Abstract
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 heat can 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.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 2084-2092 |
| Number of pages | 9 |
| Journal | Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics |
| Volume | 59 |
| Issue number | 2 PART B |
| State | Published - Feb 1999 |
| Externally published | Yes |
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ASJC Scopus subject areas
- Mathematical Physics
- Physics and Astronomy(all)
- Condensed Matter Physics
- Statistical and Nonlinear Physics
Cite this
Floppy modes and the free energy : Rigidity and connectivity percolation on Bethe lattices. / Duxbury, P. M.; Jacobs, D. J.; Thorpe, Michael; Moukarzel, C.
In: Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, Vol. 59, No. 2 PART B, 02.1999, p. 2084-2092.Research output: Contribution to journal › Article
}
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, Michael
AU - Moukarzel, C.
PY - 1999/2
Y1 - 1999/2
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 heat can 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 heat can 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.
UR - http://www.scopus.com/inward/record.url?scp=0000794168&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=0000794168&partnerID=8YFLogxK
M3 - Article
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 PART B
ER -