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Crossover in diffusion equation: anomalous and normal behaviors.

E K Lenzi1, R S Mendes, C Tsallis

  • 1Departamento de Física, Universidade Estadual de Maringá, Avenida Colombo 5790, 87020-900 Maringá-Paraná, Brazil.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|April 12, 2003
PubMed
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This study analyzes diffusion crossovers using a generalized Fokker-Planck equation. It reveals transitions between anomalous and normal diffusion regimes in various scenarios, offering insights into complex natural phenomena.

Area of Science:

  • Physics
  • Statistical Mechanics
  • Mathematical Physics

Background:

  • Natural phenomena often exhibit transitions between different diffusion behaviors over time.
  • Understanding these crossovers is crucial for modeling complex systems.

Purpose of the Study:

  • To analyze crossover phenomena between anomalous and normal diffusion regimes.
  • To investigate specific cases of the generalized Fokker-Planck equation.

Main Methods:

  • Analysis of the generalized Fokker-Planck equation with custom kernels.
  • Detailed examination of three specific cases: porous medium with normal diffusion, fractional time derivative with normal diffusion, and fractional space derivative with normal diffusion.

Main Results:

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  • All three analyzed cases demonstrate a crossover between anomalous and normal diffusion.
  • Cases (i) and (iii) show a transition from less to more diffusive regimes over time.
  • Case (ii) exhibits the opposite transition, from more to less diffusive regimes.

    • Conclusions:

    • The generalized Fokker-Planck equation effectively models diffusion crossovers.
    • Results are applicable to phenomena involving nonlinear and fractional diffusion, such as active intracellular transport and long-ranged interacting systems.
    • The framework can be extended to analyze more complex crossover scenarios.