Jamming and tiling in fragmentation of rectangles

Journal Article · Mon Sep 16 00:00:00 EDT 2019 · Physical Review E

DOI:https://doi.org/10.1103/PhysRevE.100.032122· OSTI ID:1571600

Ben-Naim, Eli ^[1]; Krapivsky, Paul L. ^[2]

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1. Los Alamos National Lab. (LANL), Los Alamos, NM (United States). Theoretical Div. and Center for Nonlinear Studies
2. Boston Univ., Boston, MA (United States). Dept. of Physics

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 cannot break any further. The average number of sticks in the jammed state, S, grows as S ≃ A/√2π 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. Here, 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.

• Research Organization: Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
• Sponsoring Organization: USDOE National Nuclear Security Administration (NNSA)
• Grant/Contract Number: 89233218CNA000001
• OSTI ID: 1571600
• Alternate ID(s): OSTI ID: 1562174
• Report Number(s): LA-UR--19-24374
• Journal Information: Physical Review E, Journal Name: Physical Review E Journal Issue: 3 Vol. 100; ISSN PLEEE8; ISSN 2470-0045
• Publisher: American Physical Society (APS)
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICS AND COMPUTING
Mathematics