Particle Packing Problems

该奖项将支持颗粒堆积问题的基本方面的工作。有许多悬而未决的问题将被追问。在d维欧几里德空间中，密度和无序度不同的球填充能被识别吗？维度大于3的球体最密集的堆积是什么？在二维和三维空间中，非球形物体最密集的堆积是什么？随机填充能比有序填充更密集地填充空间吗(意味着无序或“玻璃”基态)？对于非平铺的非球形粒子，能否推导出严格小于1的最大密度上限？具体地说，将探索以下七个一般领域：(1)密度异常低的拥塞球体堆积；(2)玻璃化转变方法中不断增长的长度尺度的识别；(3)具有大小分布的三维空间中的致密球体堆积；(4)无序多面体堆积；(5)非球形粒子的最大致密堆积；(6)高维最大致密球体堆积；以及(7)单位球在不同空间维度上的拥堵研究。在其他活动中，该奖项将用于支持攻读博士学位的研究生的研究。包装问题，如稠密的固体如何填充空间，自文明诞生以来就一直吸引着人们，并继续吸引着科学家，因为它们与物理科学、数学、工程和生物中出现的一系列问题有关。虽然最优堆积问题与凝聚态物质的固态密切相关，但无序球状堆积已被用来模拟物质和颗粒介质的玻璃态。低维非球形颗粒的密集堆积与材料科学和生物学中的问题有关。高维球形填充在通信理论中具有重要意义。纯粹的数学家对包装问题有着长期的兴趣。这是一个多学科的项目，将应用数学、统计和凝聚态物理、材料科学、工程学、生物学、通信和纯数学社区联系在一起。拟议的研究可能产生的重要科学进展和实际成果包括更深入地了解物质的低温状态(例如，晶体基态、玻璃化转变和无序基态的神秘发生)的性质、颗粒介质、机械稳定的低质量网络固体、确定新的合金晶体结构和新的高压物质相，以及关于生物细胞或细胞器堆积方式的见解。

This award will support work on fundamental aspects of particle packing problems. There are many open questions that will be pursued. Can sphere packings with a diversity of density and disorder be identified in d-dimensional Euclidean space? What are the densest packings of spheres in dimension greater than three? What are the densest packings of nonspherical objects in two and three dimensions? Can random packings ever fill space more densely than ordered packings (implying disordered or ``glassy" ground states)? For non-tiling nonspherical particles, can an upper bound on the maximal density be derived that is always strictly less than unity? Specifically, the following seven general areas will be explored: (1) jammed sphere packings with anomalously low densities; (2) identification of growing length scales in the approach to the glass transition; (3) dense spheres packings in three-dimensional space with a size distribution; (4) jammed disordered polyhedron packings; (5) maximally dense packings of nonspherical particles; (6) maximally dense sphere packings in high dimensions; and (7) studies of jamming on the unit sphere in various space dimensions. Among other activities, this award will be used to support the research of graduate students seeking their Ph.D. degrees.Packing problems, such as how densely solid objects fill space, have fascinated people since the dawn of civilization, and continue to intrigue scientists because of their connection to a host of problems that arise in the physical sciences, mathematics, engineering and biology. While optimal packing problems are intimately related to solid states of condensed matter, disordered sphere packings have been employed to model the glassy state of matter and granular media. Dense packings of nonspherical particles in low dimensions are relevant to problems in materials science and biology. Sphere packings in high dimensions is of importance in communications theory. Pure mathematicians have a longstanding interest in packing problems. This is a multidisciplinary project that links the applied mathematics, statistical and condensed-matter physics, materials science, engineering, biology, communications, and pure mathematics communities. Important scientific advances and practical outcomes that could potentially emerge from the proposed research include a deeper understanding of the nature of low-temperature states of matter (e.g., crystal ground states, the glass transition and the mysterious occurrence of disordered ground states), granular media, mechanically stable low-weight network solids, identification of new alloy crystal structures and new high-pressure phases of matter, and insights concerning the manner in which biological cells or organelles pack.