Random sequential addition of hard spheres in high Euclidean dimensions

Abstract

Sphere packings in high dimensions have been the subject of recent theoretical interest. Employing numerical and theoretical methods, we investigate the structural characteristics of random sequential addition (RSA) of congruent spheres in d -dimensional Euclidean space R{d} in the infinite-time or saturation limit for the first six space dimensions (1< or =d < or =6) . Specifically, we determine the saturation density, pair correlation function, cumulative coordination number and the structure factor in each of these dimensions. We find that for 2< or =d <or =6 , the saturation density phi{s} scales with dimension as phi{s}=c{1}/2{d}+c{2}d/2{d} , where c{1}=0.202048 and c{2}=0.973872 . We also show analytically that the same density scaling is expected to persist in the high-dimensional limit, albeit with different coefficients. A byproduct of this high-dimensional analysis is a relatively sharp lower bound on the saturation density for any d given by phi{s}> or =(d+2)(1-S{0})2;{d+1} , where S{0}[0,1] is the structure factor at k=0 (i.e., infinite-wavelength number variance) in the high-dimensional limit. We demonstrate that a Palàsti-type conjecture (the saturation density in R{d} is equal to that of the one-dimensional problem raised to the d th power) cannot be true for RSA hyperspheres. We show that the structure factor S(k) must be analytic at k=0 and that RSA packings for 1< or =d< or =6 are nearly "hyperuniform." Consistent with the recent "decorrelation principle," we find that pair correlations markedly diminish as the space dimension increases up to six. We also obtain kissing (contact) number statistics for saturated RSA configurations on the surface of a d -dimensional sphere for dimensions 2< or =d< or =5 and compare to the maximal kissing numbers in these dimensions. We determine the structure factor exactly for the related "ghost" RSA packing in R{d} and demonstrate that its distance from "hyperuniformity" increases as the space dimension increases, approaching a constant asymptotic value of 12 . Our work has implications for the possible existence of disordered classical ground states for some continuous potentials in sufficiently high dimensions.