jacobi matrices. This project concerns a surprising link between the non-equilibrium statistical mechanics of XY chains and the spectral/scattering theory of Jacobi matrices. I have several papers on this subject and I plan to continue with the exploration of this link. The immediate specific goals are the new proof of Kotani theory and study of the regularity properties of Landauer-Buttiker formula for XY chain associated to Harper's equation. *(VI) Shannon-McMillan-Breiman theorem and non-equilibrium statistical mechanics. The project concerns exploration of the link between recent developments in quantum information theory and quantum statistical mechanics.

该研究建议涉及我在十多年工作的研究计划的延续。具体目标是以下。**（i）完成熵波动计划。在过去的五年中，这项大规模的研究计划一直是我的主要重点，并且已经导致了近500页的印刷页面。该计划所谓的“量子埃文斯”一部分的完成（相对于参考状态的波动理论）需要额外的工作年份，并完成两篇主要论文（“ Pauli-fierz Systems的非平衡统计机制”（估计约100页）（估计统计机械学的量学动力学）II（估计）II（估计）II。 “局部相互作用的费米斯系统的非平衡统计力学”（估计约400页）。 *（ii）非平衡稳态的热力学。 该研究项目是熵波动计划的自然延续。它涉及远离平衡的物理系统的“熵”的问题概念。我相信，在各种特殊情况（例如开放量子系统）中，可以通过将Ruelle的几何思想与“熵和曲率”与几何参数估计理论（EFRON）（EFRON）的思想相结合的几何思想来获得令人满意的结果。该项目致力于研究一些随机PDE在数学物理学中引起的大型渐近学（尤其是大偏差理论）。主要动机是非平衡统计力学，最终目标是在数学上对随机PDE描述的物理系统的Gallavotti-Cohen波动关系的数学认识。 激励的例子是描述不可压缩粘性流体运动的Navier-Stokes方程。我还计划研究复杂的Ginzburg-Landau方程和阻尼驱动的分散PDE。在大型疾病制度中，人们对随机外部潜力的影响在随机外部电位的影响下的电子运动的随机定位进行了描述。相比之下，在不忽略电子之间的相互作用的物理上重要情况下，在物理上重要的情况下，对安德森本地化几乎一无所知。基于光谱理论的传统方法似乎不合适，需要新的想法。我计划使用最近在数学上严格的关于非平衡量子统计力学的文献中出现的思想和技术来研究这个问题。主要思想是将相互作用费物的无序样本的本地化理论与没有陆地储物的非平衡稳态转运的不存在时，当热储层附着在样品上时。该项目涉及XY链的非平衡统计力学与Jacobi矩阵的光谱/散射理论之间的惊人联系。我有几篇关于这个主题的论文，我计划继续探索该链接。直接的具体目标是Kotani理论的新证明，以及研究与Harper方程相关的XY链的Landauer-Buttiker公式的规律性特性。 *（vi）香农 - 麦克米兰 - 二元定理和非平衡统计力学。该项目涉及探索量子信息理论最新发展与量子统计力学之间的联系。