Diffusion on fractals with interacting internal boundaries

Abstract

We studied random walks interacting with the internal boundaries (borders of lacunas) of Sierpinski carpets (SC), which are infinitely ramified fractals with fractal dimensions DF between 1 and 2. The probability of steps along the borders is u=exp(−E/kBT) times the probability of steps in the bulk, where E<0 represents attraction and E>0 represents repulsion. The mean-square displacement 〈RN2〉 and the mean fraction of adsorbed steps 〈mN〉 of N-step walks were calculated using numerical simulations and extrapolations to the fractal limit considering the expected forms of finite-size corrections. The asymptotic fraction of adsorbed steps m (N→∞) is exactly calculated and confirms the accuracy of the simulations’ results. m varies continuously with u, and then there is no adsorption-desorption transition. As N→∞, 〈RN2〉≈AN2ν, thus we estimated the diffusion exponents νA and νR and the coefficients A(A) and A(R) in the attractive and in the repulsive cases, respectively, and the noninteracting exponent ν0. νR is universal and smaller than ν0, and A(R) decreases with the temperature. ν(A) is universal and larger than ν0 because the attraction helps the diffusing particles to contour the obstacles of the disordered substrate. In a SC with DF≈1.9, A(A) decreases with the temperature T, for all T; in a SC with DF≈1.7, A(A) is maximum for T∼|E|/kB, which consequently provides a condition for fastest diffusion. We suggest the investigation of these features in real systems.