Effect of dimensionality on the percolation threshold of overlapping nonspherical hyperparticles

S. Torquato, Yang Jiao

Research output: Contribution to journalArticle

19 Citations (Scopus)

Abstract

We study the effect of dimensionality on the percolation threshold ηc of identical overlapping nonspherical convex hyperparticles in d-dimensional Euclidean space Rd. This is done by formulating a scaling relation for ηc that is based on a rigorous lower bound and a conjecture that hyperspheres provide the highest threshold, for any d, among all convex hyperparticle shapes (that are not a trivial affine transformation of a hypersphere). This scaling relation also exploits the recently discovered principle that low-dimensional continuum percolation behavior encodes high-dimensional information. We derive an explicit formula for the exclusion volume vex of a hyperparticle of arbitrary shape in terms of its d-dimensional volume v, surface area s, and radius of mean curvature R̄ (or, equivalently, mean width). These basic geometrical properties are computed for a wide variety of nonspherical hyperparticle shapes with random orientations across all dimensions, including, among other shapes, various polygons for d=2, Platonic solids, spherocylinders, parallepipeds, and zero-volume plates for d=3 and their appropriate generalizations for d≥4. Using this information, we compute the lower bound and scaling relation for ηc for this comprehensive set of continuum percolation models across dimensions. We demonstrate that the scaling relation provides accurate upper-bound estimates of the threshold ηc across dimensions and becomes increasingly accurate as the space dimension increases.

Original languageEnglish (US)
Article number022111
JournalPhysical Review E - Statistical, Nonlinear, and Soft Matter Physics
Volume87
Issue number2
DOIs
StatePublished - Feb 13 2013
Externally publishedYes

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Scaling Relations
Percolation Threshold
Dimensionality
Overlapping
hyperspheres
Continuum Percolation
scaling
Hypersphere
thresholds
Platonic solid
Lower bound
continuums
Euclidean geometry
polygons
Surface area
Mean Curvature
exclusion
Polygon
Affine transformation
Euclidean space

ASJC Scopus subject areas

  • Condensed Matter Physics
  • Statistical and Nonlinear Physics
  • Statistics and Probability

Cite this

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