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Diffusion and relaxation controlled by tempered alpha-stable processes.

Aleksander Stanislavsky1, Karina Weron, Aleksander Weron

  • 1Institute of Radio Astronomy, 4 Chervonopraporna Street, 61002 Kharkov, Ukraine. alexstan@ri.kharkov.ua

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 31, 2008
PubMed
Summary
This summary is machine-generated.

We introduce tempered alpha-stable processes to explain anomalous diffusion and nonexponential relaxation. This model bridges subdiffusion and normal diffusion, yielding explicit Fokker-Planck and Cole-Davidson equations.

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Area of Science:

  • Physics
  • Mathematics
  • Statistical Mechanics

Background:

  • Anomalous diffusion and nonexponential relaxation are crucial phenomena in various scientific fields.
  • Existing models like subdiffusion (Cole-Cole law) and normal diffusion (exponential law) have limitations.
  • Tempered alpha-stable processes offer a potential framework to unify these behaviors.

Purpose of the Study:

  • To derive general properties of anomalous diffusion and nonexponential relaxation using tempered alpha-stable processes.
  • To obtain explicit forms of the Fokker-Planck equation and the Cole-Davidson relaxation function within this framework.
  • To demonstrate that the proposed model encompasses subdiffusion as a special case.

Main Methods:

  • Utilizing the theory of tempered alpha-stable processes.
  • Applying subordination by the inverse tempered alpha-stable process.
  • Deriving the Fokker-Planck equation and the Cole-Davidson relaxation function.

Main Results:

  • Tempering ensures the existence of all moments of operational time.
  • The subordination yields diffusion/relaxation intermediate between subdiffusion and normal diffusion.
  • Explicit expressions for the Fokker-Planck equation and Cole-Davidson relaxation function were obtained.

Conclusions:

  • The theory of tempered alpha-stable processes provides a unified framework for anomalous diffusion and nonexponential relaxation.
  • This model successfully bridges the gap between subdiffusion and normal diffusion.
  • The derived equations offer valuable tools for analyzing complex systems exhibiting these behaviors.