### 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

*Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics*,

*59*(2 PART B), 2084-2092.

**Floppy modes and the free energy : Rigidity and connectivity percolation on Bethe lattices.** / Duxbury, P. M.; Jacobs, D. J.; Thorpe, Michael; Moukarzel, C.

Research output: Contribution to journal › Article

*Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics*, vol. 59, no. 2 PART B, pp. 2084-2092.

}

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 -