Dynamical renormalization group study for a class of non-local interface equations

Nicoli, Matteo; Cuerno, Rodolfo; Castro, Mario

doi:10.1088/1742-5468/2011/10/p10030

Cited by 5 publications

(16 citation statements)

References 82 publications

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“…For each curve S(q, t) in figure 4, the small q behavior corresponds to an uncorrelated interface, the crossover to correlated spectra moving to smaller q (larger length scales) as time proceeds. In our case, we obtain numerically α = 1.03 ± 0.06 and β = 0.93 ± 0.07, which are equal, within error bars, to the values α = β = z = 1 predicted by renormalization group (RG) calculations on equation (1) [20,23]. In order to obtain these exponent values, we have computed the roughness using the relation…”

Section: Comparison With Experimentssupporting

confidence: 69%

“…Thus, the values α = z = 1 of the scaling exponents induce an interface which is disordered at all scales, while allowing at the same time for the identification of a 'typical' texture or motif. RG calculations and numerical simulations [21,23] both indicate the robustness of these exponent values, suggesting the universality of equation (13) as a description of a large class of nonequilibrium systems. Note, however, that interfaces developed under the same general physical principles as elucidated here, but for which the evolution equation is strongly, rather than weakly, nonlinear, may feature different morphological properties from the present cauliflower type.…”

Section: Discussionmentioning

confidence: 86%

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Universality of cauliflower-like fronts: from nanoscale thin films to macroscopic plants

et al. 2012

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Chemical vapor deposition (CVD) is a widely used technique to grow solid materials with accurate control of layer thickness and composition. Under mass-transport-limited conditions, the surface of thin films thus produced grows in an unstable fashion, developing a typical motif that resembles the familiar surface of a cauliflower plant. Through experiments on CVD production of amorphous hydrogenated carbon films leading to cauliflower-like fronts, we provide a quantitative assessment of a continuum description of CVD interface growth. As a result, we identify non-locality, non-conservation and randomness as the main general mechanisms controlling the formation of these ubiquitous shapes. We also show that the surfaces of actual cauliflower plants and combustion fronts obey the same scaling laws, proving the validity of the theory 6 2 over seven orders of magnitude in length scales. Thus, a theoretical justification is provided, which had remained elusive so far, for the remarkable similarity between the textures of surfaces found for systems that differ widely in physical nature and typical scales. Contents

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Section: Comparison With Experimentssupporting

confidence: 69%

Section: Discussionmentioning

confidence: 86%

Universality of cauliflower-like fronts: from nanoscale thin films to macroscopic plants

et al. 2012

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“…The latter breaks scale invariance at short time and length scales, which is restored back at large scales along the dynamics, as in the KS system [11]. Indeed, as borne out by numerical [30] and dynamic renormalization group [35] results, the asymptotic behavior of Eq. 2 fulfills the Family-Vicsek (FV) scaling ansatz [11].…”

