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Diffusion equations for a Markovian jumping process

T. Srokowski and A. Kamińska

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

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Phys. Rev. E 74, 021103 – Published 4 August, 2006

DOI: https://doi.org/10.1103/PhysRevE.74.021103

Abstract

We consider a Markovian jumping process which is defined in terms of the jump-size distribution and the waiting-time distribution with a position-dependent frequency, in the diffusion limit. We assume the power-law form for the frequency. For small steps, we derive the Fokker-Planck equation and show the presence of the normal diffusion, subdiffusion, and superdiffusion. For the Lévy distribution of the step size, we construct a fractional equation, which possesses a variable coefficient, and solve it in the diffusion limit. Then we calculate fractional moments and define the fractional diffusion coefficient as a natural extension to the cases with the divergent variance. We also solve the master equation numerically and demonstrate that there are deviations from the Lévy stable distribution for large wave numbers.

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Phys. Rev. E 74, 021103– Published 4 August, 2006
  • Received 28 April 2006

© 2006 American Physical Society

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