Algorithms for three-dimensional rigidity analysis and a first-order percolation transition

M. V. Chubynsky, Michael Thorpe

Research output: Contribution to journalArticle

42 Citations (Scopus)

Abstract

A fast computer algorithm, the pebble game, has been used successfully to analyze the rigidity of two-dimensional (2D) elastic networks, as well as of a special class of 3D networks, the bond-bending networks, and enabled significant progress in studies of rigidity percolation on such networks. Application of the pebble game approach to general 3D networks has been hindered by the fact that the underlying mathematical theory is, strictly speaking, invalid in this case. We construct an approximate pebble game algorithm for general 3D networks, as well as a slower but exact algorithm, the relaxation algorithm, that we use for testing the new pebble game. Based on the results of these tests and additional considerations, we argue that in the particular case of randomly diluted central-force networks on bcc and fcc lattices, the pebble game is essentially exact. Using the pebble game, we observe an extremely sharp jump in the largest rigid cluster size in bond-diluted central-force networks in 3D, with the percolating cluster appearing and taking up most of the network after a single bond addition. This strongly suggests a first-order rigidity percolation transition, which is in contrast to the second-order transitions found previously for the 2D central-force and 3D bond-bending networks. While a first order rigidity transition has been observed previously for Bethe lattices and networks with "chemical order," here it is in a regular randomly diluted network. In the case of site dilution, the transition is also first order for bcc lattices, but results for fcc lattices suggest a second-order transition. Even in bond-diluted lattices, while the transition appears massively first order in the order parameter (the percolating cluster size), it is continuous in the elastic moduli. This, and the apparent nonuniversality, make this phase transition highly unusual.

Original languageEnglish (US)
Article number041135
JournalPhysical Review E - Statistical, Nonlinear, and Soft Matter Physics
Volume76
Issue number4
DOIs
StatePublished - Oct 25 2007

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rigidity
Rigidity
First-order
Three-dimensional
games
Game
Bethe Lattice
Elastic Modulus
Exact Algorithms
Order Parameter
dilution
modulus of elasticity
Jump
Phase Transition
Strictly
Testing

ASJC Scopus subject areas

  • Physics and Astronomy(all)
  • Condensed Matter Physics
  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Algorithms for three-dimensional rigidity analysis and a first-order percolation transition. / Chubynsky, M. V.; Thorpe, Michael.

In: Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, Vol. 76, No. 4, 041135, 25.10.2007.

Research output: Contribution to journalArticle

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