Universality and dependence on initial conditions in the class of the nonlinear molecular beam epitaxy equation

Abstract

We report extensive numerical simulations of growth models belonging to the nonlinear molecular beam epitaxy (nMBE) class, on flat (fixed-size) and expanding substrates (ES). In both d=1+1 and 2+1, we find that growth regime height distributions (HDs), and spatial and temporal covariances are universal, but are dependent on the initial conditions, while the critical exponents are the same for flat and ES systems. Thus, the nMBE class does split into subclasses, as does the Kardar-Parisi-Zhang (KPZ) class. Applying the "KPZ ansatz" to nMBE models, we estimate the cumulants of the 1+1 HDs. Spatial covariance for the flat subclass is hallmarked by a minimum, which is not present in the ES one. Temporal correlations are shown to decay following well-known conjectures.

Geometry dependence in linear interface growth

“…Generalizations of this scenario for 1D circular KPZ interfaces ingrowing [27,28] or evolving out of the plane [29] have been also investigated more recently. Furthermore, a similar splitting of P (χ), F (x) and A(y) have been numerically observed in the 2D KPZ class [30][31][32][33], as well as in the nonlinear VLDS UC in both 1D and 2D [34]. Thus, nowadays it is well established that nonlinear UCs for interface growth split into subclasses depending on whether the interfaces are flat or curved.…”

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

“…On the other hand, its complexity is compensated for its generality. It is connected to a large number of stochastic processes, such as the direct polymer model [135], the weakly asymmetric simple exclusion process [136], the totally asymmetric exclusion process [137], direct d-mer diffusion [138], fire propagation [139][140][141], atomic deposition [142], evolution of bacterial colonies [143,144], turbulent liquid-crystals [145][146][147], polymer deposition in semiconductors [148], and etching [149][150][151][152][153][154][155][156][157][158][159][160].…”

“…Since the KPZ renormalization approach is valid only for 1 + 1 dimensions, questions about the validity of the Galilean invariance [164,165] for d > 1 and the existence of an upper critical dimension for KPZ [166,167] have been raised. For d > 1, the numerical simulation of the KPZ equation is not an easy task [164,165,[168][169][170][171], and the use of cellular automata models [149][150][151][152][153][154][155][156][157][158][159][160][172][173][174][175] has become increasingly common for growth simulations. Polynuclear growth (PNG), is a typical example of a discrete model that has received a lot of attention, and the outstanding works of Prähofer and Spohn [176] and Johansson [177] drive the way to the exact solution of the distributions of the heights fluctuations f (h, t) in the KPZ equation for 1 + 1 dimensions [134].…”

Local roughness exponent in the nonlinear molecular-beam-epitaxy universality class in one dimension

“…This method can be easily extended to the analysis of self-affine objects not related to surface growth such as time series modulated for seasonal changes [43]. Further enhancement of this method may include adapting it for global detrending which will allow the characterization of other features in interface growth such as properties of the underlying fluctuations in height distributions [44][45][46].…”