Simulations of deposition growth models in various dimensions: The possible importance of overhangs

Ko, D. Y. K.; Seno, Flavio

doi:10.1103/physreve.50.r1741

Cited by 18 publications

(13 citation statements)

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“…Without invoking (4) the estimation of α would be hampered by the slow power-law crossover to the asymptotic regime and underestimates of α would be obtained (as discussed above). This crossover would explain the discrepancies between previous estimates of α for BD (κ=0) and the KPZ value [6,4,9].…”

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confidence: 92%

“…Even when the interfacial dynamics are local, numerical studies have suggested that α is reduced when the bulk structure contains holes: for the ballistic deposition (BD) model values of α=0.42(3) [6] and 0.47(1) [4] were reported in early work on 1+1 dimensional BD, whereas more recent very large-scale simulations [9] yielded an estimate of α=0.45. Although BD is generally accepted as being a realization of KPZ growth, the possibility that a noncompact bulk structure implies a new non-KPZ university class has been raised [9,10].…”

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confidence: 99%

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Slow crossover to Kardar-Parisi-Zhang scaling

Blythe

Evans²

2001

Phys. Rev. E

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We study the synchronization physics of 1D and 2D oscillator lattices subject to noise and predict a dynamical transition that leads to a sudden drastic increase of phase diffusion. Our analysis is based on the widely applicable Kuramoto-Sakaguchi model, with local couplings between oscillators. For smooth phase fields, the time evolution can initially be described by a surface growth model, the Kardar-Parisi-Zhang (KPZ) theory. We delineate the regime in which one can indeed observe the universal KPZ scaling in 1D lattices. For larger couplings, both in 1D and 2D, we observe a stochastic dynamical instability that is linked to an apparent finite-time singularity in a related KPZ lattice model. This has direct consequences for the frequency stability of coupled oscillator lattices, and it precludes the observation of non-Gaussian KPZ-scaling in 2D lattices. Networks and lattices of coupled limit-cycle oscillators do not only represent a paradigmatic system in nonlinear dynamics, but are also highly relevant for potential applications. This significance derives from the fact that the coupling can serve to counteract the effects of the noise that is unavoidable in real physical systems. Synchronization between oscillators can drastically suppress the diffusion of the oscillation phases, improving the overall frequency stability. Experimental implementations of coupled oscillators include laser arrays [1] and coupled electromagnetic circuits, e.g. [2, 3], as well as the modern recent example of coupled electromechanical and op-tomechanical oscillators [4-8]. In this work, we will be dealing with the experimentally most relevant case of locally coupled 1D and 2D lattices. Naive arguments indicate that the diffusion rate of the collective phase in a coupled lattice of N synchronized oscillators is suppressed as 1/N , which leads to the improvement of frequency stability mentioned above. However , it is far from guaranteed that this ideal limit is reached in practice [9, 10]. The nonequilibrium nonlinear stochastic dynamics of the underlying lattice field theory is sufficiently complex that a more detailed analysis is called for. In this context, it has been conjectured earlier that there is a fruitful connection [11] between the synchronization dynamics of a noisy oscillator lattice and the Kardar-Parisi-Zhang (KPZ) theory of stochastic surface growth [12, 13]. We have been able to confirm that this is indeed true in a limited regime, particularly for 1D lattices. However, the most important prediction of our analysis consists in the observation that a certain dynamical instability can take the lattice system out of this regime in the course of the time evolution. As we will show, this instability is related to an apparent finite-time singularity in the evolution of the related KPZ lattice model. It has a significant impact on the phase dynamics, increasing the phase spread by several orders of magnitude. As such, ' j (t) ⇠j(t) b) a b time j x j y Figure 1. (a) Scheme of an oscillator array. We consider one-and two-d...

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confidence: 92%

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confidence: 99%

Slow crossover to Kardar-Parisi-Zhang scaling

Blythe

Evans²

2001

Phys. Rev. E

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“…In two dimensions the values of the KPZ systems are not known exactly. There is a large diver-*Electronic address: eytak@post.tau.ac.il † Electronic address: mosh@tarazan.tau.ac.il 3 + 1 0.12 [13] sity of results that range between 0.18 and 0.4 for ␣ and 0.1 and 0.25 for ␤ [16] (these results were obtained by a direct integration of the KPZ equation or the equivalent directed polymer problem). However, it usually accepted that ␣ = ϳ 0.4 and ␤ = 0.24 are the KPZ exponents in 2 + 1 dimensions.…”

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confidence: 99%

What is the connection between ballistic deposition and the Kardar-Parisi-Zhang equation?

Katzav

Schwartz

2004

Phys. Rev. E

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Ballistic deposition (BD) is believed to belong to the Kardar-Parisi-Zhang (KPZ) universality class. In this paper we study the validity of this belief by rigorously deriving a continuum equation from the BD microscopic rules, which deviates from the KPZ equation. We show that in one dimension and in the presence of noise the deviation is not important. This is not the case in the absence of noise. In more than one dimension and in the presence of noise we obtain an equation that superficially seems to be a continuum equation but in which the symmetry under rotations around the growth direction is broken.

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“…Numerical results are consistent with the proposition that RSOS and KPZ models belong to the same universality class, but the situation is more controversial when considering the BD and Eden models.For the KPZ equation in d = 1, the exact values α = 1/2 and β = 1/3 are known [3]. The estimated values obtained by various numerical works on BD in d = 1 for roughness exponent and growth exponent range from α = 0.42 to 0.506 and β = 0.3 to 0.339 [13][14][15][16][17]. Among the results, those obtained by Reis [17] are close enough to the exact KPZ values.…”

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confidence: 99%

Classification of(2+1)-dimensional growing surfaces using Schramm-Loewner evolution

2010

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Statistical behavior and scaling properties of isoheight lines in three different saturated two-dimensional grown surfaces with controversial universality classes are investigated using ideas from Schramm-Loewner evolution (SLE_{κ}). We present some evidence that the isoheight lines in the ballistic deposition (BD), Eden and restricted solid-on-solid (RSOS) models have conformally invariant properties all in the same universality class as the self-avoiding random walk (SAW), equivalently SLE_{8/3}. This leads to the conclusion that all these discrete growth models fall into the same universality class as the Kardar-Parisi-Zhang (KPZ) equation in two dimensions.

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Simulations of deposition growth models in various dimensions: The possible importance of overhangs

Cited by 18 publications

References 20 publications

Slow crossover to Kardar-Parisi-Zhang scaling

Slow crossover to Kardar-Parisi-Zhang scaling

What is the connection between ballistic deposition and the Kardar-Parisi-Zhang equation?

Classification of(2+1)-dimensional growing surfaces using Schramm-Loewner evolution

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