Carl P. Goodrich, Wouter G. Ellenbroek, Andrea J. Liu
In 2005, Wyart et al. (Europhys. Lett., 72 (2005) 486) showed that the low frequency vibrational properties of jammed sphere packings can be understood in terms of a length scale, called l*, that diverges as the system becomes marginally unstable. Despite its tremendous success, it has been difficult to connect the counting argument that defines l* to other length scales that diverge near the jamming transition. We present an alternate derivation of l* and show that it arises naturally as the minimum size of a rigid cluster of particles with free boundaries. This understanding allows us to place a quantitative upper bound on the magnitude of l*. We also present the first direct numerical calculation of l*, and show that it is close to the upper bound.
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http://arxiv.org/abs/1301.6981
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