Floppy modes and the free energy: Rigidity and connectivity percolation on Bethe lattices

Duxbury, Phillip M.; Jacobs, Donald J.; Thorpe, M. F.; Moukarzel, Cristian F.

doi:10.1103/physreve.59.2084

Cited by 65 publications

(109 citation statements)

References 27 publications

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“…Here, we have kept from the internal energy of the network only the part related to energy of the elastic deformations of the network, removing the contributions from the harmonic vibrations of the atoms and the anharmonic contributions which are irrelevant for our purpose (however, see 27 ). This idea is also consistent with the work of Duxbury and co-workers who showed that the number of floppy modes behaves as a free energy for both rigidity and connectivity percolation 28 . The entropy of the network can be evaluated as a Bragg-Williams term from the distribution of cluster probabilities p i at step l.…”

Section: B Constraint Free Energysupporting

confidence: 92%

Rings and rigidity transitions in network glasses

2003

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Three elastic phases of covalent networks, (I) floppy, (II) isostatically rigid and (III) stressed-rigid have now been identified in glasses at specific degrees of cross-linking (or chemical composition) both in theory and experiments.Here we use size-increasing cluster combinatorics and constraint counting algorithms to study analytically possible consequences of self-organization. In the presence of small rings that can be locally I, II or III, we obtain two transitions instead of the previously reported single percolative transition at the mean coordination numberr = 2.4, one from a floppy to an isostatic rigid phase, and a second one from an isostatic to a stressed rigid phase. The width of the intermediate phase ∆r and the order of the phase transitions depend on the nature of medium range order (relative ring fractions). We compare the results to the Group IV chalcogenides, such as Ge-Se and Si-Se, for which evidence of an intermediate phase has been obtained, and for which estimates of ring fractions can be made from structures of high T crystalline phases.

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Section: B Constraint Free Energysupporting

confidence: 92%

Rings and rigidity transitions in network glasses

2003

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“…The simplicity of (7) is one of the reasons leading us to study this particular (Poissonian) definition of the field. Other field definitions, like for example assuming that each site is rigidly attached to the background with probability h, or other random graph structures like Bethe lattices [20] are also tractable with the methods used here, but the algebra becomes more complicated. Clearly y = γ(H + R) plays the role of a "Weiss field" in the MF equation for a ferromagnet.…”

Section: Equation Of State In the Presence Of A Ghost Fieldmentioning

confidence: 99%

“…It seems possible that a similar mapping might exist for RP as well, although it has not been found up to now. However it has been proposed [20,45] that the number of uncanceled degrees of freedom n F is a good "free energy" candidate for RP in zero field. When g = 1, each connected cluster has one uncanceled degree of freedom, so both definitions coincide in this limit.…”

Section: Introductionmentioning

confidence: 99%

Rigidity percolation in a field

Moukarzel

2003

Phys. Rev. E

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Rigidity Percolation with g degrees of freedom per site is analyzed on randomly diluted Erdős-Renyi graphs with average connectivity γ, in the presence of a field h. In the (γ, h) plane, the rigid and flexible phases are separated by a line of first-order transitions whose location is determined exactly. This line ends at a critical point with classical critical exponents. Analytic expressions are given for the densities nF of uncanceled degrees of freedom and γr of redundant bonds. Upon crossing the coexistence line, γr and nF are continuous, although their first derivatives are discontinuous. We extend, for the case of nonzero field, a recently proposed hypothesis, namely that the density of uncanceled degrees of freedom is a "free energy" for Rigidity Percolation. Analytic expressions are obtained for the energy, entropy, and specific heat. Some analogies with a liquid-vapor transition are discussed. Particularizing to zero field, we find that the existence of a (g + 1)-core is a necessary condition for rigidity percolation with g degrees of freedom. At the transition point γc, Maxwell counting of degrees of freedom is exact on the rigid cluster and on the (g + 1)-rigid-core, i.e. the average coordination of these subgraphs is exactly 2g, although γc, the average coordination of the whole system, is smaller than 2g. γc is found to converge to 2g for large g, i.e. in this limit Maxwell counting is exact globally as well.

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“…This leads to the relationship between the probabilities p n+1 , p n of rigidity Here, (a) has all eight bonds present and is rigid with one redundant edge and has probability p 8 , (b) has any single edge missing and has probability 8p 7 (1 − p), (c) has any pair of edges missing from the three lower (shown) or the three upper ones and has probability 6p 6 (1 − p) 2 and (d) has a triple of edges missing either from the lower or upper part of the graph and has probability 2p 5 (1 − p) 3 . The number of edges in the rigid cluster is indicated by the number under each graph.…”

Section: Hierarchical Modelsmentioning

confidence: 99%

“…In this case, a single floppy mode is associated with an isolated cluster, so the free energy is just the total number of isolated clusters, and of course is an extensive quantity. Finally for connectivity and rigidity, these forms of F have been used as a free energy for percolation from a busbar onto a Cayley tree network [5].…”

Section: Hierarchical Modelsmentioning

confidence: 99%

Two exactly soluble models of rigidity percolation

Thorpe

Stinchcombe

2014

Phil. Trans. R. Soc. A.

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We summarize results for two exactly soluble classes of bond-diluted models for rigidity percolation, which can serve as a benchmark for numerical and approximate methods. For bond dilution problems involving rigidity, the number of floppy modes F plays the role of a free energy. Both models involve pathological lattices with two-dimensional vector displacements. The first model involves hierarchical lattices where renormalization group calculations can be used to give exact solutions. Algebraic scaling transformations produce a transition of the second order, with an unstable critical point and associated scaling laws at a mean coordination 〈 r 〉=4.41, which is above the ‘mean field’ value 〈 r 〉=4 predicted by Maxwell constraint counting. The order parameter exponent associated with the spanning rigid cluster geometry is β =0.0775 and that associated with the divergence of the correlation length and the anomalous lattice dimension d is dν =3.533. The second model involves Bethe lattices where the rigidity transition is massively first order by a mean coordination 〈 r 〉=3.94 slightly below that predicted by Maxwell constraint counting. We show how a Maxwell equal area construction can be used to locate the first-order transition and how this result agrees with simulation results on larger random-bond lattices using the pebble game algorithm.

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Floppy modes and the free energy: Rigidity and connectivity percolation on Bethe lattices

Cited by 65 publications

References 27 publications

Rings and rigidity transitions in network glasses

Rings and rigidity transitions in network glasses

Rigidity percolation in a field

Two exactly soluble models of rigidity percolation

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