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Fractional Langevin equation.

E Lutz1

  • 1Département de Physique Théorique, Université de Genève, 24 quai Ernest Ansermet, 1211 Genève 4, Switzerland.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 12, 2001
PubMed
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We introduce a fractional Langevin equation to study anomalous diffusion. Our findings reveal a simple criterion to distinguish fractional Brownian motion from fractal time processes based on their velocity moments.

Area of Science:

  • Statistical Physics
  • Complex Systems
  • Non-equilibrium Dynamics

Background:

  • Fractional Brownian motion and fractal time processes describe anomalous diffusion.
  • Understanding the distinctions between these non-Markovian processes is crucial for modeling complex systems.
  • Microscopic models and Langevin equations provide frameworks for studying diffusion dynamics.

Purpose of the Study:

  • To investigate fractional Brownian motion using a microscopic random-matrix model.
  • To introduce and utilize a fractional Langevin equation for analyzing subdiffusion and superdiffusion.
  • To compare fractional Brownian motion with the fractal time process and identify distinguishing features.

Main Methods:

  • Development of a microscopic random-matrix model for fractional Brownian motion.

Related Experiment Videos

  • Introduction and application of a fractional Langevin equation.
  • Analysis of subdiffusion and superdiffusion of a free particle coupled to a fractal heat bath.
  • Comparison of mean-square displacements and lower moments of fractional Brownian motion and fractal time processes.

    • Main Results:

    • Both fractional Brownian motion and fractal time processes exhibit power-law behavior in their mean-square displacements.
    • The lowest moments of these two anomalous diffusion processes are identical, with the exception of the second velocity moment.
    • A distinct criterion is identified to differentiate between these non-Markovian processes.

    Conclusions:

    • The fractional Langevin equation effectively models anomalous diffusion phenomena.
    • The second velocity moment serves as a key differentiator between fractional Brownian motion and fractal time processes.
    • This research provides a valuable tool for distinguishing between different types of anomalous diffusion in complex systems.