Langevin equations for several competitive growth models in one dimension are derived. For models with crossover from random deposition (RD) to some correlated deposition (CD) dynamics, with small probability p of CD, the surface tension νand the nonlinear coefficient λof the associated equations have linear dependence on p due solely to this random choice. However, they also depend on the regularized step functions present in the analytical representations of the CD, whose expansion coefficients scale with p according to the divergence of local height differences when p->0. The superposition of those scaling factors gives ν~ p^2 for random deposition with surface relaxation (RDSR) as the CD, and ν~ p, λ~ p^{3/2} for ballistic deposition (BD) as the CD, in agreement with simulation and other scaling approaches. For bidisperse ballistic deposition (BBD), the same scaling of RD-BD model is found. The Langevin equation for the model with competing RDSR and BD, with probability p for the latter, is also constructed. It shows linear p-dependence of λ, while the quadratic dependence observed in previous simulations is explained by an additional crossover before the asymptotic regime. The results highlight the relevance of scaling of the coefficients of step function expansions in systems with steep surfaces, which is responsible for noninteger exponents in some p-dependent stochastic equations, and the importance of the physical correspondence of aggregation rules and equation coefficients.
We present several approaches for deriving the coarse-grained continuous Langevin equation (or Edwards-Wilkinson equation) from a random deposition with surface relaxation (RDSR) model. First we introduce a novel procedure to divide the first transition moment into the three fundamental processes involved: deposition, diffusion and volume conservation. We show how the diffusion process is related to antisymmetric contribution and the volume conservation process is related to symmetric contribution, which renormalizes to zero in the coarse-grained limit. In another approach, we find the coeffic…
Usually discussions on the question of interpretation in the Langevin equation with multiplicative white noise are limited to the Ito and Stratonovich prescriptions. In this work, a Langevin equation with multiplicative white noise and its Fokker-Planck equation are considered. From this Fokker-Planck equation a connection between the stationary solution and the Tsallis distribution is obtained for different orders of prescription in discretization rule for the stochastic integrals; the Tsallis index $q$ and the prescription parameter (λ) are determined with the drift and diffusion coefficient…
We study the influence of the bulk dynamics of a growing cluster of particles on the properties of its interface. First, we define a {\it general bulk growth model} by means of a continuum Master equation for the evolution of the bulk density field. This general model just considers arbitrary addition of particles (though it can be easily generalized to consider substraction) with no other physical restriction. The corresponding Langevin equation for this bulk density field is derived where the influence of the bulk dynamics is explicitly shown. Finally, when it is assumed a well-defined inter…
We investigate a model of interacting clusters which compete for growth. For a finite assembly of coupled clusters, the largest one always wins, so that all but this one die out in a finite time. This scenario of `survival of the biggest' still holds in the mean-field limit, where the model exhibits glassy dynamics, with two well separated time scales, corresponding to individual and collective behaviour. The survival probability of a cluster eventually falls off according to the universal law $(\ln t)^{-1/2}$. Beyond mean field, the dynamics exhibits both aging and metastability, with a finit…
For the past two decades the Villain-Lai-Das Sarma equation has served as the theoretical framework for conserved surface growth processes, such as molecular-beam epitaxy. However some phenomena, such as mounding, are yet to be fully understood. In the following, we present a systematic analysis of the full, original Villain-Lai-Das Sarma equation showing that mound forming terms, which should have been included initially on symmetry grounds, are generated under renormalisation. A number of widely studied Langevin equations are recovered as limits or trivial fixed points of the full theory.
