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Title: 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 cannot break any further. The average number of sticks in the jammed state, S, grows as SA/√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.

Authors:
ORCiD logo [1];  [2]
  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
Publication Date:
Research Org.:
Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA)
OSTI Identifier:
1571600
Alternate Identifier(s):
OSTI ID: 1562174
Report Number(s):
LA-UR-19-24374
Journal ID: ISSN 2470-0045; PLEEE8
Grant/Contract Number:  
89233218CNA000001
Resource Type:
Accepted Manuscript
Journal Name:
Physical Review E
Additional Journal Information:
Journal Volume: 100; Journal Issue: 3; Journal ID: ISSN 2470-0045
Publisher:
American Physical Society (APS)
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; Mathematics

Citation Formats

Ben-Naim, Eli, and Krapivsky, Paul L. Jamming and tiling in fragmentation of rectangles. United States: N. p., 2019. Web. doi:10.1103/PhysRevE.100.032122.
Ben-Naim, Eli, & Krapivsky, Paul L. Jamming and tiling in fragmentation of rectangles. United States. doi:10.1103/PhysRevE.100.032122.
Ben-Naim, Eli, and Krapivsky, Paul L. Mon . "Jamming and tiling in fragmentation of rectangles". United States. doi:10.1103/PhysRevE.100.032122.
@article{osti_1571600,
title = {Jamming and tiling in fragmentation of rectangles},
author = {Ben-Naim, Eli and Krapivsky, Paul L.},
abstractNote = {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.},
doi = {10.1103/PhysRevE.100.032122},
journal = {Physical Review E},
number = 3,
volume = 100,
place = {United States},
year = {2019},
month = {9}
}

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