Jamming and tiling in fragmentation of rectangles

Ben-Naim, Eli; Krapivsky, Paul L.

doi:10.1103/PhysRevE.100.032122

Title: Jamming and tiling in fragmentation of rectangles

Journal Article · 16 September 2019 · Physical Review E

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

^[1]; Krapivsky, Paul L. ^[2]

Los Alamos National Lab. (LANL), Los Alamos, NM (United States). Theoretical Div. and Center for Nonlinear Studies
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.

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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

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)Copyright Statement

Country of Publication:: United States

Language:: English

References (52)

Two-dimensional monomer-dimer systems are computationally intractable Jerrum, Mark Journal of Statistical Physics, Vol. 48, Issue 1-2 https://doi.org/10.1007/BF01010403	journal	July 1987
Theory of monomer-dimer systems Heilmann, Ole J.; Lieb, Elliott H. Communications in Mathematical Physics, Vol. 25, Issue 3 https://doi.org/10.1007/BF01877590	journal	September 1972
Two-dimensional monomer-dimer systems are computationally intractable Jerrum, Mark Journal of Statistical Physics, Vol. 59, Issue 3-4 https://doi.org/10.1007/bf01025864	journal	May 1990
Theory of monomer-dimer systems Heilmann, O. J.; Lieb, E. H. Communications in Mathematical Physics, Vol. 27, Issue 2 https://doi.org/10.1007/bf01645620	journal	June 1972
A Pfaffian Formula for Monomer–Dimer Partition Functions Giuliani, Alessandro; Jauslin, Ian; Lieb, Elliott H. Journal of Statistical Physics, Vol. 163, Issue 2 https://doi.org/10.1007/s10955-016-1484-1	journal	March 2016
The statistics of dimers on a lattice Kasteleyn, P. W. Physica, Vol. 27, Issue 12 https://doi.org/10.1016/0031-8914(61)90063-5	journal	December 1961
Multiscaling in fragmentation Ben-Naim, E.; Krapivsky, P. L. Physica D: Nonlinear Phenomena, Vol. 107, Issue 2-4 https://doi.org/10.1016/S0167-2789(97)00080-8	journal	September 1997
Radical addition–fragmentation chemistry in polymer synthesis Moad, Graeme; Rizzardo, Ezio; Thang, San H. Polymer, Vol. 49, Issue 5 https://doi.org/10.1016/j.polymer.2007.11.020	journal	March 2008
Corrosion/Fragmentation of Layered Composite Cathode and Related Capacity/Voltage Fading during Cycling Process Zheng, Jianming; Gu, Meng; Xiao, Jie Nano Letters, Vol. 13, Issue 8 https://doi.org/10.1021/nl401849t	journal	July 2013
Global patterns of tropical forest fragmentation Taubert, Franziska; Fischer, Rico; Groeneveld, Jürgen Nature, Vol. 554, Issue 7693 https://doi.org/10.1038/nature25508	journal	February 2018
Aggregation-fragmentation and individual dynamics of active clusters Ginot, F.; Theurkauff, I.; Detcheverry, F. Nature Communications, Vol. 9, Issue 1 https://doi.org/10.1038/s41467-017-02625-7	journal	February 2018
Mechanics of fragmentation of crocodile skin and other thin films Qin, Zhao; Pugno, Nicola M.; Buehler, Markus J. Scientific Reports, Vol. 4, Issue 1 https://doi.org/10.1038/srep04966	journal	May 2014
Geometry and fragmentation of soft brittle impacted bodies Vandenberghe, Nicolas; Villermaux, Emmanuel Soft Matter, Vol. 9, Issue 34 https://doi.org/10.1039/c3sm50789k	journal	January 2013
Coagulation, fragmentation and radial motion of solid particles in protoplanetary disks Brauer, F.; Dullemond, C. P.; Henning, Th. Astronomy & Astrophysics, Vol. 480, Issue 3 https://doi.org/10.1051/0004-6361:20077759	journal	November 2007
Exact Solution and Multifractal Analysis of a Multivariable Fragmentation Model Boyer, D.; Tarjus, G.; Viot, P. Journal de Physique I, Vol. 7, Issue 1 https://doi.org/10.1051/jp1:1997125	journal	January 1997
