Can Disordered Sphere Packings Ever Be Maximally Dense?

Salvatore Torquato

Princeton University
ABSTRACT:
Packing problems, such as how densely nonoverlapping particles fill
d-dimensional Euclidean space R^d, are ancient and still provide
fascinating challenges for scientists and mathematicians [1,2]. Bernal
has remarked that heaps (particle packings) were the first things that
were ever measured in the form of basketfuls of grain for the purpose
of trading or of collection of taxes. While maximally dense packings
are intimately related to classicalground states of matter, disordered
sphere packings have been employed to model glassy states of matter.
There has been a resurgence of interest in maximally dense sphere
packings in high-dimensional Euclidean spaces. Interestingly, the
optimal ways of sending digital signals over noisy channels correspond
to the densest sphere packings in high-dimensional spaces. Remarkably,
no one hasbeen able to provide exponential improvement on a 100-year-
old lower bound on the maximal packing density due to Minkowski in
d-dimensional Euclidean space Rd. The asymptotic behavior of this bound
in any dimension is controlled by 2^(-d). Using an optimization
procedure and a conjecture concerning the existence of disordered
sphere packings, we obtain a conjectural lower bound onthe density
whose asymptotic behavior is controlled by 2^(-0.7786...)d, thus
providing the putative exponential improvement of Minkowski's bound
[3]. A more recent investigation reveals that this exponential
improvement is robust over a wide class of optimized functions [5]. Our
work suggests that disordered (rather than ordered) sphere packings may
be the densest for sufficiently large d, implying not only the
existence of disordered ground states for some continuous potentials
but their preponderance among all other possibilities.
1.
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups
(Springer-Verlag, New York, 1993). 2. S. Torquato, Random Heterogeneous
Materials: Microstructure and Macroscopic Proper- ties,
(Springer-Verlag, New York, 2002). 3. H. Minkowski,
Diskontinuitatsbereich fur arithmetische Aquivalenz, J. reine angew.
Math. 129, 220 (1905). 4. S. Torquato and F. H. Stillinger, New
Conjectural Lower Bounds on the Optimal Density of Sphere Packings,
Experimental Math. 15, 307 (2006). 5. A. Scardicchio, F. H. Stillinger
and S. Torquato, Estimates of the Optimal Density of Sphere Packings in
High Dimensions, J. Math. Phys., in press.