The Statistical Physics of Random Solids

该奖项支持对无序固体性质的理论研究，这是一个从橡胶到生物学的广泛适用性主题，也是一个深刻而基本的知识兴趣。 在本项目中，分析方法将应用于这些材料，以更好地了解其通用性能。 学生将接受理论物理学方法的训练。本项目涉及表现出随机固体状态的各种介质的物理特性。 这些介质的原型是硫化橡胶，其成分形成热波动的宏观随机网络的材料。像这样的系统开始时是流体，但经过充分的硫化（即随机选择的成分之间的永久化学键合），它们经历了一个相变到一个新的物质状态：随机固体状态。这种状态具有非零的弹性剪切模量，并且至少一些其组成粒子在空间中局部化。然而，硫化过程的随机性确保了至少在足够长的长度尺度上不显示结晶性。许多形式的物质显示出随机的固态。仅举几个例子，有由随机结合的小分子、生物学上重要的结构（例如肌动蛋白丝）和通过耗尽力吸引的胶体颗粒制成的化学凝胶。在聚合物环境中，组分可以是柔性的或刚性的、直链的或支链的、中性的或带电荷的、各向同性的或液晶的，并且它们可以在任何地方连接或仅通过链端连接。人们还可以交联聚合物的共混物，在这种情况下，交联甚至可以限于类似的成分，在这种情况下，出现互穿的无规固体网络。鉴于它必须面对的多层次的随机性（例如，热运动，淬火约束，涌现的结构和响应），随机固体介质的统计物理学是理论物理学中最具智力挑战性的前沿领域之一。要取得进展，仍然需要引进、发展和应用创造性地制定的概念和战略，以描述这些介质的特征，以及强大的分析技术，以提取静态和动态的关键物理信息，特别是关于其结构和弹性反应的异质性。随机固体不仅因其在自然界和技术中的丰富性而引起广泛的兴趣，而且它们也是探索物质基本无序状态的迷人实验室。该项目建立在UIUC和其他地方对随机固体形成系统理论的先前工作基础上。其目标之一是通过扩展到具有各种有序形式（如液晶或相分离）趋势的新系统，扩大所涵盖的介质和现象的范围。 重点是一般属性，即，合理地预期举行不仅为特定品牌的随机固体形成的问题，但广泛的类它。关于静态性能，中心问题包括：什么普遍的功能可以理解？人们能否设计出复制方法的有用的替代方法（例如腔，或反对易场或坐标）？ 可以调用特定于二维的方法吗？量子方面可以探索吗？如何解决随机凝固和其他有序化（如液相结晶或相分离）之间的相互作用？ 重点预计将从静态下滑到动态，无论是附近的随机凝固过渡和随机固体状态。这里的核心问题是：建立动力学理论的最佳策略和模型是什么？关于随机凝固转变附近的动态临界现象，或者关于随机固态中的流体动力学，我们能说些什么呢？没有永久随机约束的媒体也将得到解决。因此，将考虑的系统中，协会是暂时的，而不是永久的，形成和瓦解的平衡。 这就提出了一些重要的问题，例如：这些系统经历的物理胶凝在多大程度上类似于玻璃的形成，以及这些系统可以告诉我们关于玻璃的什么？为硫化物质开发的概念和技术是否可用于结构玻璃和物理胶凝系统？ 如果是，如何做到？人们能在多大程度上弥合硫化物质（具有淬火无序性）和传统玻璃态系统（显然没有）之间的差距？统计力学将是核心工具。静力学问题将通过复制场理论，腔方法和反交换变量方法来解决。动力学的问题将通过MSR形式主义和腔的方法，都阐述，以科普淬火随机constrains.Broader影响的影响来解决：随机系统的统计物理一直是其范围内的杰出。它在神经网络和信息处理，计算机科学和算法复杂性，生物学方面（例如杂聚物折叠，树状结构分析）以及主流凝聚态物质的许多方面（例如无序电子，磁性和超导介质）等领域产生了开创性的方法和结果。这一点，再加上它直接应用于不断增长的材料家族，以及它在玻璃体系等重要领域的潜力，使得随机固体统计物理研究的影响很可能远远超出其预期的领域。

This award supports theoretical research on the properties of disordered solids, a topic of wide applicability from rubbers to biology and one of deep and fundamental intellectual interest. In this project analytical methods will be applied to these materials to better understand their universal properties. Students will be trained in the methods of theoretical physics.This project addresses the physical properties of a wide range of media that exhibit random solid states. Prototypical of these media is vulcanized rubber, a material whose constituents form a thermally fluctuating, macroscopic, random network. Systems such as this start out as fluids, but upon sufficient vulcanization (i.e. permanent chemical bonding between randomly selected constituents) they undergo a phase transition to a new state of matter: the random solid state. This state has a nonzero elastic shear modulus, and at least some of its constituent particles are localized in space. However, the randomness of the vulcanization process ensures that no crystallinity is exhibited, at least on sufficiently long length-scales.Many forms of matter exhibit the random solid states. To give just a few examples, there are chemical gels made from randomly bonded small molecules, biologically vital structures (e.g. actin filaments), and colloidal particles attracting via depletion forces. In polymeric settings, the constituents may be flexible or rigid, linear or branched, neutral or charged, isotropic or liquid-crystalline, and they may be linked anywhere or solely via chain ends. One can also cross-link blends of polymers, in which case cross-linking can even be restricted to like constituents, in which case inter-penetrating random solid networks emerge.Intellectual Merit: In view of the multiple levels of randomness that it must confront (e.g. thermal motion, quenched constraints, and emergent structure and response), the statistical physics of random solid media is among the most intellectually challenging frontiers in theoretical