TY - JOUR
T1 - Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes. II. Simulation results and analyses
AU - Torquato, S.
AU - Jiao, Y.
N1 - Funding Information:
This work was supported by the Materials Research Science and Engineering Center Program of the National Science Foundation under Grant No. DMR-0820341 and and by the Division of Mathematical Sciences at the National Science Foundation under Award No. DMS-1211087.
PY - 2012/8/21
Y1 - 2012/8/21
N2 - In the first paper of this series [S. Torquato, J. Chem. Phys. 136, 054106 (2012)10.1063/1.3679861], analytical results concerning the continuum percolation of overlapping hyperparticles in d-dimensional Euclidean space R d were obtained, including lower bounds on the percolation threshold. In the present investigation, we provide additional analytical results for certain cluster statistics, such as the concentration of k-mers and related quantities, and obtain an upper bound on the percolation threshold η c. We utilize the tightest lower bound obtained in the first paper to formulate an efficient simulation method, called the rescaled-particle algorithm, to estimate continuum percolation properties across many space dimensions with heretofore unattained accuracy. This simulation procedure is applied to compute the threshold η c and associated mean number of overlaps per particle N c for both overlapping hyperspheres and oriented hypercubes for 3 d 11. These simulations results are compared to corresponding upper and lower bounds on these percolation properties. We find that the bounds converge to one another as the space dimension increases, but the lower bound provides an excellent estimate of η c and N c, even for relatively low dimensions. We confirm a prediction of the first paper in this series that low-dimensional percolation properties encode high-dimensional information. We also show that the concentration of monomers dominate over concentration values for higher order clusters (dimers, trimers, etc.) as the space dimension becomes large. Finally, we provide accurate analytical estimates of the pair connectedness function and blocking function at their contact values for any d as a function of density.
AB - In the first paper of this series [S. Torquato, J. Chem. Phys. 136, 054106 (2012)10.1063/1.3679861], analytical results concerning the continuum percolation of overlapping hyperparticles in d-dimensional Euclidean space R d were obtained, including lower bounds on the percolation threshold. In the present investigation, we provide additional analytical results for certain cluster statistics, such as the concentration of k-mers and related quantities, and obtain an upper bound on the percolation threshold η c. We utilize the tightest lower bound obtained in the first paper to formulate an efficient simulation method, called the rescaled-particle algorithm, to estimate continuum percolation properties across many space dimensions with heretofore unattained accuracy. This simulation procedure is applied to compute the threshold η c and associated mean number of overlaps per particle N c for both overlapping hyperspheres and oriented hypercubes for 3 d 11. These simulations results are compared to corresponding upper and lower bounds on these percolation properties. We find that the bounds converge to one another as the space dimension increases, but the lower bound provides an excellent estimate of η c and N c, even for relatively low dimensions. We confirm a prediction of the first paper in this series that low-dimensional percolation properties encode high-dimensional information. We also show that the concentration of monomers dominate over concentration values for higher order clusters (dimers, trimers, etc.) as the space dimension becomes large. Finally, we provide accurate analytical estimates of the pair connectedness function and blocking function at their contact values for any d as a function of density.
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U2 - 10.1063/1.4742750
DO - 10.1063/1.4742750
M3 - Article
C2 - 22920102
AN - SCOPUS:84865486322
SN - 0021-9606
VL - 137
JO - Journal of Chemical Physics
JF - Journal of Chemical Physics
IS - 7
M1 - 074106
ER -