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: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635703
• 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.

该研究计划涉及我十多年来所从事的研究计划的延续，具体目标如下：（一）完成熵涨落计划这个庞大的研究计划一直是我过去五年的主要关注点。并已导致近 500 页的期刊印刷。完成该计划的所谓“量子埃文斯-塞尔斯”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个主要课程。文件（“泡利-菲尔兹系统的非平衡统计力学”（预计约100页）和“统计力学中的熵涨落II.量子动力系统”（预计约400页））并完成研究专着“非平衡” (二)非平衡稳态热力学该研究项目是熵涨落计划的自然延续，它涉及远离平衡的物理系统的“熵”概念，我相信在各种特殊情况（例如开放量子系统）中可能会得到令人满意的结果。将Ruelle关于“熵联系和曲率”的几何思想与几何参数估计理论（Efron）的思想相结合，可以获得物理和数学意义。 （三）随机理论中的稀有事件和涨落对称性该项目致力于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论（特别是大偏差理论），其主要动机是非平衡统计力学，最终目标是对 Galavotti 的数学严格理解。 - 由随机偏微分方程描述的物理系统的科恩涨落关系 激励性的例子是描述不可压缩粘性运动的纳维-斯托克斯方程。我还计划研究复杂的Ginzburg-Landau 方程和阻尼驱动的色散偏微分方程。(IV) 晶格上相互作用的费米气体的局域化，描述电子在 影响下运动的安德森局域化。相比之下，在大无序状态下，随机外部势能被很好地理解，而在不忽略电子之间相互作用的物理重要情况下，实际上对安德森定域一无所知。谱理论似乎不合适，我计划使用最近在非平衡量子统计力学的数学严格文献中出现的思想和技术来研究这个问题，主要思想是将无序样本的局域化理论联系起来。当热储层附着在样品上时，相互作用的费米子与缺乏 Landauer-Buttiker 非平衡稳态输运有关。(V) 开放 XY 自旋链和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/散射理论之间的令人惊讶的联系，我有几篇关于这个主题的论文，我计划继续探索这个联系。 Kotani 理论的新证明以及与 Harper 方程相关的 XY 链 Landauer-Buttiker 公式的正则性质研究 (VI) Shannon-McMillan-Breiman 定理和非平衡统计。该项目探索了量子信息论和量子统计力学的最新发展之间的联系。