scholarly journals Review on Relationship Between the Universality Class of the Kardar-Parisi-Zhang Equation and the Ballistic Deposition Model

2021 ◽  
Vol 26 (4) ◽  
pp. 206-216
Author(s):  
Okhunjon Sayfidinov ◽  
Gabriella Bognar
Keyword(s):  
Universality Class ◽  
Higher Dimensions ◽  
System Size ◽  
Finite System ◽  
One Dimension ◽  
Kpz Equation ◽  
Kpz Universality Class ◽  
Deposition Model ◽  
Kpz Universality ◽  
Research Findings

Abstract We have analysed the research findings on the universality class and discussed the connection between the Kardar-Parisi-Zhang (KPZ) universality class and the ballistic deposition model in microscopic rules. In one dimension and 1+1 dimensions deviations are not important in the presence of noise. At the same time, they are very relevant for higher dimensions or deterministic evolution. Mostly, in the analyses a correction scale higher than 1280 has not been studied yet. Therefore, the growth of the interface for finite system size β ≥ 0.30 value predicted by the KPZ universality class is still predominant. Also, values of α ≥ 0.40, β ≥ 0.30, and z ≥ 1.16 obtained from literature are consistent with the expected KPZ values of α = 1/2, β = 1/3, and z = 3/2. A connection between the ballistic deposition and the KPZ equation through the limiting procedure and by applying the perturbation method was also presented.

scholarly journals Renormalization Fixed Point of the KPZ Universality Class

2015 ◽  
Vol 160 (4) ◽  
pp. 815-834 ◽  
Author(s):  
Ivan Corwin ◽  
Jeremy Quastel ◽  
Daniel Remenik
Keyword(s):  
Fixed Point ◽  
Universality Class ◽  
Kpz Universality Class ◽  
Kpz Universality

Stochastic Analysis: Around the KPZ Universality Class

2014 ◽  
Vol 11 (2) ◽  
pp. 1515-1568
Author(s):  
Alice Guionnet ◽  
Martin Hairer ◽  
Wendelin Werner
Keyword(s):  
Stochastic Analysis ◽  
Universality Class ◽  
Kpz Universality Class ◽  
Kpz Universality

scholarly journals CRITICAL LENGTH FOR THE SPREADING–VANISHING DICHOTOMY IN HIGHER DIMENSIONS

2020 ◽  
Vol 62 (1) ◽  
pp. 3-17 ◽  
Author(s):  
MATTHEW J. SIMPSON
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Moving Boundary ◽  
Numerical Solutions ◽  
Critical Length ◽  
Higher Dimensions ◽  
Moving Boundary Problem ◽  
Kolmogorov Equation ◽  
One Dimension ◽  
Spatial Dimensions

We consider an extension of the classical Fisher–Kolmogorov equation, called the “Fisher–Stefan” model, which is a moving boundary problem on $0<x<L(t)$. A key property of the Fisher–Stefan model is the “spreading–vanishing dichotomy”, where solutions with $L(t)>L_{\text{c}}$ will eventually spread as $t\rightarrow \infty$, whereas solutions where $L(t)\ngtr L_{\text{c}}$ will vanish as $t\rightarrow \infty$. In one dimension it is well known that the critical length is $L_{\text{c}}=\unicode[STIX]{x1D70B}/2$. In this work, we re-formulate the Fisher–Stefan model in higher dimensions and calculate $L_{\text{c}}$ as a function of spatial dimensions in a radially symmetric coordinate system. Our results show how $L_{\text{c}}$ depends upon the dimension of the problem, and numerical solutions of the governing partial differential equation are consistent with our calculations.

A Crossover Between Two Growth Universality Classes

1998 ◽  
Vol 09 (06) ◽  
pp. 815-819
Author(s):  
T. J. da Silva
Keyword(s):  
Numerical Simulations ◽  
Growth Models ◽  
Universality Class ◽  
Surface Relaxation ◽  
Relaxation Model ◽  
Kpz Equation ◽  
Dynamical Exponent ◽  
First Case ◽  
Universality Classes ◽  
Short Times

We investigated two simple SOS growth models where the roughness of the profile is softened by surface relaxation or by refuse. In the first case, a particle can relax until it reaches a local minimum and this process is described by a linear equation. In the second case, a particle can be evaporated and this process is described by the KPZ equation. In this work, we have mixed both the models in the same growth process. Numerical simulations in (1+1)-dimensions show a process that for short times behaves as the surface relaxation model and for intermediate times, behaves as the refuse model. Numerical simulations indicate for this crossover a dynamical exponent z≈1.4, indicating that it is in the KPZ universality class.

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