Multifractal properties of the random resistor network

Multifractal to monofractal evolution of the London street network

We perform a multifractal analysis of the evolution of London's street network from 1786 to 2010. First, we show that a single fractal dimension, commonly associated with the morphological description of cities, does not suffice to capture the dynamics of the system. Instead, for a proper characterization of such a dynamics, the multifractal spectrum needs to be considered. Our analysis reveals that London evolves from an inhomogeneous fractal structure, which can be described in terms of a multifractal, to a homogeneous one, which converges to monofractality. We argue that London's multifractal to monofractal evolution might be a special outcome of the constraint imposed on its growth by a green belt. Through a series of simulations, we show that multifractal objects, constructed through diffusion limited aggregation, evolve toward monofractality if their growth is constrained by a nonpermeable boundary.

Scaling behaviors of the voltage distribution in dielectric breakdown networks

We study the distribution of voltage drops across bonds in dielectric breakdown networks and its qth moments in the two-dimensional Euclidean space. Performing numerical simulations, we find that the distribution is composed of three different power-law regimes which are distinguished by two crossover voltages V(1) and V(2). The scaling behaviors of these crossover voltages with respect to the system size govern those of the qth moments of the voltage distribution. This feature differs from the multifractal behavior of the qth moment in random resistor networks. We discuss the implications of these scaling behaviors in relation to the application of the dielectric breakdown network to memory devices.

Advective transport in percolation clusters

We simulate advective transport in bond percolation clusters at the critical point. We compute the histogram of flow speeds in each bond of the backbone and find the multifractal spectrum for two-dimensional lattices with linear dimension L2000 and in three dimensions for L250 . We demonstrate that in the limit of large systems all the negative moments of the velocity distribution become ill-defined. However, to model transport, the velocity histogram should be weighted by the flux to obtain a well-defined mean travel time. Finally, we use continuous time random walk theory to demonstrate that anomalous transport is observed whose characteristics can be related to the multifractal properties of the system.

Postbreakthrough behavior in flow through porous media

We numerically simulate the traveling time of a tracer in convective flow between two points (injection and extraction) separated by a distance r in a model of porous media, d=2 percolation. We calculate and analyze the traveling time probability density function for two values of the fraction of connecting bonds p: the homogeneous case p=1 and the inhomogeneous critical threshold case p=p(c). We analyze both constant current and constant pressure conditions at p=p(c). The homogeneous p=1 case serves as a comparison base for the more complicated p=p(c) situation. We find several regions in the probability density of the traveling times for the homogeneous case (p=1) and also for the critical case (p=p(c)) for both constant pressure and constant current conditions. For constant pressure, the first region I(P) corresponds to the short times before the flow breakthrough occurs, when the probability distribution is strictly zero. The second region II(P) corresponds to numerous fast flow lines reaching the extraction point, with the probability distribution reaching its maximum. The third region III(P) corresponds to intermediate times and is characterized by a power-law decay. The fourth region IV(P) corresponds to very long traveling times, and is characterized by a different power-law decaying tail. The power-law characterizing region IV(P) is related to the multifractal properties of flow in percolation, and an expression for its dependence on the system size L is presented. The constant current behavior is different from the constant pressure behavior, and can be related analytically to the constant pressure case. We present theoretical arguments for the values of the exponents characterizing each region and crossover times. Our results are summarized in two scaling assumptions for the traveling time probability density; one for constant pressure and one for constant current. We also present the production curve associated with the probability of traveling times, which is of interest to oil recovery.

Self-similar Gaussian processes for modeling anomalous diffusion

We study some Gaussian models for anomalous diffusion, which include the time-rescaled Brownian motion, two types of fractional Brownian motion, and models associated with fractional Brownian motion based on the generalized Langevin equation. Gaussian processes associated with these models satisfy the anomalous diffusion relation which requires the mean-square displacement to vary with t(alpha), 0<alpha<2. However, these processes have different properties, thus indicating that the anomalous diffusion relation with a single parameter is insufficient to characterize the underlying mechanism. Although the two versions of fractional Brownian motion and time-rescaled Brownian motion all have the same probability distribution function, the Slepian theorem can be used to compare their first passage time distributions, which are different. Finally, in order to model anomalous diffusion with a variable exponent alpha(t) it is necessary to consider the multifractional extensions of these Gaussian processes.

Multifractal current distribution in random-diode networks

Recently it has been shown analytically that electric currents in a random-diode network are distributed in a multifractal manner [O. Stenull and H. K. Janssen, Europhys. Lett. 55, 691 (2001)]. In the present paper we investigate the multifractal properties of a random diode network at the critical point by numerical simulations. We analyze the currents running on a directed percolation cluster and confirm the field-theoretic predictions for the scaling behavior of moments of the current distribution. It is pointed out that a random diode network is a particularly good candidate for a possible experimental realization of directed percolation.

Noisy random resistor networks: renormalized field theory for the multifractal moments of the current distribution

We study the multifractal moments of the current distribution in randomly diluted resistor networks near the percolation threshold. When an external current is applied between two terminals x and x(') of the network, the lth multifractal moment scales as M((l))(I)(x,x(')) approximately equal /x-x'/(psi(l)/nu), where nu is the correlation length exponent of the isotropic percolation universality class. By applying our concept of master operators [Europhys. Lett. 51, 539 (2000)] we calculate the family of multifractal exponents [psi(l)] for l>or=0 to two-loop order. We find that our result is in good agreement with numerical data for three dimensions.

Flow between two sites on a percolation cluster

We study the flow of fluid in porous media in dimensions d=2 and 3. The medium is modeled by bond percolation on a lattice of L(d) sites, while the flow front is modeled by tracer particles driven by a pressure difference between two fixed sites ("wells") separated by Euclidean distance r. We investigate the distribution function of the shortest path connecting the two sites, and propose a scaling ansatz that accounts for the dependence of this distribution (i) on the size of the system L and (ii) on the bond occupancy probability p. We confirm by extensive simulations that the ansatz holds for d=2 and 3. Further, we study two dynamical quantities: (i) the minimal traveling time of a tracer particle between the wells when the total flux is constant and (ii) the minimal traveling time when the pressure difference is constant. A scaling ansatz for these dynamical quantities also includes the effect of finite system size L and off-critical bond occupation probability p. We find that the scaling form for the distribution functions for these dynamical quantities for d=2 and 3 is similar to that for the shortest path, but with different critical exponents. Our results include estimates for all parameters that characterize the scaling form for the shortest path and the minimal traveling time in two and three dimensions; these parameters are the fractal dimension, the power law exponent, and the constants and exponents that characterize the exponential cutoff functions.