Clarification of the Bootstrap Percolation Paradox

Abstract

We study the onset of the bootstrap percolation transition as a model of generalized dynamical arrest. Our results apply to two dimensions, but there is no significant barrier to extending them to higher dimensionality. We develop a new importance-sampling procedure in simulation, based on rare events around "holes", that enables us to access bootstrap lengths beyond those previously studied. By framing a new theory in terms of paths or processes that lead to emptying of the lattice we are able to develop systematic corrections to the existing theory and compare them to simulations. Thereby, for the first time in the literature, it is possible to obtain credible comparisons between theory and simulation in the accessible density range.

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

“…Such systems have not crystallized, the free energy has not been minimized, and there is no obvious order parameter for the 'transition'. This phenomenon of precipitous, and apparently collective, loss of motion has been named 'dynamical arrest' [1,2,3,4,5], or jamming [6,7,8]. It is involved in the 'glass transition' [9] and even, apparently, gellation [10,11].In the field of glassification especially, very significant advances have been made in developing our understanding [12,13].…”

“…We have sought a fundamental theoretical approach to dynamical arrest [2,3] based on a physical order parameter to describe the vicinity of dynamical arrest, keeping in mind the potential to make direct contact with experiment. In particular, the advent of new experiments [14,15], and new experimental methods [16,17,18] being developed in colloidal and soft matter science, opens up the possibility to directly connect observation of particle configurations and their fluctuations to suitably framed theoretical concepts.…”

“…We have identified dynamically available volume (DAV) [2], the ensemble of physical space available to particle dynamics, as a possible order parameter. Its use has been illustrated in a recent study of certain lattice models that exhibit dynamical arrest [2,3,19] and has been considered in models of granular matter [6].…”