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.

这个研究计划是关于我已经工作了十多年的研究项目的延续。具体目标如下。**（一）完成熵涨落程序。这个庞大的研究项目是我过去五年的主要研究重点，已经出版了近500页的期刊。要完成该计划中所谓的“量子Evans-Searles”部分（相对于参考状态的涨落理论），需要额外一年的工作，并完成两篇主要论文（“paui - fierz系统的非平衡统计力学”（估计约100页）和“统计力学中的熵涨落II”）。量子动力系统”（估计约400页）)和研究专著“局部相互作用费米子系统的非平衡统计力学”（估计约400页）的完成。*(II)非平衡稳态热力学。这个研究项目是熵涨落计划的自然延续。它涉及到远离平衡的物理系统的“熵”这个有问题的概念。我相信，在各种特殊情况下（如开放量子系统），通过将Ruelle关于“熵连接和曲率”的几何思想与几何参数估计理论（Efron）的思想相结合，可以得到一个可能具有深远物理和数学意义的令人满意的结果。*(III)随机偏微分方程理论中的稀有事件和涨落对称性。本课题主要研究数学物理中出现的一些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，最终目标是对随机偏微分方程描述的物理系统的Gallavotti-Cohen涨落关系进行数学上的严格理解。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金兹堡-朗道方程和阻尼驱动色散偏微分方程。*(IV)晶格上相互作用费米气体的局域化。描述电子在随机外部势影响下运动的随机薛定谔算符的安德森局域化在大无序状态下得到了很好的理解。相比之下，在电子之间的相互作用不容忽视的重要物理情况下，实际上对安德森局域化一无所知。传统的基于谱理论的方法显得不适应，需要新的思路。我计划使用最近在非平衡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品的局域化理论与当热储层附着在样品上时Landauer-Buttiker非平衡态输运的缺失联系起来。*(V)开XY自旋链与Jacobi矩阵的谱理论。该项目涉及XY链的非平衡统计力学与雅可比矩阵的光谱/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。当前的具体目标是对Kotani理论进行新的证明，并研究与Harper方程相关的XY链的Landauer-Buttiker公式的正则性。*(6) Shannon-McMillan-Breiman定理与非平衡统计力学。该项目旨在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。