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3D Pebble Game with Central Forces Only (No angular forces)
Although the Laman condition is not generally sufficient in three
dimensions, it has been shown that it can be generalized within
bond bending networks [1-3] via the Molecular Framework
Conjecture. Unfortunately it is not possible to rigorously extend
the Pebble Game to general three dimensional networks with central
forces only, which would represent a solution to the most general
3D case. This is because the possible existence of banana like
graphs [1, 2]. However, a very good approximation can be obtained
in generic networks using the 3D Pebble Game Fortran Central
Forces Code, which can be downloaded here. This has been
extensively tested by Mykyta Chubynsky and M.F. Thorpe for diluted
face centered cubic and body centered cubic lattices, with testing
against results on networks with a few hundred sites using direct
matrix diagonalization [3]. The errors were ~ 0.1% in the worst
cases and sometimes zero. We thus suggest that this program can be
used for the general 3D case, as long as due diligence is used,
and recognizing that the results are not strictly exact. The
development of this routine was originally due to D.J. Jacobs and
M.F. Thorpe, with subsequent input from Mykyta Chubynsky [4,
5]. The downloadable code comes with the program files, a
description of how to build the network, write the input files and
a short description of the output files. Using three pebbles, the
program finds the total number of floppy modes; gives the rigid
and over-constrained (stressed) bonds and also the fraction of
particles in the largest rigid and stressed cluster. However, no
rigid cluster decomposition is available at this time.
Files available for download.
Download a brief description of the 3D pebble game algorithm for generic networks with only central forces everywhere.3D_CF_only_description.pdf References
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