(A) study of the random walk problem on 2-dimensional fractal structures

Abstract

Using the Monte Carlo method random walk on 2-dimensional Sierpinski gasket in the presence of an external field is studied. In the isotropic case the mean end-to-end distance satisfies the power law $R\sim{t}^k$ with the exponent k $\simeq$ 0.436, and the random walk motion of the particle for the rotationally anisotropic case also shows the same behavior as that for the isotropic case. In the uniaxially anisotropic case it is observed that the random walk motion of a particle displays a crossover from anomalous diffusion to drift for a non-zero bias field, and the crossover time $t_{cr}$ is a decreasing function of the external bias field. The associated dynamic exponents obtained in our computer experiment agree with the predictions of the Stinchcombe``s scaling treatment. In addition, the random walk on 2-dimensional directed diffusion-limited aggregation clusters (non-directed, weakly directed and rather strongly directed clusters) is studied. The fractal dimension($d_f=1.71$) and the spectral dimension($d_s=1.28$ or 1.16) have the same values for the three clusters. Therefore properties of the directed and non-directed fractal clusters are in the same universality class. However it seems that our results are inconsistent with the Alexander-Orbach conjecture($d_s=4/3$).