New method to study stochastic growth equations: a cellular automata perspective

We introduce a new method based on cellular automata dynamics to study stochastic growth equations. The method defines an interface growth process which depends on height differences between neighbors. The growth rule assigns a probability $p_{i}(t)=\rho$ exp$[\kappa \Gamma_{i}(t)]$ for a site $i$ to receive one particle at a time $t$ and all the sites are updated simultaneously. Here $\rho$ and $\kappa$ are two parameters and $\Gamma_{i}(t)$ is a function which depends on height of the site $i$ and its neighbors. Its functional form is specified through discretization of the deterministic part of the growth equation associated to a given deposition process. In particular, we apply this method to study two linear equations - the Edwards-Wilkinson (EW) equation and the Mullins-Herring (MH) equation - and a non-linear one - the Kardar-Parisi-Zhang (KPZ) equation. Through simulations and statistical analysis of the height distributions of the profiles, we recover the values for roughening exponents, which confirm that the processes generated by the method are indeed in the universality classes of the original growth equations. In addition, a crossover from Random Deposition to the associated correlated regime is observed when the parameter $\kappa$ is varied.Comment: 6 pages, 7 figure

Scaling Behavior of Driven Interfaces Above the Depinning Transition

We study the depinning transition for models representative of each of the two universality classes of interface roughening with quenched disorder. For one of the universality classes, the roughness exponent changes value at the transition, while the dynamical exponent remains unchanged. We also find that the prefactor of the width scales with the driving force. We propose several scaling relations connecting the values of the exponents on both sides of the transition, and discuss some experimental results in light of these findings.Comment: Revtex 3.0, 4 pages in PRL format + 5 figures (available at ftp://jhilad.bu.edu/pub/abbhhss/ma-figures.tar.Z ) submitted to Phys Rev Let

A Northern White Family to Mr. Meredith (8 October 1962)

https://egrove.olemiss.edu/mercorr_pro/2063/thumbnail.jp

New set of measures to analyze non-equilibrium structures

We introduce a set of statistical measures that can be used to quantify non-equilibrium surface growth. They are used to deduce new information about spatiotemporal dynamics of model systems for spinodal decomposition and surface deposition. Patterns growth in the Cahn-Hilliard Equation (used to model spinodal decomposition) are shown to exhibit three distinct stages. Two models of surface growth, namely the continuous Kardar-Parisi-Zhang (KPZ) model and the discrete Restricted-Solid-On-Solid (RSOS) model are shown to have different saturation exponents

Mathematics of random growing interfaces

We establish a thermodynamic limit and Gaussian fluctuations for the height and surface width of the random interface formed by the deposition of particles on surfaces. The results hold for the standard ballistic deposition model as well as the surface relaxation model in the off-lattice setting. The results are proved with the aid of general limit theorems for stabilizing functionals of marked Poisson point processes.Comment: 12 page

Dynamic properties in a family of competitive growing models

The properties of a wide variety of growing models, generically called $X/RD$, are studied by means of numerical simulations and analytic developments. The study comprises the following $X$ models: Ballistic Deposition, Random Deposition with Surface Relaxation, Das Sarma-Tamboronea, Kim-Kosterlitz, Lai-Das Sarma, Wolf-Villain, Large Curvature, and three additional models that are variants of the Ballistic Deposition model. It is shown that after a growing regime, the interface width becomes saturated at a crossover time ($t_{x2}$) that, by fixing the sample size, scales with $p$ according to $t_{x2}(p)\propto p^{-y}, \qquad (p > 0)$, where $y$ is an exponent. Also, the interface width at saturation ($W_{sat}$) scales as $W_{sat}(p)\propto p^{-\delta}, \qquad (p > 0)$, where $\delta$ is another exponent. It is proved that, in any dimension, the exponents $\delta$ and $y$ obey the following relationship: $\delta = y \beta_{RD}$, where $\beta_{RD} = 1/2$ is the growing exponent for $RD$. Furthermore, both exponents exhibit universality in the $p \to 0$ limit. By mapping the behaviour of the average height difference of two neighbouring sites in discrete models of type $X/RD$ and two kinds of random walks, we have determined the exact value of the exponent $\delta$. Finally, by linking four well-established universality classes (namely Edwards-Wilkinson, Kardar-Parisi-Zhang, Linear-MBE and Non-linear-MBE) with the properties of both random walks, eight different stochastic equations for all the competitive models studied are derived.Comment: 23 pages, 6 figures, Submitted to Phys. Rev.

Dynamic scaling and universality in evolution of fluctuating random networks

We found that models of evolving random networks exhibit dynamic scaling similar to scaling of growing surfaces. It is demonstrated by numerical simulations of two variants of the model in which nodes are added as well as removed [Phys. Rev. Lett. 83, 5587 (1999)]. The averaged size and connectivity of the network increase as power-laws in early times but later saturate. Saturated values and times of saturation change with paramaters controlling the local evolution of the network topology. Both saturated values and times of saturation obey also power-law dependences on controlling parameters. Scaling exponents are calculated and universal features are discussed.Comment: 7 pages, 6 figures, Europhysics Letters for

Trapping mechanism in overdamped ratchets with quenched noise

A trapping mechanism is observed and proposed as the origin of the anomalous behavior recently discovered in transport properties of overdamped ratchets subject to external oscillatory drive in the presence of quenched noise. In particular, this mechanism is shown to appear whenever the quenched disorder strength is greater than a threshold value. The minimum disorder strength required for the existence of traps is determined by studying the trap structure in a disorder configuration space. An approximation to the trapping probability density function in a disordered region of finite length included in an otherwise perfect ratchet lattice is obtained. The mean velocity of the particles and the diffusion coefficient are found to have a non-monotonic dependence on the quenched noise strength due to the presence of the traps.Comment: 21 pages, 6 figures, to appear in PR