A probabilistic model for martensitic avalanches Ball, John M.; Cesana, Pierluigi; Hambly, Ben MATEC Web of Conferences, Vol. 33 https://doi.org/10.1051/matecconf/20153302008	journal	January 2015
Dimers on a Rectangular Lattice Baxter, R. J. Journal of Mathematical Physics, Vol. 9, Issue 4 https://doi.org/10.1063/1.1664623	journal	April 1968
Solution of the Dimer Problem by the Transfer Matrix Method Lieb, Elliott H. Journal of Mathematical Physics, Vol. 8, Issue 12 https://doi.org/10.1063/1.1705163	journal	December 1967
Size distribution of particles in Saturn’s rings from aggregation and fragmentation Brilliantov, Nikolai; Krapivsky, P. L.; Bodrova, Anna Proceedings of the National Academy of Sciences, Vol. 112, Issue 31 https://doi.org/10.1073/pnas.1503957112	journal	July 2015
Direct observation of antiphase boundaries in the AuCu ₃ superlattice Fisher, R. M.; Marcinkowski, M. J. Philosophical Magazine, Vol. 6, Issue 71 https://doi.org/10.1080/14786436108241233	journal	November 1961
A Brief History of Generative Models for Power Law and Lognormal Distributions Mitzenmacher, Michael Internet Mathematics, Vol. 1, Issue 2 https://doi.org/10.1080/15427951.2004.10129088	journal	January 2004
The kinetics of cluster fragmentation and depolymerisation Ziff, R. M.; McGrady, E. D. Journal of Physics A: Mathematical and General, Vol. 18, Issue 15 https://doi.org/10.1088/0305-4470/18/15/026	journal	October 1985
Jamming and tiling in aggregation of rectangles Ben-Naim, D. S.; Ben-Naim, E.; Krapivsky, P. L. Journal of Physics A: Mathematical and Theoretical, Vol. 51, Issue 45 https://doi.org/10.1088/1751-8121/aae4c0	journal	October 2018
Statistical Mechanics of Dimers on a Plane Lattice. II. Dimer Correlations and Monomers Fisher, Michael E.; Stephenson, John Physical Review, Vol. 132, Issue 4 https://doi.org/10.1103/PhysRev.132.1411	journal	November 1963
Fragmentation of particles with more than one degree of freedom Rodgers, G. J.; Hassan, M. K. Physical Review E, Vol. 50, Issue 5 https://doi.org/10.1103/PhysRevE.50.3458	journal	November 1994
Scaling and multiscaling in models of fragmentation Krapivsky, P. L.; Ben-Naim, E. Physical Review E, Vol. 50, Issue 5 https://doi.org/10.1103/PhysRevE.50.3502	journal	November 1994
Shattering transition in a multivariable fragmentation model Boyer, D.; Tarjus, G.; Viot, P. Physical Review E, Vol. 51, Issue 2 https://doi.org/10.1103/PhysRevE.51.1043	journal	February 1995
Fragmentation of thin films bonded to solid substrates: Simulations and a mean-field theory Crosby, Kevin M.; Bradley, R. Mark Physical Review E, Vol. 55, Issue 5 https://doi.org/10.1103/PhysRevE.55.6084	journal	May 1997
Transition from damage to fragmentation in collision of solids Kun, Ferenc; Herrmann, Hans J. Physical Review E, Vol. 59, Issue 3 https://doi.org/10.1103/PhysRevE.59.2623	journal	March 1999
Exponential and power-law mass distributions in brittle fragmentation Åström, J. A.; Linna, R. P.; Timonen, J. Physical Review E, Vol. 70, Issue 2 https://doi.org/10.1103/PhysRevE.70.026104	journal	August 2004
Hierarchical crack pattern as formed by successive domain divisions. Bohn, S.; Pauchard, L.; Couder, Y. Physical Review E, Vol. 71, Issue 4 https://doi.org/10.1103/PhysRevE.71.046214	journal	April 2005
Pfaffian solution of a dimer-monomer problem: Single monomer on the boundary Wu, F. Y. Physical Review E, Vol. 74, Issue 2 https://doi.org/10.1103/PhysRevE.74.020104	journal	August 2006
Transition in the pattern of cracks resulting from memory effects in paste Nakahara, Akio; Matsuo, Yousuke Physical Review E, Vol. 74, Issue 4 https://doi.org/10.1103/PhysRevE.74.045102	journal	October 2006
Driven fragmentation of granular gases Cruz Hidalgo, Raúl; Pagonabarraga, Ignacio Physical Review E, Vol. 77, Issue 6 https://doi.org/10.1103/PhysRevE.77.061305	journal	June 2008
