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Diffusion equations for a Markovian jumping process.

T Srokowski1, A Kamińska

  • 1Institute of Nuclear Physics, Polish Academy of Sciences, PL-31-342 Kraków, Poland.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|October 10, 2006
PubMed
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This study explores Markovian jumping processes, revealing normal, subdiffusion, and superdiffusion behaviors. Fractional equations and numerical solutions analyze Lévy distributions and diffusion coefficients.

Area of Science:

  • Statistical Physics
  • Stochastic Processes
  • Anomalous Diffusion

Background:

  • Markovian jumping processes are fundamental in modeling systems with random transitions.
  • Understanding diffusion dynamics, especially anomalous diffusion, is crucial in various scientific fields.
  • Position-dependent frequencies and power-law behavior introduce complexity to standard diffusion models.

Purpose of the Study:

  • To investigate the diffusion limit of a Markovian jumping process with position-dependent frequency and power-law form.
  • To derive and analyze the Fokker-Planck and fractional equations governing normal, sub-, and superdiffusion.
  • To define and calculate a fractional diffusion coefficient for systems with divergent variance.

Main Methods:

  • Derivation of the Fokker-Planck equation for small steps.

Related Experiment Videos

  • Construction and solution of a fractional equation for Lévy distributed step sizes.
  • Numerical solution of the master equation.
  • Calculation of fractional moments.

    • Main Results:

    • Demonstration of normal diffusion, subdiffusion, and superdiffusion phenomena.
    • Solution of a fractional equation with a variable coefficient in the diffusion limit.
    • Definition and calculation of a fractional diffusion coefficient.
    • Observation of deviations from Lévy stable distribution for large wave numbers in numerical simulations.

    Conclusions:

    • The study successfully characterizes different diffusion regimes (normal, sub-, super-) within a generalized Markovian jumping process.
    • Fractional equations provide a suitable framework for analyzing anomalous diffusion, particularly with Lévy-type step sizes.
    • The proposed fractional diffusion coefficient offers a robust measure for systems exhibiting divergent variances.
    • Numerical results highlight the limitations of stable Lévy distributions in describing complex stochastic processes at higher wave numbers.