Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes. II. Simulation results and analyses.
Identifieur interne : 002260 ( Main/Exploration ); précédent : 002259; suivant : 002261Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes. II. Simulation results and analyses.
Auteurs : S. Torquato [États-Unis] ; Y. JiaoSource :
- The Journal of chemical physics [ 1089-7690 ] ; 2012.
Abstract
In the first paper of this series [S. Torquato, J. Chem. Phys. 136, 054106 (2012)], 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.
DOI: 10.1063/1.4742750
PubMed: 22920102
Affiliations:
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II. Simulation results and analyses.</title><author><name sortKey="Torquato, S" sort="Torquato, S" uniqKey="Torquato S" first="S" last="Torquato">S. Torquato</name><affiliation wicri:level="4"><nlm:affiliation>Department of Chemistry, Princeton Center for Theoretical Science, and Program in Applied and Computational Mathematics, Princeton University, Princeton, New Jersey 08544, USA. torquato@electron.princeton.edu</nlm:affiliation><country xml:lang="fr">États-Unis</country><wicri:regionArea>Department of Chemistry, Princeton Center for Theoretical Science, and Program in Applied and Computational Mathematics, Princeton University, Princeton, New Jersey 08544</wicri:regionArea><orgName type="university">Université de Princeton</orgName><placeName><settlement type="city">Princeton (New Jersey)</settlement><region type="state">New Jersey</region></placeName></affiliation></author><author><name sortKey="Jiao, Y" sort="Jiao, Y" uniqKey="Jiao Y" first="Y" last="Jiao">Y. Jiao</name></author></analytic><series><title level="j">The Journal of chemical physics</title><idno type="eISSN">1089-7690</idno><imprint><date when="2012" type="published">2012</date></imprint></series></biblStruct></sourceDesc></fileDesc><profileDesc><textClass></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">In the first paper of this series [S. Torquato, J. Chem. Phys. 136, 054106 (2012)], 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.</div></front></TEI><affiliations><list><country><li>États-Unis</li></country><region><li>New Jersey</li></region><settlement><li>Princeton (New Jersey)</li></settlement><orgName><li>Université de Princeton</li></orgName></list><tree><noCountry><name sortKey="Jiao, Y" sort="Jiao, Y" uniqKey="Jiao Y" first="Y" last="Jiao">Y. Jiao</name></noCountry><country name="États-Unis"><region name="New Jersey"><name sortKey="Torquato, S" sort="Torquato, S" uniqKey="Torquato S" first="S" last="Torquato">S. Torquato</name></region></country></tree></affiliations></record>Pour manipuler ce document sous Unix (Dilib)
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