Schematic of the integration setup for computing .
Density evolution of the average bond spindle and number of nearest neighbors with density for fluid (squares, dashed line) and crystal phases (circle, solid line). The fluid behavior continuously evolves from the Poisson limit and reasonably extends to the ideal packing limit (rightmost point). The crystal phase, however, clearly sits on a different branch and even E 8 does not converge to the ideal limit. (Inset) Results for the 7:5 (long-dashed line) and 6:5 (short-dashed line) mixtures in d = 3, along with the crystal results for a monodisperse spheres (solid line).
Density evolution of the bond spindle distribution for the Poisson limit (thick line), the highest equilibrated fluid density attained (solid line – see Fig. 4 ), and a couple of intermediate densities. The narrower distribution of the 6:5 mixture compared to that of the 7:5 mixture at similarly sluggish conditions indicates that the former is closer to the ideal packing order than the latter. With increasing dimension, however, the distribution converges to that of the Poisson limit.
The diffusivity decays with increasing reduced pressure βP/ρ in a similar fashion for all dimensions. The behavior of d = 3 6:5 (short-dashed line) and 7:5 mixtures (long-dashed line) are also similar, although the former initiates its slowdown at lower pressures than the latter. (Inset) The equation of state for the different fluids. On this scale the results for the two d = 3 mixtures are indistinguishable.
Growth of the overlap with increasing pinning concentration in d = 3 for two different hard-sphere mixtures. As the system density increases, the crossover from low overlap to high overlap takes place at an ever smaller concentration of pinned particles. The average spacing between defects at a 0.4 overlap defines ξp. (Inset) Time evolution of the overlap and of the self component . The long time value of the overlap is attained when the self-part has completely decayed.
The static (point-to-set) length ξp for d=3 6:5 (short-dashed line) and 7:5 (long-dashed line), extracted from the data in Fig. 5 , both saturate the bound from Eq. (15) (thinner lines). In higher dimensions, the bound becomes continuously weaker. The ξdyn results for the 7:5 mixture (dotted-dashed line) are taken from Ref. 11 .
Illustration for the proof of Lemma 12.
Summary of the regular simplicial tilings in d ⩾ 2. The curvature reported in units of σ.
Results for . Formulae A and B are formally equivalent, but the former is much more numerically unstable than the latter as dimension increases.
Ideal packing fraction φ d in various dimensions.
Limit coordination numbers.
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