Particle Packing Problems

研究者和他的同事们研究了各种各样的包装问题。本文主要研究了以下五个方面：(1)二维和三维宽类非球形粒子密集堆积的识别和表征；(2)高欧氏维数下最大密度球填料的研究；(3)各种尺寸的低密度堵塞球体填料的识别；(4)任意维单位球上的干扰研究；(5)追求改进的顺序度量来表征包装中的随机性程度。重要的科学进步和成果可能会从拟议的研究中出现。这个非常多学科的项目加入了应用数学，统计和凝聚态物理，材料科学，工程和纯数学社区。统一这些不同的观点和目标是一项具有挑战性的任务，但总的来说，这给科学界带来了巨大的回报。包装问题，如固体物体填充空间的密度，自文明之初就吸引着人们，并继续吸引着科学家，因为它们与物理科学、数学、工程和生物学中出现的一系列问题有关。虽然最佳填充问题与凝聚态物质的固体状态密切相关，但无序球体填充已被用于模拟物质的玻璃态。高维球体封装在通信理论中具有重要意义。病毒在蛋白质容器中包装DNA的方式是一个包装问题。尺寸大于3的球体的密度是多少？在二维和三维空间中，密度最大的非球形物体是什么？随机填料能比有序填料更密集地填充空间吗（这意味着无序或“玻璃”基态）？包装的“随机性”能否以有意义和精确的方式量化？对包装问题的更深入的理解对新材料的合成、我们有效设计通信通道以远距离发送数字信号的能力以及新药的设计都有影响，这只是举几个例子。

Torquato0804431 The investigator and his colleagues study a variety ofpacking problems. The following five general areas are explored:(1) identification and characterization of dense packings ofnonspherical particles for a wide class shapes in two and threespace dimensions; (2) study of sphere packings of maximal densityin high Euclidean dimensions; (3) identification of low-densityjammed sphere packings in various dimensions; (4) studies ofjamming on the unit sphere in arbitrary dimensions; and (5)pursuit of improved order metrics to characterize the degree ofrandomness in a packing. Important scientific advances andoutcomes are likely to emerge from the proposed research. Thisvery multidisciplinary project joins the applied mathematics,statistical and condensed-matter physics, materials science,engineering, and pure mathematics communities. Unifying thesedifferent perspectives and goals is a challenging task, but onethat offers great rewards to the scientific community in general. Packing problems, such as how densely solid objects fillspace, have fascinated people since the dawn of civilization, andcontinue to intrigue scientists because of their connection to ahost of problems that arise in the physical sciences,mathematics, engineering, and biology. While optimal packingproblems are intimately related to solid states of condensedmatter, disordered sphere packings have been employed to modelthe glassy state of matter. Sphere packings in high dimensionshave relevance in communications theory. The way that virusespackage DNA in protein containers is a packing problem. What arethe densest packings of spheres in dimension greater than three? What are the densest packings of nonspherical objects in two andthree dimensions? Can random packings ever fill space moredensely than ordered packings (implying disordered or "glassy"ground states)? Can "randomness" of a packing be quantified in ameaningful and precise manner? A greater understanding ofpacking problems has implications for the synthesis of novelmaterials, our ability to efficiently design communicationschannels to send digital signals over large distances, and thedesign of new drugs, just to mention a few examples.