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Order through chaos
Adam Hopkins (GS), Frank Stillinger (fac) and Salvatore Torquato (fac)
Dept. of Chemistry, Dept. of Physics
Generally, entropy is associated with chaos and disorder; however, paradoxically, maximizing entropy can lead to highly symmetric, strongly ordered structures. The Voronoi cells depicted here correspond to a subsection of a maximally dense binary packing of nonoverlapping spheres, e.g., billiard balls of two different sizes. Each polyhedral cell represents a region within which all points are closest to a given sphere center; lavender cells correspond to small spheres and green to large spheres.

Because a sphere packing is maximally dense, it is the configuration of points (sphere centers) in which each individual point has the most radial free space to move within its Voronoi cell. The most free space per particle is synonymous with maximal structural entropy, and thus for dense systems exhibiting strong radial repulsion, the maximally dense sphere packing is the configuration of points that exhibits maximal entropy. We have found a wealth of previously unknown maximally dense binary sphere packings, nearly all of which exhibit high symmetry. These are all cases where, paradoxically, high symmetry and order result from maximizing chaos.