Dynamic scaling behaviors of the discrete growth models on fractal substrates

Abstract

The dynamic scaling behaviors of the Family model and the Etching model on different fractal substrates are studied by means of Monte Carlo simulations, so as to discuss the microscopic mechanisms influencing the dynamic behavior of growth interfaces by changing the structure of the substrates. The Sierpinski arrowhead, crab lattice and dual Sierpinski gasket are employed as the substrates of the growth. These substrates have same fractal dimensions (d f ≈ 1.585), but with different morphologies. It is shown that the structure of the substrates can affect the dynamic scaling properties of the surfaces and interfaces. Although the standard Family-Vicsek scaling is still satisfied in describing the scaling behavior of the growth on fractal substrates, the original continuum equations are invalid. The dynamic behavior of the Family model satisfies the fractional Edwards-Wilkinson equation introduced by Lee and Kim, and the dynamic behavior of the Etching model implies that α + z > 2, which is different from the analytical results of the Kardar-Parisi-Zhang equation. The fractal character of the substrate also affects the lateral behavior of the Etching growth. Interestingly, the same fractal dimensions lead to different scaling exponents. The scaling exponents of the growth models on fractal substrates are determined by not only the fractal dimensions of the substrates, but also the spectral dimensions. Fortunately, it seems that the fractal dimension and the spectral dimension are sufficient to determine the scaling exponents of the growth model on fractal substrates.

Absorbing phase transitions in diluted conserved threshold transfer process

“…Therefore, the critical exponents might be similar or, if not, might weakly depend on the embedding lattice dimension. Similar observations were made in the equilibrium surface growth, for which the growth exponents β, defined by the rms surface width as W (t) ∼ t β , were similar on a percolation network in two and higher dimensions [42] and were found to depend only on the spectral dimensions on geometrical fractal lattices [43].…”

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

“…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].…”

“…For a given growth model, the first question to be answered is if the model belongs to the same universality class as KPZ. In this context, we expose here the etching model [149], which has attracted considerable attention in recent years [150][151][152][153][154][155][156][157][158][159][160]. These studies suggest a close relation between the etching model and KPZ.…”

Dynamic scaling behaviors of linear fractal Langevin-type equation driven by nonconserved and conserved noise

“…(ii) The scaling relation 2α + d f = z is satisfied, independent of linear diffusion type (this can be seen from different values of factor m) on the surfaces and interfaces. (iii) These results of the general linear fractal Langevin-type equation can be tested by numerical works[13,16,19].…”

“…Among previous theoretical investigations of continuum equations, as well as numerical simulations of discrete atomistic models, much more were performed on regular or Euclidean substrates with integer dimension, however, less were devoted to fractal substrates. As a result, there is no simple and clear understanding about the interplay between the dynamical growth rules of the system and the self-similarity of fractal structures until the recent works [12][13][14][15][16][17][18][19][20].…”