Transfer across Random versus Deterministic Fractal Interfaces

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 o...

We derive equation describing distribution of energy losses of the particle propagating in fractal medium with quenched and dynamic heterogeneities. We show that in the case of the medium with fractal dimension $2<D<3$ the losses of energy are described by the Mittag-Leffler renewal process. The average energy loss of the particle experiences anomalous drift $\Delta\sim x^{\alpha}$ with power-law dependence on the distance $x$ from the surface and exponent $\alpha=D-2$.

Fractal and fractal-rate stochastic point processes (FSPPs and FRSPPs) provide useful models for describing a broad range of diverse phenomena, including electron transport in amorphous semiconductors, computer-network traffic, and sequences of neuronal action potentials. A particularly useful statistic of these processes is the fractal exponent $\alpha$, which may be estimated for any FSPP or FRSPP by using a variety of statistical methods. Simulated FSPPs and FRSPPs consist...

The `plate tectonics' is an observed fact and most models of earthquake incorporate that through the frictional dynamics (stick-slip) of two surfaces where one surface moves over the other. These models are more or less successful to reproduce the well known Gutenberg-Richter type power law in the (released) energy distribution of earthquakes. During sticking period, the elastic energy gets stored at the contact area of the surfaces and is released when a slip occurs. Therefo...

This paper proposes a simple model of anomalous diffusion, in which a particle moves with the velocity field induced by a single "dipole" (a doublet or a pair of source and sink), whose moment is modulated randomly at each time step. A motivation to introduce such a model is that it may serve as a toy model to investigate an anomalous diffusion of fluid particles in turbulence. We perform a numerical simulation of the fractal dimension of the trajectory using periodic boundar...

Fully Developed Turbulence (FDT) is a theoretical asymptotic phenomenon which can only be approximated experimentally or computationally, so its defining characteristics are hypothetical. It is considered to be a chaotic stationary flow field, with self-similar fractalline features. A number of approximate models exist, often exploiting this self-similarity. The idealized mathematical model of Fractal Potential Flows is hereby presented, and linked philosophically to the phen...

When a finite volume of an etching solution comes in contact with a disordered solid, a complex dynamics of the solid-solution interface develops. Since only the weak parts are corroded, the solid surface hardens progressively. If the etchant is consumed in the chemical reaction, the corrosion dynamics slows down and stops spontaneously leaving a fractal solid surface, which reveals the latent percolation criticality hidden in any random system. Here we introduce and study, b...

A computer simulation technique, suited to replicate real adsorption experiments, was applied to pure simulated silica in order to gain insight into the fractal regime of its surface. The previously reported experimental fractal dimension was closely approached and the hitherto uncharted lower limit of fractal surface behaviour is reported herein.

In many low-dimensional dynamical systems transport coefficients are very irregular, perhaps even fractal functions of control parameters. To analyse this phenomenon we study a dynamical system defined by a piece-wise linear map and investigate the dependence of transport coefficients on the slope of the map. We present analytical arguments, supported by numerical calculations, showing that both the Minkowski-Bouligand and Hausdorff fractal dimension of the graphs of these fu...

We investigate a contaminant transport in fractal media with randomly inhomogeneous diffusion barrier. The diffusion barrier is a low permeable matrix with extremely rare high permeability pathways (punctures). At times, less than a characteristic matrix diffusion time, the problem is effectively barrier-free with an effective source acting during the time t<<teff. The punctures result in a precursor contaminant concentration at short times and additional stage of the asympto...