Anomalous Transport on Random Fractal Structures: Stretched Gaussians, Power‐Laws and Logarithmic Time Dependences

We study diffusion of Brownian particles on random fractal structures, self‐avoiding random walks and percolation clusters at criticality, which serve as model systems for polymers in a good solvent and random two‐component mixtures, respectively. Using numerical simulations and scaling arguments we find that the distribution function P ( r, t ) of the particles is a stretched Gaussian and scales as log[ P ( r, t )/ P ( r , 0)] ∼ —[ r/R ( t )] u , where R ( t ) ∼ t 1/dw is the root‐mean‐square displacement, u = d w /( d w ‐1), and d w = 2 d f for self‐avoiding walks and d w = 3 d f /2 for percolation; d f is the fractal dimension of the structure. In the presence of an external constant bias field, diffusion is drastically reduced and R ( t ) ∼ log t evolves logarithmically in time, in exactly the same way in both fractal structures.