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Jamming and tiling in fragmentation of rectangles.
1Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA.
Physical Review. E
|October 24, 2019
Summary
This study models rectangle fragmentation, revealing that discrete sizes lead to a jammed state of minimal-width sticks. The number of sticks scales with rectangle area, and their length distribution exhibits power-law behavior, independent of aspect ratio.
Area of Science:
- Statistical Mechanics
- Materials Science
- Fracture Mechanics
Background:
- Investigating fragmentation processes is crucial for understanding material failure and granular dynamics.
- Previous models often assume continuous variables, limiting applicability to systems with discrete fragmentation constraints.
Purpose of the Study:
- To analyze a stochastic fragmentation process of rectangles with discrete dimensions.
- To characterize the resulting 'jammed state' composed of minimal-width 'sticks'.
- To examine the influence of asymmetric fragmentation probabilities on system behavior.
Main Methods:
- Stochastic modeling of fragmentation events (horizontal and vertical).
- Analysis of discrete variable constraints leading to a jammed state.
- Derivation of scaling laws for the number of sticks and analysis of stick length distributions.
Main Results:
- The average number of sticks (S) in the jammed state scales with rectangle area (A) as S ≃ A/sqrt[2πlnA] in the large-area limit.
- Stick length distribution exhibits a power-law tail with nonlinear scaling exponents.
- Asymmetric fragmentation introduces a phase transition affecting length distribution dependence on system size.
Conclusions:
- Discrete fragmentation constraints lead to a universal jammed state of sticks, irrespective of aspect ratio.
- The system's behavior under asymmetric fragmentation is characterized by a phase transition, altering statistical properties.
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