Particle Packing Problems

扭矩0804431 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的问题：（1）二维和三维空间中各种形状的非球形粒子的稠密堆积的识别和表征;（2）高欧几里德维数下最大密度的球形堆积的研究;（3）不同维数下低密度堵塞球形堆积的识别;（4）任意维数下单位球上堵塞的研究;（5）寻求改进的序度量来表征packing中的随机程度。 拟议的研究可能会产生重要的科学进步和成果。 这个多学科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。 将这些不同的观点和目标统一起来是一项具有挑战性的任务，但这对科学界来说是一个巨大的回报。 包装问题，例如固体物体如何填充空间，自文明之初就吸引了人们，并继续吸引科学家，因为它们与物理科学，数学，工程和生物学中出现的许多问题有关。 虽然最佳packingproblems是密切相关的凝聚物质的固体状态，无序的球体包装已被用来模拟玻璃态的物质。 高维球填充与通信理论有关。 病毒将DNA包装在蛋白质容器中的方式是一个包装问题。 大于3维的球体的密排是什么？在二维和三维中，非球形物体的密度填充是什么？ 无规堆积能比有序堆积（意味着无序或“玻璃态“基态）更致密地填充空间吗？ 包装的“随机性”是否可以用有意义和精确的方式来量化？ 更好地理解包装问题对新材料的合成、我们有效设计远距离发送数字信号的通信渠道的能力以及新药的设计都有意义，这里仅举几个例子。

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.