Exact Solution of a Jamming Transition: Closed Equations for a Bootstrap Percolation Problem

Details Exact Solution of a Jamming Transition: Closed Equations for a Bootstrap Percolation Problem Exact Solution of a Jamming Transition: Closed Equations for a Bootstrap Percolation Problem

2005-05-12

arxiv.org/abs/cond-mat/0505327v1

Proc. Nat. Acad. Sci. USA 102, 5669, 2005. http://www.pnas.org/cgi/content/abstract/102/16/5669

Physics

Condensed Matter

Statistical Mechanics

17 pages, 4 figures

Scientific paper

10.1073/pnas.0408756102

Jamming, or dynamical arrest, is a transition at which many particles stop moving in a collective manner. In nature it is brought about by, for example, increasing the packing density, changing the interactions between particles, or otherwise restricting the local motion of the elements of the system. The onset of collectivity occurs because, when one particle is blocked, it may lead to the blocking of a neighbor. That particle may then block one of its neighbors, these effects propagating across some typical domain of size named the dynamical correlation length. When this length diverges, the system becomes immobile. Even where it is finite but large the dynamics is dramatically slowed. Such phenomena lead to glasses, gels, and other very long-lived nonequilibrium solids. The bootstrap percolation models are the simplest examples describing these spatio-temporal correlations. We have been able to solve one such model in two dimensions exactly, exhibiting the precise evolution of the jamming correlations on approach to arrest. We believe that the nature of these correlations and the method we devise to solve the problem are quite general. Both should be of considerable help in further developing this field.

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