Floppy modes and the free energy

Rigidity and connectivity percolation on Bethe lattices

P. M. Duxbury, D. J. Jacobs, Michael Thorpe, C. Moukarzel

Research output: Contribution to journalArticle

60 Citations (Scopus)

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 languageEnglish (US)
Pages (from-to)2084-2092
Number of pages9
JournalPhysical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
Volume59
Issue number2 PART B
StatePublished - Feb 1999
Externally publishedYes

Fingerprint

Bethe Lattice
rigidity
Rigidity
Free Energy
Connectivity
free energy
Specific Heat
Second derivative
Length Scale
Counting
counting
specific heat
First-order
Calculate
Demonstrate
Model

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 journalArticle

@article{3b6d68f6bdd748c9b9070e6c3dd9cfd9,
title = "Floppy modes and the free energy: Rigidity and connectivity percolation on Bethe lattices",
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.",
author = "Duxbury, {P. M.} and Jacobs, {D. J.} and Michael Thorpe and C. Moukarzel",
year = "1999",
month = "2",
language = "English (US)",
volume = "59",
pages = "2084--2092",
journal = "Physical Review E - Statistical, Nonlinear, and Soft Matter Physics",
issn = "1539-3755",
publisher = "American Physical Society",
number = "2 PART B",

}

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 -