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Erosion by a one-dimensional random walk.

Rebecca H Chisholm1, Barry D Hughes1, Kerry A Landman1

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Summary
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This study models a random walker that expands its allowed territory by converting blocked sites into allowed ones. Researchers derived exact results for the walker's position and the expansion rate.

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Area of Science:

  • Statistical Physics
  • Stochastic Processes
  • Computational Physics

Background:

  • Lattice random walks are fundamental models in statistical physics.
  • Understanding domain expansion dynamics is crucial in various fields, including material science and biology.
  • Previous models often lack the continuous interpolation between different random walker behaviors.

Purpose of the Study:

  • To analyze a novel random walker model that erodes boundaries.
  • To investigate the continuous interpolation between Pólya and "blind" walkers.
  • To derive exact results for walker location and erosion rate.

Main Methods:

  • A single lattice random walker model was employed.
  • The walker expands its domain by converting adjacent blocked sites to allowed sites with a specific probability.
  • Mathematical analysis was used to obtain exact results.

Main Results:

  • Exact results were obtained for the walker's position over time.
  • The rate of domain erosion was precisely determined.
  • The model successfully interpolates between known random walker behaviors.

Conclusions:

  • The proposed random walker model provides a new framework for studying domain expansion.
  • The derived exact results offer valuable insights into the dynamics of boundary erosion.
  • This model bridges the gap between different types of lattice random walkers.