Random-sequential-packing simulations in three dimensions for spheres

The goal of one three-dimensional random-parking (occupation) -limit problem is to determine the mean fraction of space that would be occupied by fixed equal-size spheres, created at random locations sequentially until no more can be added. No analytical solution has yet been found for this problem. Our earlier simulations, done for ratios of cubic-region side length (L) to sphere radius (R) up to L/R=20, predicted a parking-limit volume fraction (as volume becomes very large) of 0.37–0.40, using regression equations that indicated the approach to this limit as a function of the ratio of the volume of spheres tried to the volume of the region and the ratio L/R. Our results for random parking in a volume with penetrable walls can be adjusted with a multiplicative correction factor to give the results for the same volume with impenetrable walls. An improved algorithm, almost an order of magnitude faster than our earlier one, was used to extend our simulations to L/R=40 and confirm the original predictions, for a series of six runs totaling 9×${10}^6$ attempts at sphere placement. The results supported a narrower estimate of the parking-limit volume fraction: 0.385±0.010.