5,194 research outputs found
Hierarchical models of rigidity percolation
We introduce models of generic rigidity percolation in two dimensions on
hierarchical networks, and solve them exactly by means of a renormalization
transformation. We then study how the possibility for the network to self
organize in order to avoid stressed bonds may change the phase diagram. In
contrast to what happens on random graphs and in some recent numerical studies
at zero temperature, we do not find a true intermediate phase separating the
usual rigid and floppy ones.Comment: 20 pages, 8 figures. Figures improved, references added, small
modifications. Accepted in Phys. Rev.
Algorithms for 3D rigidity analysis and a first order percolation transition
A fast computer algorithm, the pebble game, has been used successfully to
study rigidity percolation on 2D elastic networks, as well as on a special
class of 3D networks, the bond-bending 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 for Bethe lattices and networks
with ``chemical order'', this is the first time it has been seen for a regular
randomly diluted network. In the case of site dilution, the transition is also
first order for BCC, but results for FCC 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 non-universality, make this phase
transition highly unusual.Comment: 28 pages, 19 figure
Self-organization with equilibration: a model for the intermediate phase in rigidity percolation
Recent experimental results for covalent glasses suggest the existence of an
intermediate phase attributed to the self-organization of the glass network
resulting from the tendency to minimize its internal stress. However, the exact
nature of this experimentally measured phase remains unclear. We modify a
previously proposed model of self-organization by generating a uniform sampling
of stress-free networks. In our model, studied on a diluted triangular lattice,
an unusual intermediate phase appears, in which both rigid and floppy networks
have a chance to occur, a result also observed in a related model on a Bethe
lattice by Barre et al. [Phys. Rev. Lett. 94, 208701 (2005)]. Our results for
the bond-configurational entropy of self-organized networks, which turns out to
be only about 2% lower than that of random networks, suggest that a
self-organized intermediate phase could be common in systems near the rigidity
percolation threshold.Comment: 9 pages, 6 figure
Self-organized criticality in the intermediate phase of rigidity percolation
Experimental results for covalent glasses have highlighted the existence of a
new self-organized phase due to the tendency of glass networks to minimize
internal stress. Recently, we have shown that an equilibrated self-organized
two-dimensional lattice-based model also possesses an intermediate phase in
which a percolating rigid cluster exists with a probability between zero and
one, depending on the average coordination of the network. In this paper, we
study the properties of this intermediate phase in more detail. We find that
microscopic perturbations, such as the addition or removal of a single bond,
can affect the rigidity of macroscopic regions of the network, in particular,
creating or destroying percolation. This, together with a power-law
distribution of rigid cluster sizes, suggests that the system is maintained in
a critical state on the rigid/floppy boundary throughout the intermediate
phase, a behavior similar to self-organized criticality, but, remarkably, in a
thermodynamically equilibrated state. The distinction between percolating and
non-percolating networks appears physically meaningless, even though the
percolating cluster, when it exists, takes up a finite fraction of the network.
We point out both similarities and differences between the intermediate phase
and the critical point of ordinary percolation models without
self-organization. Our results are consistent with an interpretation of recent
experiments on the pressure dependence of Raman frequencies in chalcogenide
glasses in terms of network homogeneity.Comment: 20 pages, 18 figure
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