Universal behavior of the coefficients of the continuous equation in competitive growth models

Muraca, Diego; Braunstein, Lidia A.; Buceta, R. C.

doi:10.1103/physreve.69.065103

Cited by 22 publications

(36 citation statements)

References 17 publications

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“…This is fundamentally different in more complex systems where the initial regime can extend over very large times [12,13,14,15,16,17,18,19,20,21,22,23,25]. As already mentioned, the Family-Vicsek scaling relation (3) assigns a new set of coordinates to the second crossover point.…”

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confidence: 98%

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Parameter-free scaling relation for nonequilibrium growth processes

Chou

Pleimling

2009

Phys. Rev. E

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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 limitations of the latter are highlighted.PACS numbers: 64.60.Ht,68.35.Ct,05.70.Np The study of growing interfaces has been a very active field for many years [1,2,3]. Many studies focus on the technologically relevant growth of thin films or nanostructures, but growing interfaces are also encountered in various other physical, chemical, or biological systems, ranging from bacterial growth to diffusion fronts. Over the years important insights into the behavior of nonequilibrium growth processes have been gained through the study of simple model systems that capture the most important aspects of real experimental systems [4,5].In their seminal work, Edwards and Wilkinson investigated surface growth phenomena generated by particle sedimentation under the influence of gravity [6]. They proposed to describe this process in (d + 1) dimensions by the following stochastic equation of motion for the surface height h(x, t), now called the Edwards-Wilkinson (EW) equation,where ν is the diffusion constant (surface tension), whereas η(x, t) is a Gaussian white noise with zero mean and covariance η(x, t)η(y, s) = Dδ d (x − y)δ(t − s). Since Eq. 1 is linear, it can be solved exactly by Fourier transformations [2,4,6]. Later, Family [7] discussed the random deposition (RD) and random deposition with surface relaxation (RDSR) processes. RD [3,7] is one of the simplest surface growth processes. In this lattice model particles drop from randomly chosen sites over the surface and stick directly on the top of the selected surface site. Since there is no surface diffusion, the independently growing columns yield an uncorrelated and never-saturated surface. The RDSR process is realized by adding surface diffusion which allows particles just deposited on the surface to jump to the neighboring site with lowest height. This diffusion step smoothes the surface and limits the maximum interface width W (t), defined at deposition time t as the standard deviation of the surface height h from its mean value h: W (t) = h − h 2 . Starting from an initially flat surface, RDSR yields at very early times, with t < t 1 ∼ 1 (we assume here that one layer is deposited per unit time), a surface growing in the same way as for the RD process since no (or only very few) diffusion steps occur in that regime. For t > t 1 the width increases as a power law of time with a growth exponent β before entering a saturation regime after a crossover time t 2 , see Fig. 1. Both...

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“…[12,13,14,15,16,17,18,19,20,21,22,23]. In a competitive growth model one considers a mixture of two different deposition processes where one of them takes place with probability p whereas the other takes place with probability 1−p.…”

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confidence: 99%

Parameter-free scaling relation for nonequilibrium growth processes

Chou

Pleimling

2009

Phys. Rev. E

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“…Consequently, many competitive growth models were already proposed, with microscopic aggregation rules representing the atomistic dynamics. They are usually defined on lattices, such as those with aggregation of different species of particles [6][7][8] and those mixing different aggregation rules for the same species [9][10][11][12][13][14][15][16][17][18][19]. They usually show crossover effects from one dynamics at small times t and short length scales L to another dynamics at long t or large L, and in special cases anomalous roughening is present [20,21].…”

Section: Introductionmentioning

confidence: 99%

“…[17,18]. However, the claim on the universal quadratic form of vanishing coefficients [15] is ruled out by the study of a restricted solid-on-solid (RSOS) [31] model with deposition and erosion, which shows linear p dependence of the nonlinear term in the KPZ equation [30], and by the RD-BD model [17,18], which shows p 3/2 scaling of that term.…”

Section: Introductionmentioning

confidence: 99%

Langevin equations for competitive growth models

Silveira

Reis

2012

Phys. Rev. E

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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 ν and the nonlinear coefficient λ of 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.

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“…However, more complex processes cannot be described by a single growth model and are better represented as the competition between two or more simple mechanisms. The competition between models on euclidean lattices [20][21][22][23] was studied by few researchers, in spite of the fact that it is more realistic in describing grow processes on real materials. Pellegrini et.…”

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Competition between surface relaxation and ballistic deposition models in scale-free networks

Rocca¹,

Macri²,

Braunstein³

2013

EPL

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In this paper we study the scaling behavior of the fluctuations in the steady state WS with the system size N for a surface growth process given by the competition between the surface relaxation (SRM) and the ballistic deposition (BD) models on degree uncorrelated scale-free (SF) networks, characterized by a degree distribution P (k) ∼ k −λ , where k is the degree of a node. It is known that the fluctuations of the SRM model above the critical dimension (dc = 2) scale logarithmically with N on Euclidean lattices. However, Pastore y Piontti et al. (Phys. Rev. E, 76 (2007) 046117) found that the fluctuations of the SRM model in SF networks scale logarithmically with N for λ < 3 and as a constant for λ 3. In this letter we found that for a pure ballistic deposition model on SF networks WS scales as a power law with an exponent that depends on λ. On the other hand when both processes are in competition, we find that there is a continuous crossover between a SRM behavior and a power law behavior due to the BD model that depends on the occurrence probability of each process and the system size. Interestingly, we find that a relaxation process contaminated by any small contribution of ballistic deposition will behave, for increasing system sizes, as a pure ballistic one. Our findings could be relevant when surface relaxation mechanisms are used to synchronize processes that evolve on top of complex networks.

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Universal behavior of the coefficients of the continuous equation in competitive growth models

Cited by 22 publications

References 17 publications

Parameter-free scaling relation for nonequilibrium growth processes

Parameter-free scaling relation for nonequilibrium growth processes

Langevin equations for competitive growth models

Competition between surface relaxation and ballistic deposition models in scale-free networks

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