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Relation between generalized diffusion equations and subordination schemes.

A Chechkin1, I M Sokolov2

  • 1Institute of Physics and Astronomy, Potsdam University, Karl-Liebknecht-Strasse 24/25, 14476 Potsdam-Golm, Germany and Akhiezer Institute for Theoretical Physics, Akademicheskaya Strasse 1, 61108 Kharkow, Ukraine.

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Summary
This summary is machine-generated.

This study explores generalized diffusion equations and subordination schemes, clarifying when these mathematical tools accurately describe non-Fickian diffusion processes across various scientific fields.

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Area of Science:

  • Physics
  • Biology
  • Earth Sciences
  • Mathematical Physics

Background:

  • Generalized diffusion equations and subordination schemes are key for modeling non-Fickian diffusion.
  • These models are applied across physics, biology, and earth sciences.
  • Some processes fit both descriptions, while others fit only one.

Purpose of the Study:

  • To establish conditions for the equivalence between generalized diffusion equations and subordination schemes.
  • To identify processes uniquely described by one method over the other.
  • To analyze the applicability of both descriptive frameworks.

Main Methods:

  • Analysis of generalized (non-Markovian) diffusion equations with memory kernels.
  • Investigation of subordination schemes based on random time changes in Brownian diffusion.
  • Comparative study of different random processes and their descriptive models.

Main Results:

  • Conditions are defined for a generalized diffusion equation to correspond to a subordination scheme.
  • Conditions are established for a subordination scheme to possess a corresponding generalized diffusion equation.
  • Examples illustrate processes uniquely suited to one descriptive approach.

Conclusions:

  • The relationship between generalized diffusion equations and subordination schemes is clarified.
  • The study provides criteria for selecting appropriate models for non-Fickian diffusion.
  • Understanding these equivalences enhances the modeling of complex diffusion phenomena.