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: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=588723
• 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.

这项研究计划涉及我已经工作了十多年的研究计划的继续。具体目标如下。 (I)完成熵波动程序。在过去的五年里，这个大规模的研究项目一直是我的主要关注点，已经导致了近500页的期刊印刷。完成所谓的“量子埃文斯-塞尔”部分的计划（波动理论相对于参考状态）需要额外的一年的工作和完成两个主要文件（“非平衡统计力学的保利-菲耶茨系统”（估计约100页）和“熵波动统计力学II。量子动力系统”（估计约400页）和完成一本研究专著“局部相互作用费米子系统的非平衡统计力学”（估计约400页）。 (II)非平衡定态热力学。 这个研究项目是熵波动计划的自然延续。它涉及到远离平衡的物理系统的“熵”的问题概念。我相信，在各种特殊情况下（如开放的量子系统），一个令人满意的结果，可能深远的物理和数学的影响，可以通过结合的几何思想Ruelle关于“熵连接和曲率”的几何参数估计理论（埃夫隆）的想法。 (III)随机偏微分方程理论中的稀有事件和涨落对称性。本计画致力于研究数学物理中随机偏微分方程的大时间渐近性（特别是大偏差理论）。主要动机是非平衡统计力学和最终目标是数学上严格的理解随机PDE的描述的物理系统的Gallavotti-Cohen波动关系。 激励的例子是描述不可压缩粘性流体的运动的Navier-Stokes方程。我还计划研究复杂的金-朗道方程和阻尼驱动色散偏微分方程。 (IV)格点上相互作用费米气体的定域化。描述电子在随机外势作用下运动的随机薛定谔算符的安德森定域性在大无序区中得到了很好的理解。相比之下，在电子之间的相互作用不被忽略的物理重要情况下，对安德森局域化几乎一无所知。基于谱理论的传统方法似乎不合适，需要新的想法。我计划使用最近在非平衡态量子统计力学的数学严谨文献中出现的思想和技术来研究这个问题。其主要思想是将相互作用费米子的无序样本的定域理论与朗道-布蒂克的缺失联系起来 非平衡稳态传输时，热库连接到样品。 (V)开XY自旋链与雅可比矩阵的谱理论。该项目涉及XY链的非平衡统计力学和雅可比矩阵的光谱/散射理论之间的惊人联系。我有几篇关于这个主题的论文，我计划继续探索这个联系。当前的具体目标是Kotani理论的新证明和与哈珀方程相关的XY链Landauer-Buttiker公式的正则性研究。 (VI)Shannon-McMillan-Breiman定理与非平衡统计力学该项目涉及探索量子信息理论和量子统计力学的最新发展之间的联系。