A competitive growth model (CGM) describes aggregation of a single type of particle under two distinct growth rules with occurrence probabilities $p$ and $1-p$. We explain the origin of scaling behaviors of the resulting surface roughness with respect to $p$ for two CGMs which describe random deposition (RD) competing with ballistic deposition (BD) and RD competing with the Edward Wilkinson (EW) growth rule. Exact scaling exponents are derived and are in agreement with previously conjectured values. Using this analytical result we are able to derive theoretically the scaling behaviors of the c…
We investigate the motility of a growing population of cells in a idealized setting: we consider a system of hard disks in which new particles are added according to prescribed growth kinetics, thereby dynamically changing the number density. As a result, the expected Brownian motion of the hard disks is modified. We compute the density-dependent friction of the hard disks and insert it in an effective Langevin equation to describe the system, assuming that the inter-collision time is smaller than the timescale of the growth. We find that the effective Langevin description captures the changes…
In order to characterise non-equilibrium growth processes, we study the behaviour of global quantities that depend in a non-trivial way on two different times. We discuss the dynamical scaling forms of global correlation and response functions and show that the scaling behaviour of the global response can depend on how the system is perturbed. On the one hand we derive exact expressions for systems characterised by linear Langevin equations (as for example the Edwards-Wilkinson and the noisy Mullins-Herring equations), on the other hand we discuss the influence of non-linearities on the scalin…
We study the space-time correlation and response functions in nonequilibrium growth processes described by linear stochastic Langevin equations. Exploiting exclusively the existence of space and time dependent symmetries of the noiseless part of these equations, we derive expressions for the universal scaling functions of two-time quantities which are found to agree with the exact expressions obtained from the stochastic equations of motion. The usefulness of the space-time functions is illustrated through the investigation of two atomistic growth models, the Family model and the restricted Fa…
We investigate the effects of Hamiltonian and Langevin microscopic dynamics on the growth laws of domains in coarsening. Using a one-dimensional class of generalized $ϕ^4$ models with power-law decaying interactions, we show that the two dynamics exhibit scaling regimes characterized by different scaling laws for the coarsening dynamics. For Langevin dynamics, it concurs with the exponent of defect dynamics, while Hamiltonian dynamics reveals new scaling laws with distinct early-time and a late-time regimes. This new behaviour can be understood as an effect of energy conservation, which induce…
There is nowadays considerable interest on ways to quantify the dynamics of research activities, in part due to recent changes in research and development (R&D) funding. Here, we seek to quantify and analyze university research activities, and compare their growth dynamics with those of business firms. Specifically, we analyze five distinct databases, the largest of which is a National Science Foundation database of the R&D expenditures for science and engineering of 719 United States (US) universities for the 17-year period 1979--1995. We find that the distribution of growth rates displays a …
Lattice growth models where uncorrelated random deposition competes with some aggregation dynamics that generates correlations are studied with rates of the correlated component decreasing as a power law. These models have anomalous roughening, with anomalous exponents related to the normal exponents of the correlated dynamics, to an exponent characterizing the aggregation mechanism and to that power law exponent. This is shown by a scaling approach extending the Family-Vicsek relation previously derived for the models with time-independent rates, thus providing a connection of normal and anom…
We extend the study of a model of competitive cluster growth in an active medium to a basis of networks; this is done by adding nonlocal connections with probability $p$ to sites on a regular lattice, thus enabling one to interpolate between regularity and full randomness. The model on networks demonstrates high sensitivity to small changes in initial configurations, which we characterize using damage spreading. The main focus of this paper is, however, the devising of survival strategies through selective networking, to alter the the fate of an arbitrarily chosen cluster: whether this be to r…
We present a variational solution of the Langevin field equation describing the nonequilibrium dynamics of a harmonically trapped Bose-Einstein condensate. If the thermal cloud remains in equilibrium at all times, we find that the equation of motions for the parameters in our variational ansatz are equivalent to the Langevin equations describing the motion of a massive Brownian particle in an external potential. Moreover, these equations are coupled to a stochastic rate equation for the number of atoms in the condensate. As applications of our approach, we have calculated the collisional dampi…
