Anisotropic anomalous diffusion modulated by log-periodic oscillations

We introduce finite ramified self-affine substrates in two dimensions with a set of appropriate hopping rates between nearest-neighbor sites, where the diffusion of a single random walk presents an anomalous {\it anisotropic} behavior modulated by log-periodic oscillations. The anisotropy is revealed by two different random walk exponents, $\nu_x$ and $\nu_y$, in the {\it x} and {\it y} direction, respectively. The values of these exponents, as well as the period of the oscillation, are analytically obtained and confirmed by Monte Carlo simulations.Comment: 7 pages, 7 figure

No self-similar aggregates with sedimentation

Two-dimensional cluster-cluster aggregation is studied when clusters move both diffusively and sediment with a size dependent velocity. Sedimentation breaks the rotational symmetry and the ensuing clusters are not self-similar fractals: the mean cluster width perpendicular to the field direction grows faster than the height. The mean width exhibits power-law scaling with respect to the cluster size, ~ s^{l_x}, l_x = 0.61 +- 0.01, but the mean height does not. The clusters tend to become elongated in the sedimentation direction and the ratio of the single particle sedimentation velocity to single particle diffusivity controls the degree of orientation. These results are obtained using a simulation method, which becomes the more efficient the larger the moving clusters are.Comment: 10 pages, 10 figure

Phase transitions in diluted negative-weight percolation models

We investigate the geometric properties of loops on two-dimensional lattice graphs, where edge weights are drawn from a distribution that allows for positive and negative weights. We are interested in the appearance of spanning loops of total negative weight. The resulting percolation problem is fundamentally different from conventional percolation, as we have seen in a previous study of this model for the undiluted case. Here, we investigate how the percolation transition is affected by additional dilution. We consider two types of dilution: either a certain fraction of edges exhibit zero weight, or a fraction of edges is even absent. We study these systems numerically using exact combinatorial optimization techniques based on suitable transformations of the graphs and applying matching algorithms. We perform a finite-size scaling analysis to obtain the phase diagram and determine the critical properties of the phase boundary. We find that the first type of dilution does not change the universality class compared to the undiluted case whereas the second type of dilution leads to a change of the universality class.Comment: 8 pages, 7 figure

Particle Survival and Polydispersity in Aggregation

We study the probability, $P_S(t)$, of a cluster to remain intact in one-dimensional cluster-cluster aggregation when the cluster diffusion coefficient scales with size as $D(s) \sim s^\gamma$. $P_S(t)$ exhibits a stretched exponential decay for $\gamma < 0$ and the power-laws $t^{-3/2}$ for $\gamma=0$, and $t^{-2/(2-\gamma)}$ for $0<\gamma<2$. A random walk picture explains the discontinuous and non-monotonic behavior of the exponent. The decay of $P_S(t)$ determines the polydispersity exponent, $\tau$, which describes the size distribution for small clusters. Surprisingly, $\tau(\gamma)$ is a constant $\tau = 0$ for $0<\gamma<2$.Comment: submitted to Europhysics Letter

Finite-element analysis of contact between elastic self-affine surfaces

Finite element methods are used to study non-adhesive, frictionless contact between elastic solids with self-affine surfaces. We find that the total contact area rises linearly with load at small loads. The mean pressure in the contact regions is independent of load and proportional to the rms slope of the surface. The constant of proportionality is nearly independent of Poisson ratio and roughness exponent and lies between previous analytic predictions. The contact morphology is also analyzed. Connected contact regions have a fractal area and perimeter. The probability of finding a cluster of area $a_c$ drops as $a_c^{-\tau}$ where $\tau$ increases with decreasing roughness exponent. The distribution of pressures shows an exponential tail that is also found in many jammed systems. These results are contrasted to simpler models and experiment.Comment: 13 pages, 15 figures. Replaced after changed in response to referee comments. Final two figures change

Dynamic Scaling in One-Dimensional Cluster-Cluster Aggregation

We study the dynamic scaling properties of an aggregation model in which particles obey both diffusive and driven ballistic dynamics. The diffusion constant and the velocity of a cluster of size $s$ follow $D(s) \sim s^\gamma$ and $v(s) \sim s^\delta$, respectively. We determine the dynamic exponent and the phase diagram for the asymptotic aggregation behavior in one dimension in the presence of mixed dynamics. The asymptotic dynamics is dominated by the process that has the largest dynamic exponent with a crossover that is located at $\delta = \gamma - 1$. The cluster size distributions scale similarly in all cases but the scaling function depends continuously on $\gamma$ and $\delta$. For the purely diffusive case the scaling function has a transition from exponential to algebraic behavior at small argument values as $\gamma$ changes sign whereas in the drift dominated case the scaling function decays always exponentially.Comment: 6 pages, 6 figures, RevTeX, submitted to Phys. Rev.

Level Crossing Analysis of Growing surfaces

We investigate the average frequency of positive slope $\nu_{\alpha}^{+}$, crossing the height $\alpha = h- \bar h$ in the surface growing processes. The exact level crossing analysis of the random deposition model and the Kardar-Parisi-Zhang equation in the strong coupling limit before creation of singularities are given.Comment: 5 pages, two column, latex, three figure

Controlling surface statistical properties using bias voltage: Atomic force microscopy and stochastic analysis

The effect of bias voltages on the statistical properties of rough surfaces has been studied using atomic force microscopy technique and its stochastic analysis. We have characterized the complexity of the height fluctuation of a rough surface by the stochastic parameters such as roughness exponent, level crossing, and drift and diffusion coefficients as a function of the applied bias voltage. It is shown that these statistical as well as microstructural parameters can also explain the macroscopic property of a surface. Furthermore, the tip convolution effect on the stochastic parameters has been examined.Comment: 8 pages, 11 figures

Stable propagation of an ordered array of cracks during directional drying

We study the appearance and evolution of an array of parallel cracks in a thin slab of material that is directionally dried, and show that the cracks penetrate the material uniformly if the drying front is sufficiently sharp. We also show that cracks have a tendency to become evenly spaced during the penetration. The typical distance between cracks is mainly governed by the typical distance of the pattern at the surface, and it is not modified during the penetration. Our results agree with recent experimental work, and can be extended to three dimensions to describe the properties of columnar polygonal patterns observed in some geological formations.Comment: 8 pages, 4 figures, to appear in PR