Collaborative Research: Mathematical Studies of Short-Ranged Spin Glasses

NSF奖摘要- DMS-0102587数学科学：合作研究：短程自旋玻璃的数学研究摘要DMS-0102587纽曼在此资助下的工作涉及称为自旋玻璃的无序磁体理论的数学基础。 虽然存在理想化（和不现实的）自旋玻璃模型的解决方案，主要研究人员的兴趣集中在回答基本的统计力学问题，承担的行为真实的实验室自旋玻璃。 这些问题，其中许多仍然存在争议，尽管二十年的深入研究，包括在平衡自旋玻璃相的有序性，理解异常行为，如缓慢松弛和老化引起的非平衡动力学，并证明存在或不存在的相变在有限的尺寸。 所使用的方法和引入的概念应该不仅与自旋玻璃有关，而且与其他无序系统有关，其中许多系统仍然知之甚少。 此外，主要研究人员对动力学的研究方法的普遍性应该会在均匀和无序系统中的非平衡动力学的某些方面取得进展。我们对固态和液态有序系统的深刻物理和数学理解-例如，晶体，铁磁体，超导体，液晶，和许多其他的-都具有基本的科学重要性，并在上个世纪的后半叶激发了深刻的技术变革。 然而，存在着许多系统，无论是熟悉的还是不熟悉的，其中随机性或无序性起着关键作用，而我们的数学和物理理解仍然相对原始。 一个熟悉的例子是普通的窗户玻璃，其中的原子或分子被“粘”在随机的位置（与在冰中发现的规则的晶体阵列相反）。 自旋玻璃是无序的磁性系统，被认为是这种宏观“冻结”无序的原型。一般来说，无序系统既提出了基本的科学挑战，同时又有很大的应用前景。 后者不仅包括新材料和新设备的可能性，还包括新算法的创建和在生物科学和其他科学中的应用。 因此，非常需要在理解这些系统方面取得进展。 自旋玻璃可能比这类材料更适合数学分析。 然而，即使在这方面也没有取得什么根本性的进展。 主要研究人员的工作旨在解决有关这些材料的基本数学和物理问题，并为各种无序系统提供一般的理论方法。

NSF Award Abstract - DMS-0102587 Mathematical Sciences: Collaborative Research: Mathematical Studies of Short-Ranged Spin Glasses Abstract DMS-0102587 Newman The work under this grant addresses the mathematical foundations of the theory of disordered magnets known as spin glasses. Although there exist proposed solutions of idealized (and unrealistic) spin glass models, the principal investigators' interests center on answering fundamental statistical mechanical questions that bear on the behavior of real laboratory spin glasses. These questions, many of which remain controversial despite two decades of intensive study, include the nature of ordering in the equilibrium spin glass phase, understanding anomalous behavior, such as slow relaxation and aging arising from nonequilibrium dynamics, and proving the presence or absence of a phase transition in finite dimensions. The methods used and concepts introduced should be relevant not only for spin glasses but also for other disordered systems, many of which remain poorly understood. Moreover, the generality of the principal investigators' approach to dynamics should yield progress in certain aspects of nonequilibrium dynamics in both homogeneous and disordered systems.Our deep physical and mathematical understanding of ordered systems in the solid and liquid state --- for example, crystals, ferromagnets, superconductors, liquid crystals, and many others --- has been both of fundamental scientific importance and has spurred profound technological change throughout the second half of the last century. However, there exist many systems, both familiar and unfamiliar, in which randomness or disorder plays a key role, and in which our mathematical and physical understanding remains comparatively primitive. One familiar example is ordinary window glass, where the atoms or molecules are "stuck" in random locations (as opposed to a regular crystalline array as would be found, for example, in ice). Spin glasses are disordered magnetic systems which are thought to be prototypes for this kind of macroscopic "frozen-in" disorder. Disordered systems in general present both fundamental scientific challenges and at the same time hold great promise for applications. The latter includes not only the possibility of new materials and devices but also the creation of new algorithms and applications to the biological and other sciences. Progress in understanding these systems is therefore greatly desirable. Spin glasses may be more amenable to mathematical analysis than other materials in this class. Nevertheless, little fundamental progress has been made even here. The principal investigators' work is aimed at resolving basic mathematical and physical issues concerning these materials and at providing a general theoretical approach for a wide variety of disordered systems.