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Numerical study of diffusion on a random-mixed-bond lattice.

Devora Holder1, Harvey Scher, Brian Berkowitz

  • 1Department of Environmental Sciences and Energy Research, Weizmann Institute of Science, Rehovot, Israel. devora.holder@weizmann.ac.il

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
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Summary

This study investigates anomalous superdiffusion in disordered lattices using a random walk algorithm. Below the percolation threshold, diffusion exhibits time-dependent behavior, indicating superdiffusion for specific bond configurations.

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Area of Science:

  • Physics
  • Statistical Mechanics
  • Condensed Matter Physics

Background:

  • Diffusion on disordered lattices is crucial for understanding transport phenomena.
  • Previous studies focused on pure percolation, leaving gaps in understanding mixed-bond systems.
  • The master equation and random walk (RW) algorithms are standard tools for diffusion analysis.

Purpose of the Study:

  • To investigate diffusion on lattices with random mixed bonds in two and three dimensions.
  • To analyze the anomalous diffusion behavior below the percolation threshold.
  • To characterize the time-dependent diffusion constant and its relation to system parameters.

Main Methods:

  • Utilized a random walk (RW) algorithm, equivalent to the master equation.
  • Focused on lattices with two distinct transition rates (W(1), W(2)).
  • Analyzed the mean square displacement (r^2) as a function of time (t) for varying probabilities (p) and rate ratios (Delta).

Main Results:

  • For probabilities below the percolation threshold (p < p(c)) and Delta > 1, diffusion exhibits anomalous superdiffusion (eta > 0).
  • The mean square displacement is a nonlinear function of time, r^2 proportional to t^(1+eta).
  • The anomalous diffusion behavior becomes more pronounced with increased time regimes and parameter ranges.

Conclusions:

  • Diffusion on random mixed-bond lattices can be anomalous superdiffusion under specific conditions.
  • The observed superdiffusion is linked to percolation cluster geometry and Lévy walks.
  • The RW algorithm provides a robust method for revealing the time-dependent nature of diffusion in disordered systems.