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Anomalous diffusion, stable processes, and generalized functions.

Barry D Hughes1

  • 1Department of Mathematics and Statistics, University of Melbourne, Victoria 3010, Australia.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|March 23, 2002
PubMed
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This study examines anomalous diffusion processes and derives evolution equations for stable processes using generalized functions. Novel interpretations of known results and a specific evolution law for unity-order stable processes are presented.

Area of Science:

  • Mathematical Physics
  • Stochastic Processes
  • Non-Fickian Transport

Background:

  • Anomalous diffusion processes deviate from standard Fickian diffusion.
  • Understanding their evolution is crucial for modeling complex systems.
  • Existing theories often lack a unified framework for diverse anomalous behaviors.

Purpose of the Study:

  • To derive and analyze evolution equations for a class of anomalous diffusion processes.
  • To investigate special cases, specifically stable processes.
  • To provide a new interpretation of known results within generalized function theory.

Main Methods:

  • Analysis of evolution equations in real space and time.
  • Application of generalized function theory.

Related Experiment Videos

  • Derivation of specific cases for stable processes.

    • Main Results:

    • Established evolution equations for anomalous diffusion.
    • Derived evolution equations for stable processes, including order unity.
    • Recovered and reinterpreted known results using a novel theoretical approach.

    Conclusions:

    • The generalized function theory provides a robust framework for anomalous diffusion.
    • The derived equations offer new insights into the behavior of stable processes.
    • This work unifies the understanding of certain anomalous diffusion phenomena.