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

Maxwell- theory

Regular vs. generic lattices

The Pebble game for 2d

Rigidity percolation thresholds

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

Critical exponents

Exponent	2d	3d	4d	5d	6d
$\alpha$	0.48(5),^[2]
$\beta$	0.175(20),^[1]
$\nu$	1.21(6),^[2]

These satisfy the following scaling relations.

Bethe and hierarchical lattices

On the Bethe lattice,

References

External links

^ ^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.
^ ^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.

[JacobsThorpe96-1] 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.

[JacobsThorpe95-2] 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.

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