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User:Rziff/Rigidity percolation

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Rigidity percolation is the phenomenon where disordered system become rigid to various forces or stresses when a sufficient number of connections between particles become formed. In analogy to the conventional percolation transition where long-range connectivity through a random system takes place when a sufficient number of connections between neighbors are made, the word percolation is used to describe this transition. A lattice-based model for rigidity percolation was introduced by M. F. Thorpe in 1983 and has been studied by many researchers ever since. The behavior of the model is very rich and has numerous applications to colloidal systems, gels, the glass transition, and polymerization.

Background

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Maxwell- theory

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Regular vs. generic lattices

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The Pebble game for 2d

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Rigidity percolation thresholds

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Lattice z Site percolation threshold Bond percolation threshold
generic triangular 3 0.69755(30),[1] 0.6602(3),[2]

Critical exponents

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Exponent 2d 3d 4d 5d 6d
0.48(5),[2]
0.175(20),[1]
1.21(6),[2]

These satisfy the following scaling relations.


Bethe and hierarchical lattices

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On the Bethe lattice,

See also

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References

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  1. ^ a b Jacobs, D. J.; M. F. Thorpe (1996). "Generic rigidity percolation in two dimensions". Physical Review E. 53 (3): 3682–3693. Bibcode:1996PhRvE..53.3682J. doi:10.1103/PhysRevE.53.3682. PMID 9964678.
  2. ^ a b c Jacobs, D. J.; M. F. Thorpe (1995). "Generic Rigidity Percolation: The Pebble Game". Physical Review Letters. 75 (22): 4051–4054. Bibcode:1995PhRvL..75.4051J. doi:10.1103/PhysRevLett.75.4051. PMID 10059802.