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This study uses effective-medium approximation (EMA) to model random walks with disorder. It finds that Lévy-flights can emerge, explaining anomalous diffusion in complex systems.

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Area of Science:

  • Statistical Physics
  • Condensed Matter Physics
  • Stochastic Processes

Background:

  • Random walks are fundamental models for diffusion.
  • Disordered systems and long-range jumps lead to complex dynamics.
  • Anomalous diffusion deviates from standard Brownian motion.

Purpose of the Study:

  • To apply effective-medium approximation (EMA) to random walks with disordered transition rates and long-range jumps.
  • To understand the coarse-grained dynamics of such systems.
  • To identify conditions leading to anomalous diffusion.

Main Methods:

  • Employing Bruggeman's effective-medium approximation (EMA).
  • Self-consistently replacing the disordered system with a reference system.
  • Analyzing power-law scaling of transition rates with distance.

Main Results:

  • The EMA provides a method to coarse-grain disordered random walk dynamics.
  • Lattice variants of Lévy-flights emerge as the effective medium under specific conditions.
  • Analytical solutions for effective anomalous diffusivity were derived.

Conclusions:

  • The study successfully models anomalous diffusion in disordered lattices using EMA.
  • Lévy-flight-like behavior is a key mechanism for anomalous diffusion in these systems.
  • The findings show good agreement with numerical simulations.