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

TL;DR

This paper presents an exact solution to a bootstrap percolation model in two dimensions, revealing detailed evolution of jamming correlations as the system approaches dynamical arrest, with implications for understanding glasses and gels.

Contribution

We provide a novel exact solution to a bootstrap percolation model, offering insights into the evolution of jamming correlations near the transition point.

Findings

Exact solution for a 2D bootstrap percolation model

Detailed characterization of jamming correlation evolution

Potential generalization of methods to other models

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.