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Jamming and tiling in fragmentation of rectangles.

E Ben-Naim1, P L Krapivsky2

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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.

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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.