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Transient anomalous diffusion with Prabhakar-type memory.

Aleksander Stanislavsky1, Aleksander Weron2

  • 1Institute of Radio Astronomy, Ukrainian National Academy of Sciences, 4 Mystetstv St., 61002 Kharkiv, Ukraine.

The Journal of Chemical Physics
|August 3, 2018
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Summary
This summary is machine-generated.

This study explores anomalous diffusion and relaxation using a generalized fractional integral operator within the Fokker-Planck equation. Findings reveal changing time scaling exponents, offering a more comprehensive view of natural diffusion processes.

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Area of Science:

  • Physics
  • Mathematics
  • Physical Chemistry

Background:

  • Anomalous diffusion and non-exponential relaxation are complex phenomena observed in various natural systems.
  • Existing models often struggle to capture the transient and multi-scale characteristics of these processes.

Purpose of the Study:

  • To derive general properties of anomalous diffusion and non-exponential relaxation.
  • To develop a probabilistic description of anomalous diffusion using a generalized fractional integral operator.
  • To analyze the temporal behavior and scaling exponents of mean-squared displacement.

Main Methods:

  • Utilizing the Fokker-Planck equation with a memory function.
  • Employing the Prabhakar integral operator, a generalization of the Riemann-Liouville fractional integral.
  • Analyzing transient anomalous diffusion with two-scale features.

Main Results:

  • The study derives general properties of anomalous diffusion and non-exponential relaxation.
  • The Prabhakar integral operator enables the study of transient anomalous diffusion with two-scale features.
  • The temporal behavior shows changes in time scaling exponents of mean-squared displacement.

Conclusions:

  • The developed framework provides a more general probabilistic description of anomalous diffusion.
  • This approach offers a deeper understanding of complex diffusion dynamics observed in nature.
  • The findings are applicable to systems exhibiting anomalous diffusion and non-exponential relaxation.