TY - JOUR

T1 - Floppy modes and the free energy

T2 - Rigidity and connectivity percolation on Bethe lattices

AU - Duxbury, P. M.

AU - Jacobs, D. J.

AU - Thorpe, M. F.

AU - Moukarzel, C.

N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 1999

Y1 - 1999

N2 - We show that the negative of the number of floppy modes behaves as a free energy for both connectivity and rigidity percolation, and we illustrate this result using Bethe lattices. The rigidity transition on Bethe lattices is found to be first order at a bond concentration close to that predicted by Maxwell constraint counting. We calculate the probability of a bond being on the infinite cluster and also on the overconstrained part of the infinite cluster, and show how a specific heatcan be defined as the second derivative of the free energy. We demonstrate that the Bethe lattice solution is equivalent to that of the random bond model, where points are joined randomly (with equal probability at all length scales) to have a given coordination, and then subsequently bonds are randomly removed.

AB - We show that the negative of the number of floppy modes behaves as a free energy for both connectivity and rigidity percolation, and we illustrate this result using Bethe lattices. The rigidity transition on Bethe lattices is found to be first order at a bond concentration close to that predicted by Maxwell constraint counting. We calculate the probability of a bond being on the infinite cluster and also on the overconstrained part of the infinite cluster, and show how a specific heatcan be defined as the second derivative of the free energy. We demonstrate that the Bethe lattice solution is equivalent to that of the random bond model, where points are joined randomly (with equal probability at all length scales) to have a given coordination, and then subsequently bonds are randomly removed.

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U2 - 10.1103/PhysRevE.59.2084

DO - 10.1103/PhysRevE.59.2084

M3 - Article

AN - SCOPUS:0000794168

VL - 59

SP - 2084

EP - 2092

JO - Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

JF - Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

SN - 1539-3755

IS - 2

ER -