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

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

1998-07-05

arxiv.org/abs/cond-mat/9807073v1

Physics

Condensed Matter

Statistical Mechanics

RevTeX 11 pages + epsfig embedded figures. Submitted to Phys. Rev. E

Scientific paper

10.1103/PhysRevE.59.2084

We show that 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.

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