19 research outputs found
Random walk through fractal environments
We analyze random walk through fractal environments, embedded in
3-dimensional, permeable space. Particles travel freely and are scattered off
into random directions when they hit the fractal. The statistical distribution
of the flight increments (i.e. of the displacements between two consecutive
hittings) is analytically derived from a common, practical definition of
fractal dimension, and it turns out to approximate quite well a power-law in
the case where the dimension D of the fractal is less than 2, there is though
always a finite rate of unaffected escape. Random walks through fractal sets
with D less or equal 2 can thus be considered as defective Levy walks. The
distribution of jump increments for D > 2 is decaying exponentially. The
diffusive behavior of the random walk is analyzed in the frame of continuous
time random walk, which we generalize to include the case of defective
distributions of walk-increments. It is shown that the particles undergo
anomalous, enhanced diffusion for D_F < 2, the diffusion is dominated by the
finite escape rate. Diffusion for D_F > 2 is normal for large times, enhanced
though for small and intermediate times. In particular, it follows that
fractals generated by a particular class of self-organized criticality (SOC)
models give rise to enhanced diffusion. The analytical results are illustrated
by Monte-Carlo simulations.Comment: 22 pages, 16 figures; in press at Phys. Rev. E, 200