Particle Packing Problems

该奖项将支持粒子堆积问题的基本方面的工作。有许多悬而未决的问题将继续探讨。在d维欧氏空间中，具有密度多样性和无序性的球填充能被识别吗？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）更密集地填充空间吗？对于非平铺的非球形粒子，是否可以推导出最大密度的上界总是严格小于1？具体地说，将探讨以下七个一般领域：（1）具有异常低密度的堵塞球填料;（2）在接近玻璃化转变过程中的增长长度尺度的识别;（3）具有尺寸分布的三维空间中的致密球填料;（4）堵塞无序多面体填料;（5）非球形颗粒的最大致密填料;（6）高维空间中的最大稠密球填充;（7）不同空间维数下单位球上的干扰研究。除其他活动外，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，如固体物体如何密集地填充空间，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一系列问题有关。虽然最佳填充问题与凝聚态物质的固态密切相关，但无序球填充已被用于模拟物质和颗粒介质的玻璃态。低维非球形粒子的密集堆积与材料科学和生物学中的问题有关。高维球填充在通信理论中具有重要的意义。纯数学家对填充问题有着长期的兴趣。这是一个多学科的项目，链接应用数学，统计和凝聚态物理，材料科学，工程，生物学，通信和纯数学社区。拟议研究可能产生的重要科学进步和实际成果包括更深入地了解物质低温状态的性质（例如，晶体基态、玻璃化转变和无序基态的神秘发生）、颗粒介质、机械稳定的低重量网络固体、新合金晶体结构和物质新高压相的鉴定，以及关于生物细胞或细胞器包装方式的见解。

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.