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.

该研究计划涉及我十多年来从事的研究计划的延续。具体目标如下：**（一）完成熵涨落计划。这个庞大的研究项目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。该计划的所谓“量子埃文斯-塞尔斯”部分（相对于参考态的涨落理论）的完成需要额外一年的工作并完成两篇主要论文（“泡利-菲尔兹的非平衡统计力学”）系统”（预计约 100 页）和“统计力学中的熵涨落 II”（预计约 400 页））以及完成一本研究专着。 “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) 非平衡稳态的热力学。 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中存在问题的“熵”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“熵联系和曲率”的几何思想与几何参数估计理论的思想相结合，可以获得令人满意的结果，并且可能具有深远的物理和数学意义（ Efron).*(III) 随机偏微分方程理论中的罕见事件和涨落对称性。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）。主要动机是非平衡统计力学，最终目标是在数学上严格理解随机偏微分方程描述的物理系统的加拉沃蒂-科恩涨落关系。 激动人心的例子是描述不可压缩粘性流体运动的纳维-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。描述电子在随机外部电势影响下运动的随机薛定谔算子的安德森定位在大无序状态下得到了很好的理解。相比之下，在不忽略电子之间相互作用的物理重要情况下，实际上对安德森局域化一无所知。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。主要思想是将相互作用费米子无序样本的局域化理论与热储层附着到样本时不存在 Landauer-Buttiker 非平衡稳态输运联系起来。*(V) 开放 XY 自旋链和谱理论雅可比矩阵。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链接。近期的具体目标是 Kotani 理论的新证明以及与 Harper 方程相关的 XY 链的 Landauer-Buttiker 公式的正则性研究。 *（六）香农-麦克米伦-布雷曼定理和非平衡统计力学。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。