Studies in Statistical Mechanics

这笔资金由材料研究、物理和数学科学部联合资助。这项研究的目的是更好地理解宏观系统的性质，这些系统起源于其微观组成部分的集体行为。重点放在非平衡系统上，使用的方法从严格的数学分析到计算机模拟。我们对弥合严格的结果和应用之间的差距特别感兴趣。它的智慧价值包括：1)将玻尔兹曼的熵和H定理推广到非局部热平衡的系统。有人建议将最近关于稠密流体的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，熵因素发挥着重要作用。这将需要适当地将玻尔兹曼熵推广到非平衡量子系统。2)最近获得的精确大偏差函数，描述了模型系统静态非平衡态的涨落，具有意想不到的形式。它们最显著的特征，宏观尺度上的非局域性，可以追溯到这种系统中存在的长程关联，这种关联可以通过中子散射实验来测量。3)对于开放的量子系统，如载流纳米线，经典的随机相互作用的水库模型是有问题的。通常的方法是使用自由场或理想气体作为储气库。这在许多情况下并不令人满意，我们计划继续研究替代方法，如使用由随机矩阵表示的强耦合系统作为水库。4)我们开发的处理量子系统对依赖于时间的强外场的响应的技术已经产生了关于简单模型系统的严格结果。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)我们得到了关于粒子数涨落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。6)发展流体近似格式的一个重要问题是构造具有特定密度和对关联的颗粒分布的可能性。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。一般的存在标准正在调查中。7)我们将统计力学方法应用到流行病的数学研究中，考虑了网络的相关性和饱和效应。描述地方性感染状态演变和流行的新的宏观方程改进了与更详细的微观模型的一致性。计划将这些技术扩展到种群动态和生态模型。更广泛的影响：拟议的研究是高度跨学科的，将物理学家、数学家、化学家以及生物和社会科学理论领域的工作人员聚集在一起。预期的应用领域包括材料科学、复杂流体以及生物和社会系统。我们的计划还包括每年组织两次会议，在大学气氛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，并与该领域的领导人互动。这些会议还充当了立场的交流中心，并经常导致新的合作。

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.