Mathematical Theory of Non-Equilibrium Statistical Mechanics

这项研究提案涉及我工作了十多年的研究计划的继续。具体目标如下：(一)完成熵涨落方案。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，已经出版了近500页的期刊。要完成程序中所谓的“量子Evans-Searles”部分(相对于参考态的涨落理论)，需要额外一年的工作，并完成两篇主要论文(《Pauli-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.量子动力学系统》(估计约400页))，以及完成一本研究专著《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)非平衡稳态的热力学。这项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”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)，把Ruelle关于“熵联系和曲率”的几何思想与几何参数估计理论(Efron)的思想结合起来，可以得到令人满意的结果，可能具有深远的物理和数学意义。(Iii)随机偏微分方程论中的罕见事件和涨落对称性。本课题致力于研究数学物理中出现的一些随机偏微分方程解的大时间渐近性(特别是大偏差理论)。其主要动机是非平衡统计力学，最终目的是从数学上严格理解随机偏微分方程组所描述的物理系统的Gallavotti-Cohen涨落关系。激励的例子是描述不可压缩粘性流体运动的Navier-Stokes方程。我还计划研究复的Ginzburg-Landau方程和阻尼色散偏微分方程组。(Iv)格子上费米气体相互作用的局域化。描述电子在随机外势作用下运动的随机薛定谔算符的Anderson局部化在大无序区被很好地理解。相比之下，在物理上重要的情况下，电子之间的相互作用不被忽略，关于安德森局域化几乎一无所知。传统的基于谱理论的方法显得不太合适，需要新的思路。我计划使用最近在关于非平衡量子统计力学的数学严谨文献中出现的想法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附在样品上时不存在Landauer-Buttiker非平衡稳态输运联系起来。(V)开放的XY自旋链和雅可比矩阵的谱理论。这个项目涉及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链接。直接的具体目标是Kotani理论的新证明和与Harper方程相关的XY链的Landauer-Buttiker公式的正则性的研究。(Vi)Shannon-McMillan-Breiman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。

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.