Impact fragmentation of model flocks Miller, Pearson W.; Ouellette, Nicholas T. Physical Review E, Vol. 89, Issue 4 https://doi.org/10.1103/PhysRevE.89.042806	journal	April 2014
Minimal fragmentation of regular polygonal plates Dias, Laércio; Parisio, Fernando Physical Review E, Vol. 90, Issue 3 https://doi.org/10.1103/PhysRevE.90.032405	journal	September 2014
Geometrical model for martensitic phase transitions: Understanding criticality and weak universality during microstructure growth Torrents, Genís; Illa, Xavier; Vives, Eduard Physical Review E, Vol. 95, Issue 1 https://doi.org/10.1103/PhysRevE.95.013001	journal	January 2017
Aggregation-fragmentation-diffusion model for trail dynamics Kawagoe, Kyle; Huber, Greg; Pradas, Marc Physical Review E, Vol. 96, Issue 1 https://doi.org/10.1103/PhysRevE.96.012142	journal	July 2017
Scaling Theory of Fragmentation Cheng, Z.; Redner, S. Physical Review Letters, Vol. 60, Issue 24 https://doi.org/10.1103/PhysRevLett.60.2450	journal	June 1988
Asymptotic results for the random sequential addition of unoriented objects Tarjus, G.; Viot, P. Physical Review Letters, Vol. 67, Issue 14 https://doi.org/10.1103/PhysRevLett.67.1875	journal	September 1991
Kinematic Scaling and Crossover to Scale Invariance in Martensite Growth Rao, Madan; Sengupta, Surajit; Sahu, H. K. Physical Review Letters, Vol. 75, Issue 11 https://doi.org/10.1103/PhysRevLett.75.2164	journal	September 1995
Comment on “Kinematic Scaling and Crossover to Scale Invariance in Martensite Growth” Ben-Naim, E.; Krapivsky, P. L. Physical Review Letters, Vol. 76, Issue 17 https://doi.org/10.1103/PhysRevLett.76.3234	journal	April 1996
Dynamic Buckling and Fragmentation in Brittle Rods Gladden, J. R.; Handzy, N. Z.; Belmonte, A. Physical Review Letters, Vol. 94, Issue 3 https://doi.org/10.1103/PhysRevLett.94.035503	journal	January 2005
Habitat fragmentation causes immediate and time-delayed biodiversity loss at different trophic levels: Immediate and time-delayed biodiversity loss Krauss, Jochen; Bommarco, Riccardo; Guardiola, Moisès Ecology Letters, Vol. 13, Issue 5 https://doi.org/10.1111/j.1461-0248.2010.01457.x	journal	April 2010
On the Distribution of the Sizes of Particles which Undergo Splitting Filippov, A. F. Theory of Probability & Its Applications, Vol. 6, Issue 3 https://doi.org/10.1137/1106036	journal	January 1961
Ecological Responses to Habitat Fragmentation Per Se Fahrig, Lenore Annual Review of Ecology, Evolution, and Systematics, Vol. 48, Issue 1 https://doi.org/10.1146/annurev-ecolsys-110316-022612	journal	November 2017
Kinetic description of coagulation and fragmentation in dilute granular particle ensembles Spahn, F.; Albers, N.; Sremčević, M. Europhysics Letters (EPL), Vol. 67, Issue 4 https://doi.org/10.1209/epl/i2003-10301-2	journal	August 2004
diamond Cohn, Henry; Elkies, Noam; Propp, James Duke Mathematical Journal, Vol. 85, Issue 1 https://doi.org/10.1215/S0012-7094-96-08506-3	journal	October 1996
Log-normal Distributions across the Sciences: Keys and Clues Limpert, Eckhard; Stahel, Werner A.; Abbt, Markus BioScience, Vol. 51, Issue 5 https://doi.org/10.1641/0006-3568(2001)051[0341:LNDATS]2.0.CO;2	journal	January 2001
Minimal Fragmentation of Regular Polygonal Plates Dias, Laercio; Parisio, Fernando arXiv https://doi.org/10.48550/arxiv.1404.6106	text	January 2014
Jamming and Tiling in Aggregation of Rectangles Ben-Naim, D. S.; Ben-Naim, E.; Krapivsky, P. L. arXiv https://doi.org/10.48550/arxiv.1808.03714	text	January 2018
Scaling and Multiscaling in Models of Fragmentation Krapivsky, P. L.; Ben-Naim, E. arXiv https://doi.org/10.48550/arxiv.cond-mat/9407084	text	January 1994

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