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.

17 pages, 4 figures

References in corpus (8)

• Glassy dynamics of kinetically constrained models
• On nonlinear susceptibility in supercooled liquids
• Spatial structures and dynamics of kinetically constrained models for glasses
• Statistical Physics of Structural Glasses
• Clarification of the Bootstrap Percolation Paradox
• Dynamical arrest, tracer diffusion and Kinetically Constrained Lattice Gases
• Geometry of Empty Space is the Key to Near-Arrest Dynamics
• Threshold value of three dimensional bootstrap percolation

Cited by in corpus (18)

• Jamming Percolation and Glass Transitions in Lattice Models
• Hilbert space shattering and dynamical freezing in the quantum Ising model
• Asymmetric exclusion processes with constrained dynamics
• Correlated percolation and tricriticality
• $1/d$ Expansion for $k$-Core Percolation
• Geometry of Empty Space is the Key to Near-Arrest Dynamics
• Newman-Ziff algorithm for the bootstrap percolation: application to the Archimedean lattices
• Thresholds for contagious sets in random graphs
• Bootstrap and diffusion percolation transitions in three-dimensional lattices
• Finite-density effects in the Fredrickson-Andersen and Kob-Andersen kinetically-constrained models
• The time of bootstrap percolation in two dimensions
• The time of bootstrap percolation with dense initial sets for all thresholds
• A sharper threshold for bootstrap percolation in two dimensions
• Lattice Model of Glasses
• Locality approach to the bootstrap percolation paradox
• Local Bootstrap Percolation
• Slow Convergence in Bootstrap Percolation
• Phase transition in the diffusion and bootstrap percolation models on regular random and Erdős-Rényi networks