Multifractal scaling in Sinai diffusion (original) (raw)

Diffusion on two-dimensional random walks

Physical Review Letters, 1987

Analysis of Monte Carlo enumerations for diAusion on the fractal structure generated by the random walk on a two-dimensional lattice allows us to predict a behavior &r)n "(1nn)' with v=0. 325~0.01 and a =0.35~0.03. This leads to the conjecture that v=a = -, ' . This value of v, and the presence of logarithmic corrections, are strongly supported by heuristic arguments based on Flory theory and on plausible assumptions.

Generalization of the Sinai Anomalous Diffusion Law

Sinai has considered a novel one-dimensional walk with a random bias field E on each site. He has shown that when the field is taken with equal probability to be +E,, or -E , the R M S displacement R [(x2)]"' increases with time t by the anomalously slow law R -(log t)'. Here we introduce long-range correlation between the random fields on each site by choosing a 'string' of k sites to have the same value of E, where k is chosen from the power law distribution P ( k ) = k-O. We find that the Sinai law is generalised to the form R -(log I ) ' , where J sticks at the Sinai value y = 2 for p 2. However, for 1 < p < 2, y varies continuously with p as y = p / ( p -1). We interpret this result physically

Biased diffusion in percolation systems: indication of multifractal behaviour

Journal of Physics A: Mathematical and General, 1987

We study diffusion in percolation systems at criticality in the presence of a constant bias field E. Using the exact enumeration method we show that the mean displacement of a random walker varies as (r(f))-log f / A (E) where A (€) = I n [ (l + E) / (l-E) ] for small E. More generally, diffusion on a given configuration is characterised by the probability P (r , t) that the random walker is on site r at time t. We find that the corresponding configurational average shows simple scaling behaviour and is described by a single exponent. In contrast, our numerical results indicate that the averaged moments (P q (t)) = (Z , P 4 (r , t)) are described by an infinite hierarchy of exponents. For zero bias field, however, all moments are determined by a single gap exponent.

Where two fractals meet: The scaling of a self-avoiding walk on a percolation cluster

Physical Review E, 2004

The scaling properties of self-avoiding walks on a d-dimensional diluted lattice at the percolation threshold are analyzed by a field-theoretical renormalization group approach. To this end we reconsider the model of Y. Meir and A. B. Harris (Phys. Rev. Lett., 63, 2819 (1989)) and argue that via renormalization its multifractal properties are directly accessible. While the former first order perturbation did not agree with the results of other methods, we find that the asymptotic behavior of a self-avoiding walk on the percolation cluster is governed by the exponent νp = 1/2+ε/42+110ε 2 /21 3 , ε = 6 − d. This analytic result gives an accurate numeric description of the available MC and exact enumeration data in a wide range of dimensions 2 ≤ d ≤ 6.

Diffusion in one-dimensional multifractal porous media

1998

We examine the scaling properties of one-dimensional random walks on media with multifractal diffusivities, which is a simple model for transport in scaling porous media. We find both theoretically and numerically that the anomalous scaling exponent of the walk is d w ϭ 2 ϩ K(Ϫ1) where K(Ϫ1) is the scaling exponent of the reciprocal spatially averaged ("dressed") resistance to diffusion. Since K(Ϫ1) Ͼ 0, the walk is subdiffusive; the walkers are effectively trapped in a hierarchy of barriers. The trapping is dominated by contributions from a specific order of singularity associated with a phase transition between anomalous and normal diffusion. We discuss the implications for transport in porous media.

Biased random walk on a deterministic fractal

Physical Review A, 1990

Anisotropic diffusion on a Sierpi'nski gasket is studied using renormalization equations for the probability densities of waiting times. The effect of an external field on a random walk is described in terms of the dependence of hopping probabilities, mean waiting times, and standard deviations on the hopping distance and on the intensity of the field. Our procedure can also be applied to other deterministic fractals.

Multifractal spectra of mean first-passage-time distributions in disordered chains

Physical Review E, 2003

The multifractal characterization of the distribution over disorder of the mean first-passage time in a finite chain is revisited. Both, absorbing-absorbing and reflecting-absorbing boundaries are considered. Two models of dichotomic disorder are compared and our analysis clarifies the origin of the multifractality. The phenomenon is only present when the diffusion is anomalous.

Fractal to Euclidean crossover and scaling for random walks on percolation clusters. II. Three-dimensional lattices

The Journal of Chemical Physics, 1985

We perform random walk simulations on binary three-dimensional simple cubic lattices covering the entire ratio of open/closed sites (fractionp) from the critical percolation threshold to the perfect crystal. We observe fractal behavior at the critical point and derive the value of the number-of-sites-visited exponent, in excellent agreement with previous work or conjectures, but with a new and imprOVed computational algorithm that extends the calculation to the long time limit. We show the crossover to the classical Euclidean behavior in these lattices and discuss its onset as a function of the fractionp. We compare the observed trends with the two-dimensional case.