Studies in Statistical Mechanics

该基金由材料研究部、物理部和数学科学部联合支持。本研究的目的是更好地理解起源于其微观成分的集体行为的宏观系统的性质。重点是非平衡系统和使用的方法范围从严格的数学分析到计算机模拟。我们特别感兴趣的是弥合严谨的结果和应用之间的差距。理论价值包括：1)将玻尔兹曼熵和h定理推广到非局部热平衡系统。建议将致密流体的最新工作推广到更复杂的系统，包括纳米和生物系统，在这些系统中熵的考虑起着重要的作用。这需要将玻尔兹曼熵适当地推广到非平衡量子系统。最近得到的精确的大偏差函数，描述了模型系统的平稳非平衡状态的波动是一种意想不到的形式。它们最显著的特征是宏观尺度上的非局部性，这可以追溯到存在于此类系统中的中子散射实验测量的远程相关性。计划将这些结果推广和应用到更实际的系统中。3)对于开放量子系统，如载流纳米线，经典的随机相互作用储层模型是有问题的。通常的方法是利用自由场或理想气体作为储层。在许多情况下，这是不令人满意的，我们计划继续研究替代方法，例如使用由随机矩阵表示的强耦合系统作为储层。4)我们开发的用于处理量子系统对强时间相关外场的响应的技术已经在简单模型系统上产生了严格的结果。计划将其应用于分子中量子跃迁的最优控制和固体中的外场。5)我们获得了关于粒子数波动减少的系统结构的新结果。这些在宇宙学和排队论中都有应用，这将被进一步探索。发展流体近似格式的一个重要问题是构造具有指定密度和对相关的粒子分布的可能性。在某些情况下，这可以通过通过行列式和更新来显式构造点过程来实现。目前正在调查一般存在标准。我们将统计力学方法应用于流行病的数学研究，并考虑了网络上的相关性和饱和效应。新的宏观方程描述的演变和流行的地方病感染状态改进协议与更详细的微观模型。计划将这些技术推广到种群动态和生态学模型中。更广泛的影响：拟议的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。预期的应用是在材料科学，复杂流体和生物和社会系统。我们的计划还包括每年组织两次会议，在学术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。这些会议还起到了交换意见的作用，并经常促成新的合作。

This grant is supported jointly by the Divisions of Materials Research, Physics and Mathematical Sciences. The aim of this research is a better understanding of the properties of macroscopic systems originating in the collective behavior of their microscopic constituents. The emphasis is on non-equilibrium systems and the methods used range from rigorous mathematical analysis to computer simulations. We are particularly interested in bridging the gap between rigorous results and applications.Intellectual Merit includes: 1) An extension of Boltzmann's entropy and H-theorem to systems not in local thermal equilibrium. It is proposed to generalize recent work for dense fluids to more complex systems, including nano and biological ones, where entropic considerations play an important role. This will require an appropriate generalization of Boltzmann's entropy to non-equilibrium quantum systems.2) Recently-obtained exact large deviation functions, describing fluctuations in stationarynon-equilibrium states of model systems are of an unexpected form. Their most striking feature, non-locality on the macroscopic scale, can be traced to the long-range correlations, measurable experimentally by neutron scattering, which exist in such systems. Extension and application of these results to more realistic systems is planned.3) For open quantum systems, such as current-carrying nano-wires, the classical modeling of the reservoirs by stochastic interactions is problematic. The usual approach is to use free fields, or ideal gases, as reservoirs. This is unsatisfactory in many cases and we plan to continue our investigation of alternative approaches such as the use of strongly coupled systems, represented by random matrices, as reservoirs.4) Techniques we developed for dealing with the response of quantum systems to strong time dependent external fields have yielded rigorous results about simple model systems. Applications to the optimal control of quantum transitions in molecules and to external fields in solids are planned. 5) We have obtained novel results about the structure of systems with reduced particle number fluctuations. These have applications to cosmology and to queuing theory, which will be explored further. 6) An important question in developing approximation schemes for fluids is the possibility of constructing particle distributions having specified densities and pair correlations. This can be achieved in some cases by the explicit construction of point processes via determinants and renewals. General existence criteria are being investigated. 7) We have applied statistical mechanical methods to the mathematical study of epidemics taking into account correlations as well as saturation effects on networks. New macroscopic equations for the description of the evolution and prevalence of an endemic infected state improve agreement with more detailed microscopic models. Extension of these techniques to models of population dynamics and ecology is planned.Broader Impact: The proposed research is 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 biological and social systems. Our program also includes the organization of two conferences every year in which both core subjects and newdevelopments 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. The conferences also serve as a clearing house for positions and often lead to new collaborations.