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confidence: 87%

Dimensional fragility of the Kardar–Parisi–Zhang universality class

Nicoli¹,

Cuerno²,

Castro³

2013

J. Stat. Mech.

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We assess the dependence on substrate dimensionality of the asymptotic scaling behavior of a whole family of equations that feature the basic symmetries of the Kardar-Parisi-Zhang (KPZ) equation. Even for cases in which, as expected from universality arguments, these models display KPZ values for the critical exponents and limit distributions, their behavior deviates from KPZ scaling for increasing system dimensions. Such a fragility of KPZ universality contradicts naive expectations, and questions straightforward application of universality principles for the continuum description of experimental systems.One of the most powerful concepts in contemporary Statistical Mechanics is the idea of universality, by which microscopically dissimilar systems show the same large scale behavior, provided they are controlled by interactions that share dimensionality, symmetries, and conservation laws. Being rooted in the behavior of equilibrium critical systems [1], universality has more recently allowed to describe scaling behavior far from equilibrium [2-4], as for e.g. the stock market [5], crackling-noise [6], or random networks [7]. In complex systems like these, universality provides an enormously simplifying framework, as significant descriptions can be put forward on the basis of the general principles just mentioned.Celebrated non-equilibrium systems include those with generic scale invariance, displaying criticality throughout parameter space [8]. Examples are self-organizedcritical [9] and driven-diffusive systems [10], or surface kinetic roughening [11]. Indeed, the paradigmatic KardarParisi-Zhang (KPZ) equation for a rough interface [12]is very recently proving itself as a remarkable instance of universality. Here, h(x, t) is a height field above substrate position x ∈ R d at time t, and η is Gaussian white noise with zero mean and variance 2D. Thus, the exact asymptotic height distribution function has been very recently obtained for d = 1 [13][14][15]: it is given by the largest-eigenvalue distribution of large random matrices in the Gaussian unitary (GUE) (orthogonal, GOE) ensemble, the Tracy-Widom (TW) distribution, for globally curved (flat) interfaces, as proposed in [16], see reviews in [17,18]. Besides elucidating fascinating connections with probabilistic and exactly solvable systems, these results are showing that, not only are the critical exponent values common to members of this universality class, but also the distribution functions and limiting processes are shared by discrete models and continuum equations [19], and by experimental systems, from turbulent liquid crystals [20] to drying colloidal suspensions [21].In view of the success for d = 1 (1D) substrates, a natural important step is to assess the behavior of the KPZ universality class when changing space dimension, analogous to e.g. the experimental change from 2D to 1D behavior for ferromagnetic nanowires, that nonetheless occurs within the creeping-domain-wall class [22]. Thus, for discrete models and the continuous equation itself, indeed...

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“…More recently, it has been successfully applied to understand e.g. the multiscale nature of fluctuating interfaces [45], kinetic roughening in surfaces controlled by unstable nonlocal interactions [46,47], or the interplay between noise and morphological instabilities in anisotropic pattern-forming systems [48], to cite a few examples.…”

Section: Dynamic Renormalization Groupmentioning

confidence: 99%

“…The reasons behind such a simplicity are: (i) Given that in Eq. (24) λ y = 0 to begin with, parameter renormalization can be only due to the remaining nonlinearity λ x , which does not contribute to k 2 y order, hence ν y does not renormalize; (ii) at one-loop order there is a vertex cancellation [47] by which λ x does not renormalize either; (iii) as standard for conserved equations with non-conserved noise, since the lowest-order non-linear modification of the noise propagator is O(k 2 x ), the variance D is not affected and it does not renormalize either. Finally, the fixed points of the RG flow control the scaling behavior.…”

Section: A Systems With Shift Invariance: Conserved Akpz Equationmentioning

confidence: 99%

Strong anisotropy in two-dimensional surfaces with generic scale invariance: Nonlinear effects

2014

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We expand a previous study [Phys. Rev. E 86, 051611 (2012)] on the conditions for occurrence of strong anisotropy (SA) in the scaling properties of two-dimensional surfaces displaying generic scale invariance. There, a natural Ansatz was proposed for SA, which arises naturally when analyzing data from e.g. thin-film production experiments. The Ansatz was tested in Gaussian (linear) models of surface dynamics and in non-linear models, like the Hwa-Kardar (HK) equation [Phys. Rev. Lett. 62, 1813], which are susceptible of accurate approximations through the former. In contrast, here we analyze non-linear equations for which such type of approximations fail. Working within generically-scale-invariant situations, and as representative case studies, we formulate and study a generalization of the HK equation for conserved dynamics, and reconsider well-known systems, such as the conserved and the non-conserved anisotropic Kardar-Parisi-Zhang equations. Through the combined use of Dynamic Renormalization Group analysis and direct numerical simulations, we conclude that the occurrence of SA in two-dimensional surfaces requires dynamics to be conserved. We find that, moreover, SA is not generic in parameter space but requires, rather, specific shapes from the terms appearing in the equation of motion, whose justification needs detailed information on the dynamical process that is being modeled in each particular case.

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Dynamical renormalization group study for a class of non-local interface equations

Cited by 5 publications

References 82 publications

Universality of cauliflower-like fronts: from nanoscale thin films to macroscopic plants

Universality of cauliflower-like fronts: from nanoscale thin films to macroscopic plants

Dimensional fragility of the Kardar–Parisi–Zhang universality class

Strong anisotropy in two-dimensional surfaces with generic scale invariance: Nonlinear effects

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