Studies in Statistical Mechanics

技术概述材料研究部、数学科学部和物理部为该奖项提供资金。该奖项支持理论研究和教育，旨在促进对宏观系统起源于其微观成分的集体行为的特性的理解。所用的方法从精确的数学分析到计算机模拟都有。它们为平衡和非平衡现象提供了严谨结果和应用之间的桥梁。这项研究是高度跨学科的，汇集了物理学家、数学家、化学家以及生物和社会科学理论领域的工作人员。研究主题包括：1)将阐明量子系统中随机性的影响。这将扩展最近的证明，即在2维或更少的空间中，在量子系统中加入随机性会使共轭序参数的一阶相变四维或更少的空间中，或涉及到连续对称的破坏的情况下。将探讨随机系统的新不等式。2)研究了具有空间不对称远程相互作用的一般系统的相图，在一维上扩展了三种ABC模型的精确解。3)扩展了具有多自旋相互作用的晶格体系相图的结果。4)在平衡和非平衡系统中，由于短程吸引和远程排斥而形成的条纹等图案将被阐明。5)研究由自治方程给出的开放系统宏观状态的演化。与孤立系统中的玻尔兹曼熵类似，相对于非平衡稳态的平稳测度的大偏差函数是这种演化的李雅普诺夫函数。这为非线性扩散方程提供了新的Lyapunov函数。偏差函数作为开放系统的相对自由能，将从微观和宏观两个角度进行探讨。6)严格确定输运系数是本研究项目的重要目标。将阐明傅立叶定律在各种系统中的有效性域，例如具有随机质量的弱非调和晶体。当在动力学中加入噪声时，傅立叶定律严格成立，但是少量的非谐波会破坏声子的局域化吗？将研究一维和二维动量守恒模型中违反傅里叶定律的情况。7)将研究现实系统中“典型”的标准：对于一个“典型”的大型孤立量子系统，来自能量壳的每个初始波函数都以这样一种方式演化，在大多数时候，它在宏观上等同于微正则密度矩阵。该奖项支持PI每年组织两次会议，在学院氛围中讨论统计力学的核心主题和新发展。鼓励研究生、博士后和少数族裔科学家就他们的工作发表演讲，并与该领域的领导者进行互动。它们还可以作为职位的交换所，并经常导致新的合作。除了在许多科学会议上应邀演讲外，PI还进行了公开演讲，并为Scholarpedia撰写了一篇关于时间之箭的文章。该奖项支持统计力学各个分支领域的理论研究和教育。这项工作由材料研究部和数学科学部共同支持。这项工作的中心主题是更好地理解源于其基本原子成分集体行为的材料系统的特性。所用的方法从精确的数学分析到计算机模拟都有。这些方法弥合了严谨结果和应用之间的差距。研究主题范围从经典物理到量子物理，从高度正式到技术相关的应用，甚至到疾病传播的动力学。该项目旨在推进对生物系统、生物材料和更广泛的材料产生影响的远离平衡平衡的系统理论。研究活动是高度跨学科的，汇集了物理学家，数学家，化学家和那些在生物和社会科学理论领域工作的人。预期的应用是在材料科学，复杂流体和生物系统。该项目还包括每年组织两次会议，在学院气氛中讨论统计力学的核心主题和新发展。研究生、博士后和少数族裔科学家参与其中，就他们的工作发表演讲，并与该领域的知名研究人员进行互动。这些会议还提供了一个建立专业网络的机会，并可能导致新的合作。

TECHNICAL SUMMARYThe Division of Materials Research, the Division of Mathematical Sciences, and the Physics Division contribute funds to this award. This award supports theoretical research and education aimed at advancing understanding of the properties of macroscopic systems originating in the collective behavior of their microscopic constituents. The methods used range from exact mathematical analysis to computer simulation. They provide a bridge between rigorous results and applications, for both equilibrium and nonequilibrium phenomena.The research is highly interdisciplinary, bringing together physicists, mathematicians, chemists and those working in theoretical areas of the biological and social sciences. Topics of study include:1) The effects of randomness in quantum systems will be elucidated. This will extend the recent proof that the addition of randomness to a quantum system rounds a first-order phase transition in the conjugate order parameter in 2 dimensions or less, or for cases involving the breaking of a continuous symmetry in 4 dimensions or less. New inequalities for random systems will be explored.2) The phase diagram of general systems with spatially asymmetric long range interactions will be investigated, extending the exact solution of the three species ABC model in 1D.3) Results on the phase diagram of lattice systems with multispin interactions will be extended.4) Pattern formation such as stripes, in equilibrium and nonequilibrium systems, due to short range attractions and long range repulsions will be elucidated.5) The evolution of the macrostate of an open system given by an autonomous equation willbe investigated. In analogy to the Boltzmann entropy in an isolated system, the large deviation function, with respect to the stationary measure of the nonequilibrium stationary state, is a Lyapunov function for this evolution. This yields new Lyapunov functions for nonlinear diffusion equations. The deviation function, considered as a relative free energy of an open system, will be explored from both a microscopic and macroscopic point of view.6) A rigorous determination of transport coefficients is an important objective of this research project. Domains of validity of Fourier's law in various systems, for example weakly anharmonic crystals with random masses, will be elucidated. When noise is added to the dynamics Fourier's law holds rigorously, but will a small amount of anharmonicity destroy phonon localization? Violations of Fourier's law for momentum conserving models in 1D and 2D will be investigated.7) Criteria for "typicality" in realistic systems will be investigated: for a "typical" large isolated quantum system every initial wave function from an energy shell evolves in such a way that, for most time it is macroscopically equivalent to the micro-canonical density matrix.This award supports the PI's efforts to organize two conferences every year in which both core subjects and new developments in statistical mechanics are discussed in a collegial atmosphere. Graduate students, postdocs and minority scientists are encouraged to present talks on their work and interact with leaders in the field. They also serve as a clearing house for positions and often lead to new collaborations. In addition to many invited lectures at scientific conferences the PI has given public lectures and has written an article for Scholarpedia on Time's Arrow.NONTECHNICAL SUMMARYThis award supports theoretical research and education in a variety of subfields of statistical mechanics. This work is jointly supported by the Division of Materials Research and the Division of Mathematical Sciences. The central theme of the effort is a better understanding of the properties of material systems originating in the collective behavior of their elementary atomic constituents. The methods used range from exact mathematical analysis to computer simulations. These approaches bridge the gap between rigorous results and applications. Topics of study range from classical physics to quantum physics, from highly formal to applications of technological relevance and even to dynamics of disease propagation. The project aims to advance the theory of systems that are far from the balance of equilibrium with an impact on biological systems, biomaterials, and materials more generally. The research activities are highly interdisciplinary, bringing together physicists, mathematicians, chemists and those working in theoretical areas of the biological and social sciences. The expected applications are in material science, complex fluids and in biological systems. The project also includes the organization of two conferences every year in which both core subjects and new developments in statistical mechanics are discussed in a collegial atmosphere. Graduate students, postdocs and minority scientists are involved and present talks on their work and interact with established researchers in the field. The conferences also serve as an opportunity for professional networking and can lead to new collaborations.