Langevin description of superdiffusive Lévy processes.

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.