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

M V Chubynsky1, M F Thorpe

  • 1Département de Physique, Université de Montréal, Case Postale 6128, Succursale Centre-Ville, Montréal, Québec, Canada H3C 3J7. mykyta.chubynsky@umontreal.ca

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A new approximate pebble game algorithm enables analysis of 3D network rigidity. It reveals a first-order rigidity percolation transition in diluted 3D networks, unlike previous 2D studies.

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Area of Science:

  • Physics
  • Materials Science
  • Computer Science

Background:

  • The pebble game algorithm efficiently analyzes rigidity in 2D elastic networks and some 3D networks.
  • Its application to general 3D networks was limited by theoretical constraints.
  • Rigidity percolation studies are crucial for understanding network properties.

Purpose of the Study:

  • To develop and validate an approximate pebble game algorithm for general 3D networks.
  • To investigate the nature of rigidity percolation transitions in randomly diluted 3D networks.

Main Methods:

  • Construction of an approximate pebble game algorithm for general 3D networks.
  • Development of an exact relaxation algorithm for testing the approximate pebble game.
  • Analysis of bond-diluted and site-diluted central-force networks on bcc and fcc lattices.

Main Results:

  • The approximate pebble game is found to be highly accurate for diluted central-force networks.
  • An abrupt, first-order rigidity percolation transition is observed in bond-diluted 3D networks.
  • Site dilution shows a first-order transition for bcc lattices and a second-order for fcc lattices.

Conclusions:

  • The pebble game is a reliable tool for studying rigidity in general 3D networks.
  • The observed first-order transition in regular diluted networks is unusual and warrants further investigation.
  • The phase transition exhibits unique characteristics, including continuity in elastic moduli and apparent nonuniversality.