Event Information
Optimal Particle Packings: Problems for the Ages
- Abstract:
- Dense particle packings have fascinated people since the dawn of civilization and the fascination persists. Resurgent interest comes from the recent proof [1] of the Kepler conjecture that the face-centered-cubic lattice provides the densest packing of congruent spheres in three-dimensional Euclidean space and from the problem of "random" sphere packings, which holds the clue to the structure of liquids and formation of glasses [2]. Sphere packings in high dimensions have relevance in communications theory [3].
I discuss why the venerable 50-year old notion of "random close packing" (RCP) of spheres (the putative "random" analog of Kepler's conjecture) is mathematically ill-defined. To replace this traditional notion, we introduce a new concept called the maximally random jammed (MRJ) state, which can be made precise [4]. This state can viewed as a "perfect" glass, and rests on devising precise meanings for "jamming" and "randomness," which I describe. We have defined three hierarchical categories of jamming [5], which relates to the mechanical stability of the packing, and a linear program that enables us to test a particle packing for any of the jamming categories. We show that the density of MRJ packings of ellipsoids in three dimensions closely approaches that of the densest lattice packing [6]. This has interesting implications for the existence of a thermodynamically stable glass. Moreover, we have discovered a periodic ellipsoid packing with the highest known density and one in which the ellipsoids are not highly eccentric in shape [7]. Finally, we provide strong evidence that the densest sphere packings in high Euclidean dimensions might be disordered, implying the existence of disordered (rather than periodic) ground states of matter. This leads to a lower bound on the maximal density of sphere packings in high dimensions that putatively provides the first exponential improvement on the 100-year-old lower bound due to Minkowski [8].
1. T. Hales, "The Kepler conjecture," arXiv:math.MG/9811078 (1998).
2. S. Torquato, Random Heterogeneous Materials: Microstructure and Macroscopic Properties, (Springer-Verlag, New York, 2002).
3. J. H. Conway and N. Sloane, Sphere Packings, Lattices and Groups, (Springer-Verlag, New York, 1998).
4. S. Torquato, T. M. Truskett and P. G. Debenedetti, "Is Random Close Packing of Spheres Well Defined?", Phys. Rev. Lett., 84, 2064 (2000).
5. S. Torquato and F. H. Stillinger, "Multiplicity of Generation, Selection, and Classification Procedures for Jammed Hard-Particle Packings," J. Phys. Chem., 105, 11849 (2001).
6. A. Donev, I. Cisse, D. Sachs, E. A. Variano, F. H. Stillinger, R. Connelly, S. Torquato, and P. M. Chaikin, "Improving the Density of Jammed Disordered Packings using Ellipsoids," Science, 33, 690 (2004).
7. A. Donev, F. H. Stillinger, P. M. Chaikin and S. Torquato, "Unusually Dense Crystal Ellipsoid Packings," Phys. Rev. Lett., 92, 255506 (2004).
8. S. Torquato and F. H. Stillinger, "New Conjectural Lower Bounds on the Optimal Density of Sphere Packings," Experimental Math., in press; arXiv:math.MG/0508381 (2005).
Host: Weitao Yang
Theory Seminar