Erratum: “Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes. II. Simulation results and analyses” [J. Chem. Phys. 137, 074106 (2012)]

Continuum percolation expressed in terms of density distributions

We present an approach to derive the connectivity properties of pairwise interacting n-body systems in thermal equilibrium. We formulate an integral equation that relates the pair connectedness to the distribution of nearest neighbors. For one-dimensional systems with nearest-neighbor interactions, the nearest-neighbor distribution is in turn related to the pair-correlation function g through a simple integral equation. As a consequence, for those systems, we arrive at an integral equation relating g to the pair connectedness, which is readily solved even analytically if g is specified analytically. We demonstrate the procedure for a variety of pair potentials including fully penetrable spheres as well as impenetrable spheres, the only two systems for which analytical results for the pair connectedness exist. However, the approach is not limited to nearest-neighbor interactions in one dimension. Hence, we also outline the treatment of external fields and long-range interactions and we illustrate how the formalism can applied to higher-dimensional systems using the three-dimensional ideal gas as an example.

Continuum percolation of polydisperse hyperspheres in infinite dimensions

We analyze the critical connectivity of systems of penetrable d-dimensional spheres having size distributions in terms of weighed random geometrical graphs, in which vertex coordinates correspond to random positions of the sphere centers, and edges are formed between any two overlapping spheres. Edge weights naturally arise from the different radii of two overlapping spheres. For the case in which the spheres have bounded size distributions, we show that clusters of connected spheres are treelike for d→∞ and they contain no closed loops. In this case, we find that the mean cluster size diverges at the percolation threshold density η(c)→2(-d), independently of the particular size distribution. We also show that the mean number of overlaps for a particle at criticality z(c) is smaller than unity, while z(c)→1 only for spheres with fixed radii. We explain these features by showing that in the large dimensionality limit, the critical connectivity is dominated by the spheres with the largest size. Assuming that closed loops can be neglected also for unbounded radii distributions, we find that the asymptotic critical threshold for systems of spheres with radii following a log-normal distribution is no longer universal, and that it can be smaller than 2(-d) for d→∞.