physics. Making progress continues to require the introduction, development and application of creatively crafted concepts and strategies for characterizing these media, as well as powerful analytical techniques for extracting pivotal physical information both static and dynamic especially about the heterogeneity of their structure and elastic response. Not only are random solids of broad interest due to their abundance in nature and technology, but also they serve as fascinating laboratories for exploring fundamentally disordered states of matter.The project builds upon prior work done at UIUC and elsewhere on the theory of random-solid-forming systems. Among its aims is the widening of the range of media and phenomena encompassed, via extensions to novel systems with tendencies towards various forms of ordering, such as liquid-crystalline or phase-separational. The focus is on generic properties, i.e., ones reasonably expected to hold not just for specific brands of random-solid-forming matter but for wide classes it.Regarding static properties, central questions include: What universal features can be understood? Can one devise useful alternatives to the replica method (e.g. cavities, or anti-commuting fields or coordinates)? Can methods specific to two-dimensions be invoked? Can quantal aspects be explored? How is the interplay between random solidification and other ordering (e.g. liquid-crystallization or phase separation) resolved? Emphasis is anticipated to glide from statics to dynamics, both near the random solidification transition and in the random solid state. Here, the central questions are: What are the optimal strategies and models for building a dynamical theory? What can one say about dynamic critical phenomena near the random solidification transition, or about hydrodynamics in the random solid state?Media without permanent random constraints will be addressed, too. Thus, consideration will be given to systems in which associations are temporary rather than permanent, forming and disintegrating in equilibrium. This raises important questions, such as: To what extent does the physical gelling that such systems undergo resemble glass formation, and what can such systems teach us about glasses? Can concepts and techniques developed for vulcanized matter be useful for structural glasses and physically gelling systems? If so, how? To what extent can one bridge the gap between vulcanized matter (with its quenched disorder) and conventional glassy systems (which apparently have none)?Statistical mechanics will be the central tool. Statics issues will be tackled via replica field theory, the cavity method and anti-commuting variables methods. Issues of dynamics will be addressed via the MSR formalism and the cavity method, both elaborated to cope with the impact of quenched random constraints.Broader Impact: The statistical physics of random systems has been outstanding for its reach. It has engendered seminal approaches and results in fields as diverse as neural networks and information processing, computer science and algorithmic complexity, aspects of biology (e.g. heteropolymer folding, analysis of treelike structures), and many aspects of mainstream condensed matter (such as disordered electronic, magnetic and superconducting media). This, coupled with the growing family of materials to which it directly applies, and its potential for shedding light on important areas such as glassy systems, makes it highly likely that the impact of research on the statistical physics of random solids will reach far beyond its intended domain.