We discuss a parameter free scaling relation that yields a complete data collapse for large classes of nonequilibrium growth processes. We illustrate the power of this new scaling relation through various growth models, as for example the competitive growth model RD/RDSR (random deposition/random deposition with surface diffusion) and the RSOS (restricted solid-on-solid) model with different nearest-neighbor height differences, as well as through a new deposition model with temperature dependent diffusion. The new scaling relation is compared to the familiar Family-Vicsek relation and the limi…
To construct continuum stochastic growth equations for competitive nonequilibrium surface-growth processes of the type RD+X that mixes random deposition (RD) with a correlated-growth process X, we use a simplex decomposition of the height field. A distinction between growth processes X that do and do not create voids in the bulk leads to the definition of the {\it effective probability} $p_{\mathrm{eff}}$ of the process X that is a measurable property of the bulk morphology and depends on the {\it activation probability} $p$ of X in the competitive process RD+X. The bulk morphology is reflecte…
We present models of electrodeposition of ZnO films with organic additives, with focus on the growth of hybrid films with eosin Y. First we propose a rate equation model which assumes that the additives form branches with an exposed part above the ZnO deposit, growing with larger rate than the pure film, and that the rate of production of ZnO near those branches is proportional to the height exposed to the solution. This accounts for the production of OH- ions near the branches and the reactions with Zn++ ions. The steady state solution shows both species growing with the rate of the branches,…
We study relaxation dynamics of a three dimensional elastic manifold in random potential from a uniform initial condition by numerically solving the Langevin equation.We observe growth of roughness of the system up to larger wavelengths with time.We analyze structure factor in detail and find a compact scaling ansatz describing two distinct time regimes and crossover between them. We find short time regime corresponding to length scale smaller than the Larkin length $L_c$ is well described by the Larkin model which predicts a power law growth of domain size $L(t)$. Longer time behavior exhibit…
We show that in the construction of continuum equations for competitive growth processes that are a mixture of random deposition and a correlation process, a distinction must be made within a \textit{single universality class} between depositions that do and do not create voids in the bulk. Within these subclasses the bulk morphology is reflected in the surface roughening via \textit{nonuniversal} prefactors in the universal scaling of the surface width.
The article [Phys. Rev. E {\bf 73}, 031111 (2006)] by Horowitz and Albano reports on simulations of competitive surface-growth models RD+X that combine random deposition (RD) with another deposition X that occurs with probability $p$. The claim is made that at saturation the surface width $w(p)$ obeys a power-law scaling $w(p) \propto 1/p^δ$, where $δ$ is only either $δ=1/2$ or $δ=1$, which is illustrated by the models where X is ballistic deposition and where X is RD with surface relaxation. Another claim is that in the limit $p \to 0^+$, for any lattice size $L$, the time evolution of $w(t)$…
In the paper [Phys.Rev.E Vol.79, 051605 (2009)] by Chou and Pleimling a claim is made that a parameter-free scaling that gives data collapse for some simulation models would replace universal Family-Vicsek (FV) scaling. Here, by giving the explicit form of this scaling for competitive growth models, it is shown that data collapse in this procedure is obtained by a shift-and-scale operator that gives no information about stochastic dynamics and has no relation with FV function.
We study the growth dynamics of the size of manufacturing firms considering competition and normal distribution of competency. We start with the fact that all components of the system struggle with each other for growth as happened in real competitive bussiness world. The detailed quantitative agreement of the theory with empirical results of firms growth based on a large economic database spanning over 20 years is good .Further we find that this basic law of competition leads approximately a power law scaling over a wide range of parameters. The empirical datas are in accordance with present …
The non-stationary relaxation and physical ageing in the diffusion-limited erosion process ({\sc dle}) is studied through the exact solution of its Langevin equation, in $d$ spatial dimensions. The dynamical exponent $z=1$, the growth exponent $β=\max(0,(1-d)/2)$ and the ageing exponents $a=b=d-1$ and $λ_C=λ_R=d$ are found. In $d=1$ spatial dimension, a new representation of the meta-conformal Lie algebra, isomorphic to $\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{sl}(2,\mathbb{R})$, acts as a dynamical symmetry of the noise-averaged {\sc dle} Langevin equation. Its infinitesimal generators are…
We introduce and investigate the stochastic dynamics of the density of local extrema (minima and maxima) of non-equilibrium surface fluctuations. We give a number of exact, analytic results for interface fluctuations described by linear Langevin equations, and for on-lattice, solid-on-solid surface growth models. We show that in spite of the non-universal character of the quantities studied, their behavior against the variation of the microscopic length scales can present generic features, characteristic to the macroscopic observables of the system. The quantities investigated here present us …
