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

TL;DR

This paper demonstrates that the negative number of floppy modes acts as a free energy in connectivity and rigidity percolation on Bethe lattices, revealing a first-order transition and linking to the random bond model.

Contribution

It introduces a free energy framework for rigidity and connectivity percolation on Bethe lattices, connecting floppy modes to thermodynamic concepts and analyzing phase transitions.

Findings

Rigidity transition is first order near Maxwell prediction

Free energy can be derived from floppy modes

Bethe lattice results match the random bond model

Abstract

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.