### Abstract

We study the effect of dimensionality on the percolation threshold η_{c} of identical overlapping nonspherical convex hyperparticles in d-dimensional Euclidean space Rd. This is done by formulating a scaling relation for η_{c} 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 v_{ex} 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 η_{c} for this comprehensive set of continuum percolation models across dimensions. We demonstrate that the scaling relation provides accurate upper-bound estimates of the threshold η_{c} across dimensions and becomes increasingly accurate as the space dimension increases.

Original language | English (US) |
---|---|

Article number | 022111 |

Journal | Physical Review E - Statistical, Nonlinear, and Soft Matter Physics |

Volume | 87 |

Issue number | 2 |

DOIs | |

State | Published - Feb 13 2013 |

Externally published | Yes |

### Fingerprint

### ASJC Scopus subject areas

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

### Cite this

**Effect of dimensionality on the percolation threshold of overlapping nonspherical hyperparticles.** / Torquato, S.; Jiao, Yang.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Effect of dimensionality on the percolation threshold of overlapping nonspherical hyperparticles

AU - Torquato, S.

AU - Jiao, Yang

PY - 2013/2/13

Y1 - 2013/2/13

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84874280553&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84874280553&partnerID=8YFLogxK

U2 - 10.1103/PhysRevE.87.022111

DO - 10.1103/PhysRevE.87.022111

M3 - Article

VL - 87

JO - Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

JF - Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

SN - 1539-3755

IS - 2

M1 - 022111

ER -