Studies in Statistical Mechanics

统计力学研究

• 批准号: 0442066
• 负责人: Joel Lebowitz
• 金额: $ 48万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-15 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0442066&HistoricalAwards=false
• 关键词: Studies; Statistical; Mechanics

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.

该资助由材料研究、物理和数学科学部门共同支持。 这项研究的目的是更好地理解源自其微观成分集体行为的宏观系统的特性。重点是非平衡系统，所使用的方法范围从严格的数学分析到计算机模拟。我们对弥合严格结果和应用之间的差距特别感兴趣。智力优点包括： 1) 将玻尔兹曼熵和 H 定理扩展到不处于局部热平衡的系统。建议将最近关于致密流体的工作推广到更复杂的系统，包括纳米和生物系统，其中熵的考虑发挥着重要作用。这将需要将玻尔兹曼熵适当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的稳态非平衡状态的波动，其形式是意想不到的。它们最显着的特征是宏观尺度上的非定域性，可以追溯到此类系统中存在的长程相关性，可以通过中子散射进行实验测量。计划将这些结果扩展到更现实的系统。3) 对于开放量子系统，例如载流纳米线，通过随机相互作用对储层进行经典建模是有问题的。通常的方法是使用自由场或理想气体作为储层。在许多情况下，这并不令人满意，我们计划继续研究替代方法，例如使用以随机矩阵为代表的强耦合系统作为储存库。4）我们开发的用于处理量子系统对强时间依赖外部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。计划应用于分子量子跃迁的最佳控制和固体中的外部场。 5）我们获得了关于减少粒子数波动的系统结构的新颖结果。这些在宇宙学和排队论中都有应用，将进一步探讨。 6) 开发流体近似方案的一个重要问题是构建具有指定密度和配对相关性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建点过程来实现。一般存在标准正在调查中。 7）我们将统计力学方法应用于流行病的数学研究，考虑到相关性以及对网络的饱和效应。用于描述地方性感染状态的演变和流行的新宏观方程提高了与更详细的微观模型的一致性。计划将这些技术扩展到人口动态和生态学模型。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，在学术氛围中讨论统计力学的核心主题和新发展。我们鼓励研究生、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互动。这些会议还充当职位信息交换所，并经常促成新的合作。