Jamming and Tiling in Fragmentation of Rectangles

Abstract

We investigate a stochastic process where a rectangle breaks into smaller rectangles through a series of horizontal and vertical fragmentation events. We focus on the case where both the vertical size and the horizontal size of a rectangle are discrete variables. Because of this constraint, the system reaches a jammed state where all rectangles are sticks, that is, rectangles with minimal width. Sticks are frozen as they can not break any further. The average number of sticks in the jammed state, $S$, grows as $S\simeq A/\sqrt{2\pi \ln A}$ with rectangle area $A$ in the large-area limit, and remarkably, this behavior is independent of the aspect ratio. The distribution of stick length has a power-law tail, and further, its moments are characterized by a nonlinear spectrum of scaling exponents. We also study an asymmetric breakage process where vertical and horizontal fragmentation events are realized with different probabilities. In this case, there is a phase transition between a weakly asymmetric phase where the length distribution is independent of system size, and a strongly asymmetric phase where this distribution depends on system size.

arXiv comment13 pages, 15 figures

References in corpus (14)

• Coagulation, fragmentation and radial motion of solid particles in protoplanetary disks
• A variational principle for domino tilings
• Transition from damage to fragmentation in collision of solids
• Local statistics for random domino tilings of the Aztec diamond
• Size distribution of particles in Saturn's rings from aggregation and fragmentation
• Dynamic buckling and fragmentation in brittle rods
• Scaling and Multiscaling in Models of Fragmentation
• Transition in the pattern of cracks resulting from memory effects in paste
• The Pfaffian solution of a dimer-monomer problem: Single monomer on the boundary
• A Pfaffian formula for monomer-dimer partition functions
• Aggregation-fragmentation-diffusion model for trail dynamics
• Minimal Fragmentation of Regular Polygonal Plates
• Jamming and Tiling in Aggregation of Rectangles
• A probabilistic model for interfaces in a martensitic phase transition

Cited by in corpus (4)

• Non-extensive super-cluster states in aggregation with fragmentation
• Revisiting log-periodic oscillations
• On multidimensional record patterns
• Scaling in simple continued fraction