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Langevin description of superdiffusive Lévy processes.

S Eule1, V Zaburdaev, R Friedrich

  • 1Max Planck Institute for Dynamics and Self-Organization, Göttingen, Germany.

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Summary

This study unifies random walk, Fokker-Planck, and Langevin equations for anomalous diffusion. It establishes a complementary framework for describing superdiffusive behavior in complex systems.

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Area of Science:

  • Physics
  • Statistical Mechanics
  • Complex Systems

Background:

  • Classical diffusion is well-understood across random walk, Fokker-Planck, and Langevin equation frameworks.
  • Anomalous diffusion, particularly superdiffusion, lacks a unified description across these methods.
  • Understanding the interplay between different diffusion models is crucial for complex systems.

Purpose of the Study:

  • To unify three distinct frameworks for describing anomalous superdiffusive behavior.
  • To establish the relationships and complementary nature of random walk, Fokker-Planck, and Langevin equations in anomalous diffusion.
  • To provide a more flexible and synergetic approach for modeling real-world systems exhibiting anomalous diffusion.

Main Methods:

  • Utilizing the method of subordination to connect different diffusion formalisms.
  • Demonstrating how Langevin equations can model anomalous diffusion in Lévy-walk-type systems.
  • Establishing the equivalence between random walk models and generalized Kramers-Fokker-Planck equations.

Main Results:

  • The Langevin equation is shown to describe anomalous diffusion characteristic of Lévy-walk models.
  • Equivalence is demonstrated between random walk models and the generalized Kramers-Fokker-Planck equation.
  • A synergetic and complementary description of anomalous diffusion is achieved.

Conclusions:

  • The study provides a unified footing for disparate anomalous diffusion models.
  • The developed framework offers enhanced flexibility for applications in diverse scientific and engineering fields.
  • This work bridges the gap between theoretical models and the empirical observation of anomalous diffusion.