Renormalization of stochastic lattice models: Epitaxial surfaces

Haselwandter, Christoph A.; Vvedensky, Dimitri D.

doi:10.1103/physreve.77.061129

Cited by 32 publications

(26 citation statements)

References 101 publications

(380 reference statements)

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“…By construction, h → h − h , then f (h, t) has zero mean, so its skewness and kurtosis are the most important quantities to observe [147,[178][179][180][181]. In addition, Langevin equations for growth models have been discussed by some authors [182][183][184]. Several works have been done in the weakly asymmetric simple exclusion process [136], the totally asymmetric exclusion process [137,185], and the direct d-mer diffusion model [138]: for a review see [170,181,186,187].…”

Section: Scaling Invariancementioning

confidence: 99%

Anomalous Diffusion: A Basic Mechanism for the Evolution of Inhomogeneous Systems

et al. 2019

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In this article we review classical and recent results in anomalous diffusion and provide mechanisms useful for the study of the fundamentals of certain processes, mainly in condensed matter physics, chemistry and biology. Emphasis will be given to some methods applied in the analysis and characterization of diffusive regimes through the memory function, the mixing condition (or irreversibility), and ergodicity. Those methods can be used in the study of small-scale systems, ranging in size from single-molecule to particle clusters and including among others polymers, proteins, ion channels and biological cells, whose diffusive properties have received much attention lately.

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Section: Scaling Invariancementioning

confidence: 99%

Anomalous Diffusion: A Basic Mechanism for the Evolution of Inhomogeneous Systems

et al. 2019

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“…For long times (t ≫ L z ) f → constant, so the saturated roughness W sat ∼ L α . The connection to a lattice model is based on comparing exponents and invoking universality [2].The foregoing paradigm can be justified for many models [2,8], but outstanding issues persist in some cases, most notably, ballistic deposition (BD). Originally formulated as a model for aggregation and sedimentation [9,10], BD is the prototypical model of nonconserved growth, in which the volume of material over the substrate is not equal to that deposited, in this case because of void formation.…”

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confidence: 99%

Large-scale simulations of ballistic deposition: The approach to asymptotic scaling

Farnudi

Vvedensky

2011

Phys. Rev. E

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Extensive kinetic Monte Carlo simulations are presented for ballistic deposition (BD) in (1 + 1) dimensions. Asymptotic scaling is found only for lattice sizes L 2 12 . Such a large system size for the onset of scaling explains the widespread discrepancies of previous reports for exponents of BD in one and likely higher dimensions. The exponents obtained from our simulations, α = 0.499 ± 0.004 and β = 0.336 ± 0.004, are in excellent agreement with the exact values α = 1 2 and β = 1 3 for the one-dimensional Kardar-Parisi-Zhang equation. Our findings enable a more informed exploration of exponents for BD in higher dimensions, accurate estimates of which have proven to be elusive. Fluctuations of growing surfaces are often described by idealized models [1][2][3] wherein the complex interactions between atoms or molecules are replaced by simple transition rules on a lattice that abstract the essence of these interactions. The appeal of such models stems from the fact that their rules can be easily implemented in efficient kinetic Monte Carlo (KMC) algorithms. This enables a comprehensive analysis of their statistical characteristics, which can be compared directly with experiments [4][5][6].A complementary approach is based on postulating a stochastic differential equation. Solutions of such equations typically focus on the asymptotic kinetic roughening regime, where the standard deviation W (L, t) of the surface profile exhibits dynamic scaling [7]:Here, h(x, t) is the surface height at position x and time t, L is the lateral viewing scale, α is the roughness exponent, z is the dynamic exponent, and f is a scaling function. At early times (t ≪ L z ), f (x) ∼ x β and W ∼ t β , where β is the growth exponent and z = α/β. For long times (t ≫ L z ) f → constant, so the saturated roughness W sat ∼ L α . The connection to a lattice model is based on comparing exponents and invoking universality [2].The foregoing paradigm can be justified for many models [2,8], but outstanding issues persist in some cases, most notably, ballistic deposition (BD). Originally formulated as a model for aggregation and sedimentation [9,10], BD is the prototypical model of nonconserved growth, in which the volume of material over the substrate is not equal to that deposited, in this case because of void formation. In classic BD [9, 10] a particle impinges on a randomly chosen lattice site and irreversibly attaches to the first vertical or lateral nearest neighbor encountered. The updating algorithm for the integer heights h i (n) at site i after n depositions isfor i = 1, 2, . . . , L, where max(x, y, z) yields the maximum of the three arguments.The continuum formulation of BD is thought to be the Kardar-Parisi-Zhang (KPZ) equation [11],where u(x, t) is the deviation of the height from its mean at position x and time t on a d-dimensional surface, ν is the surface tension, λ is the "excess velocity," and ξ is a Gaussian noise with mean zero and covarianceAlthough among the first surface growth models to be studied with KMC simulations [7], dis...

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“…where the expansion coefficients have to be chosen according to the rules of the lattice model to be represented [27][28][29]44]. Equation (19) is inserted in Eq.…”

Section: -3mentioning

confidence: 99%

“…However, the corresponding jump moment expression [Eq. (29)] is correct, since ω (3) i and ω (4) i are multiplied by h i−1 − h i and h i+1 − h i , respectively. Also note that ω (3) i and ω (4) i account for lateral aggregation, while ω (5) i refers to aggregation at a local maximum (see Fig.…”

Section: The Rd-bd and The Bbd Modelmentioning

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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Renormalization of stochastic lattice models: Epitaxial surfaces

Cited by 32 publications

References 101 publications

Anomalous Diffusion: A Basic Mechanism for the Evolution of Inhomogeneous Systems

Anomalous Diffusion: A Basic Mechanism for the Evolution of Inhomogeneous Systems

Large-scale simulations of ballistic deposition: The approach to asymptotic scaling

Langevin equations for competitive growth models

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