AMC-SS: Mathematical and Computational in Nonequilibrium Statistical Mechanics.

Luc Rey-Bellet在统计力学的几个方向上工作，包括平衡和非平衡。(a)本文研究了由一个或多个热源耦合的非线性Klein-Gordon方程模型所产生的随机偏微分方程定态的构造及其遍历性。 （B）涨落定理在某些非一致双曲系统（如台球）中的熵产生的有效性。这涉及到为这样的系统开发大偏差技术。(c)经典和量子开放系统中熵产生涨落定理的有效性。 (d)Monte-Carlo方法的高阶粗粒化数值格式的发展。 非平衡物理力学中的许多基本问题在概念和数学层面上仍然知之甚少：例如，驱动开放系统的非平衡定态的表征。Gallavotti和Cohen的涨落定理是这些态的一个新的普适性质，它在各种系统（确定性、随机和量子）中的研究将是研究者工作的主要主题之一。多尺度数值方法和从复杂系统中求取相关自由度的一般问题是现代应用数学中一个极其重要的问题。 研究者建议使用统计力学中的概率技术（团簇膨胀和重整化群）来开发有效的数值方案，用于蒙特-卡罗方法的粗粒度化。 研究者的建议，除了其理论方面，有许多应用到各种具体的物理模型。这些应用程序被集成（通过数值或分析工作）到研究生研究项目。 该项目涉及与美国和国外机构的研究人员的几次合作。该项目还有助于传播现代数学工具，特别是概率的，到应用科学。统计力学领域是物理和数学理论，试图连接微观和宏观世界。微观世界，原子和分子的世界，是由牛顿定律或量子力学，其中涉及大量的方程描述。相反，宏观世界通常由几个参数或方程描述，如压力、温度、电导率和热导率等。这种简化来自于这样一个事实，即通过宏观透镜观察，大量的粒子以非常规则的方式运动。例如，在一个隔离良好的房间里，整个房间的温度几乎是恒定的。另一个例子是一块金属，一端加热，另一端冷却：能量会从热的部分流向冷的部分，但（几乎）不会反向流动。这种现象在精神上和数学上都与以下情况相似：如果一个人多次投掷一枚无偏差的硬币，那么正面的比例将非常接近一半，并且（几乎）永远不会出现显著的偏差。事实证明，为了研究这种典型行为，研究和表征与非典型行为相对应的罕见事件是非常有用的，即，“大偏差”。事实上，对大量粒子的非典型行为有一个详细的了解，是对什么是真正典型的基本理解的线索。这些想法，可以追溯到物理学和概率论的创始人，以及他们在各种物理情况下的实施，构成了该提案的核心。

Luc Rey-Bellet works in several directions in statistical mechanics,both in equilibrium and non-equilibrium. (a) The construction and theergodic properties of stationary states for stochastic partialdifferential equations arising from a model of a nonlinearKlein-Gordon equation coupled to one or several heat reservoirs. (b)The validity of the fluctuation theorem for entropy production in someclass of non-uniformly hyperbolic systems, such as billiards. Thisinvolves the development of large deviations techniques for suchsystems. (c) The validity of the fluctuation theorem for the entropyproduction in classical and quantum open systems. (d) The developmentof higher-order coarse-graining numerical schemes for Monte-Carlomethods. Many fundamental problems in non-equilibrium statisticalmechanics remain poorly understood, both at the conceptual andmathematical level: for example the characterization of non-equilibriumstationary states of driven open systems. The fluctuation theorem ofGallavotti and Cohen is a new universal property of these states andits study in various systems (deterministic, random and quantum) willbe one of the main theme in the work of the investigator. Multiscalenumerical methods and the general question of extracting the relevantdegrees of freedom out of complex systems is a problem of paramountimportance in modern applied mathematics. The investigator proposesto use probabilistic techniques from statistical mechanics (clusterexpansion and renormalization group) to develop efficient numericalschemes for the coarse-graining of Monte-Carlo methods. The proposalof the investigator, besides its theoretical aspects, has a number ofapplications to various concrete physical models. These applicationsare integrated (via numerical or analytical work) into graduateresearch projects. The project involves several collaborations withresearchers in U.S. institutions and abroad. The project also helps tothe dissemination of modern mathematical tools, in particularprobabilistic ones, into applied sciences.The field of statistical mechanics is the physical and mathematicaltheory which attempts to link the microscopic and macroscopic worlds.The microscopic world, the world of atoms and molecules, is describedby the laws of Newtonian or Quantum mechanics which involve a hugenumber of equations. The macroscopic world on the contrary is usuallydescribed by a few parameters or equations, such as pressure,temperature, electrical and thermal conductivity, etc... Thisreduction comes from the fact that a very large number of particles,seen from through macroscopic lenses, behave in a very regularfashion. For example, in a well isolated room, the temperaturethroughout the room will be nearly constant. Another example is apiece of metal heated at one end and cooled at the other end: therewill be a flow of energy from the hot part to the cold part but(almost) never in the opposite direction. This phenomena are similar,both in spirit and in mathematical terms to the following: if one throws an unbiased coin very many times then the proportion of headwill be extremely close to one half and (almost) never exhibitsignificant deviations. It turns out that to study this typicalbehavior it is very useful to study and characterize the rare eventscorresponding to untypical behavior, i.e., "large deviations". Havinga detailed understanding of the atypical behavior of a very largenumber of particles is in fact the clue to a fundamental understandingof what is really typical. Theses ideas, which go back to the foundingfathers of physics and probability theory, and their implementationsin various physical situations form the core of the proposal.