Exact solution of a jamming transition: Closed equations for a bootstrap percolation problem

Abstract

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.T here exists within nature a whole class of systems that exhibit a geometrical percolation transition at which they become spanned by a single infinite cluster extending across the whole system (1-3). Such transitions may be observed, for example, by randomly occupying lattice sites at some prescribed density. Spatio-temporal particle correlations implied by simple dynamical models may also be studied by using percolation ideas. Indeed, since its introduction (4, 5), the potential of the bootstrap percolation problem (6, 7) to analyze the dynamics of a system of highly coupled and locally interacting units has been recognized. The range of applications has continued to grow (8)(9)(10)(11)(12).This problem is of particular interest because of a growing focus on, and appreciation of, the unifying role of dynamical arrest (13-17) or jamming (18) in the formation of complex condensed states of matter. Despite many advances, there is as yet no complete and fundamental conceptual framework to describe the phenomena. In comparable situations it has been an important lesson of critical phenomena (19,20) that an exact solution, even of a 2D model system, can be of great assistance in broader efforts to understand the issues. Thus, an exact closed solution of one bootstrap problem (with all of the implications of strong packing-induced coupling and divergent correlated domains) would represent, even without direct access to transport coefficients, a solution of a nontrivial (and non-mean field) jamming or arrest scenario. We will present such a solution in this article.That such a treatment is possible must be considered surprising, for there have been no prior indications of such simplification, to our knowledge.The connection of the bootstrap percolation problem to jamming phenomena is clear. Thus, particles, process...

Asymmetric Exclusion Processes with Constrained Dynamics

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Heterogeneities in aging models of granular compaction

“…In particular, this issue was recently investigated in the KA model [49,55] without gravity using fourth-order correlation functions [45,46]. In Ref.…”

“…On the other hand, nonconnected holes leave a backbone of blocked particles. Close to dynamical arrest, the density of connected holes decreases with the density of particles and is related with the inverse of the bootstrap length [45,46], the average distance between two connected holes, in bootstrap percolation. In turn, this length is associated with the transport properties of the system [47][48][49].…”

Geometry of dynamically available empty space is the key to near-arrest dynamics

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