Bulletin of the Korean Chemical Society

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Fractional Diffusion Equation Approach to the Anomalous Diffusion on Fractal Lattices

  • Huh, Dann (Department of Chemistry, Seoul National University) ;
  • Lee, Jin-Uk (Department of Chemistry, Seoul National University) ;
  • Lee, Sang-Youb (Department of Chemistry, Seoul National University)
  • Published : 2005.11.20
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Abstract

A generalized fractional diffusion equation (FDE) is presented, which describes the time-evolution of the spatial distribution of a particle performing continuous time random walk (CTRW) on a fractal lattice. For a case corresponding to the CTRW with waiting time distribution that behaves as $\psi(t) \sim (t) ^{-(\alpha+1)}$, the FDE is solved to give analytic expressions for the Green’s function and the mean squared displacement (MSD). In agreement with the previous work of Blumen et al. [Phys. Rev. Lett. 1984, 53, 1301], the time-dependence of MSD is found to be given as < $r^2(t)$ > ~ $t ^{2\alpha/dw}$, where $d_w$ is the walk dimension of the given fractal. A Monte-Carlo simulation is also performed to evaluate the range of applicability of the proposed FDE.

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References

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