Universal scaling relations for growth phenomena

Rodrigues, Evandro A; Mozo Luis, Edwin E; de Assis, Thiago A; Oliveira, Fernando A

doi:10.1088/1742-5468/ad1d57

Cited by 4 publications

(5 citation statements)

References 54 publications

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“…Furthermore, the upper bound of d − 1 ≤ d n ≤ d gives the exact result α = 1/2 for d = 1 as already mentioned. Thus, Equation ( 16) establishes the appropriate bounds and we do not need relation (17).…”

Section: Upper Critical Dimensionmentioning

confidence: 99%

“…The main relationships between exponents are the result of scaling, Equation ( 5), and renormalization approaches, Equation (6). Recent results [17] generalizing the Family-Vicsek relation to all d dimensions would be a hopeful starting point for a generalization of a renormalization group (RG) approach to KPZ. Thus, a new approach involving a suitable renormalization with a fractal dimension for the noise would be desired.…”

Section: Renormalizationmentioning

confidence: 99%

“…with w s ∼ L α and t × ∼ L z , where t × is a crossover time. The critical exponents z, α and β satisfy the scaling relation [16,17]…”

Section: Introductionmentioning

confidence: 99%

“…We are interested in physical systems in which the roughness grows with time and then saturates at a maximum value

[ 9 ]:

with

and

, where

is a crossover time. The critical exponents z ,

and

satisfy the scaling relation [ 16 , 17 ]

Also, the one-loop renormalization group approach preserves Galilean invariance, which results in [ 6 ]

and therefore there is only one independent exponent.…”

Section: Introductionmentioning

confidence: 99%

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Restoring the Fluctuation–Dissipation Theorem in Kardar–Parisi–Zhang Universality Class through a New Emergent Fractal Dimension

Gomes-Filho,

de Castro,

Liarte

et al. 2024

Entropy

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The Kardar–Parisi–Zhang (KPZ) equation describes a wide range of growth-like phenomena, with applications in physics, chemistry and biology. There are three central questions in the study of KPZ growth: the determination of height probability distributions; the search for ever more precise universal growth exponents; and the apparent absence of a fluctuation–dissipation theorem (FDT) for spatial dimension d>1. Notably, these questions were answered exactly only for 1+1 dimensions. In this work, we propose a new FDT valid for the KPZ problem in d+1 dimensions. This is achieved by rearranging terms and identifying a new correlated noise which we argue to be characterized by a fractal dimension dn. We present relations between the KPZ exponents and two emergent fractal dimensions, namely df, of the rough interface, and dn. Also, we simulate KPZ growth to obtain values for transient versions of the roughness exponent α, the surface fractal dimension df and, through our relations, the noise fractal dimension dn. Our results indicate that KPZ may have at least two fractal dimensions and that, within this proposal, an FDT is restored. Finally, we provide new insights into the old question about the upper critical dimension of the KPZ universality class.

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Section: Upper Critical Dimensionmentioning

confidence: 99%

Section: Renormalizationmentioning

confidence: 99%

“…with w s ∼ L α and t × ∼ L z , where t × is a crossover time. The critical exponents z, α and β satisfy the scaling relation [16,17]…”

Section: Introductionmentioning

confidence: 99%

“…We are interested in physical systems in which the roughness grows with time and then saturates at a maximum value

[ 9 ]:

with

and

, where

is a crossover time. The critical exponents z ,

and

satisfy the scaling relation [ 16 , 17 ]

Also, the one-loop renormalization group approach preserves Galilean invariance, which results in [ 6 ]

and therefore there is only one independent exponent.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Restoring the Fluctuation–Dissipation Theorem in Kardar–Parisi–Zhang Universality Class through a New Emergent Fractal Dimension

Gomes-Filho,

de Castro,

Liarte

et al. 2024

Entropy

View full text Add to dashboard Cite

show abstract

“…The only case where the exact solution is known is taking λ = 0 in the equation (4), resulting in the linear Edwards-Wilkinson (EW) equation [4]. If we consider the particular case of the KPZ class, which has been one of the most studied growth models, determining the exact form of W(t) is still an open issue [48].…”

Section: Introductionmentioning

confidence: 99%

Machine learning method for roughness prediction

Makhoul,

Simas Filho,

de Assis

2024

Surf. Topogr.: Metrol. Prop.

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This work aims to employ machine-learning models, specifically neural networks, to predict the time evolution of the global surface roughness in a lattice model that represents a film growing on a $d$-dimensional substrate. We analyze the well-known ballistic deposition (BD) model for $d=1,2$ since it presents strong corrections to the scaling, making it difficult to observe directly, via effective scaling exponents, its correspondence with the Kadar-Parisi-Zhang (KPZ) universality class. As an alternative to overcome this difficulty, we first intend to learn the time evolution of the global roughness for substrate sizes that are computationally viable to simulate. To test the learning, we apply two different methodologies for $d=1$: the first one learns the Family-Vicsek scaling relation, and by doing the reverse transformation, we get the global roughness as a function of the time, and the second one learns the kinetic roughening directly from the time series data. For growth in $d=2$ where applications arise and no exact KPZ scaling exponents are known, we apply the second methodology. However, we employ a more resilient learning model tailored for time series problems. Hence, the time required to generate the same amount of data, showing the evolution of global roughness, is reduced dramatically. Importantly, machine learning techniques capture the scaling corrections of the BD model, predicting an effective global roughness exponent, $\alpha$, calculated from the learned data extracted from very large lateral sizes and times that cannot be simulated using lattice models. Our prediction is consistent with accurate estimates of the KPZ roughness exponent reported in the literature for $d=2$.

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