Universality class of isotropic on-lattice Eden clusters

Abstract

The shape of large on-lattice Eden clusters grown from a single seed is ruled by the underlying lattice anisotropy. This is reflected on the linear growth with time of the interface width (w ∼ t), in contrast with the KPZ universality class (w ∼ t 1/3 ) observed when the Eden model is grown on flat substrates. We propose an extended Eden model, in which the growth probability has a power law dependence with the number of occupied nearest neighbors. Large scale simulations (N 4 × 10 9 particles) were used to determine the time evolution of w. We found that a suitable choice of the power exponent removes the lattice-induced cluster anisotropy and provides a growth exponent in very good agreement with the KPZ universality class. Also, the present model corroborates the results found in off-lattice simulations, in which the center of mass fluctuations are considered in the interface scaling analysis.

New dynamic scaling in increasing systems

“…At large radius, r > 1000, for circular clusters grown over a square lattice, a crossover comes out. Several authors [12,13] this increase in the exponent is the lattice anisotropy effect and the origin of the reference system [14]. In this figure, we show two insets that represent cluster interfaces (r vs. θ) in two different regions: Left inset (r < 1000) does not show any characteristic length scale.…”

Linear surface roughness growth and flow smoothening in a three-dimensional biofilm model

“…Bacterial [13,14]a n df u n g a l [ 15,16]c o l o n i e sh a v eb e e n investigated within the framework of fractal surface growth [17]. However, the relevance of these findings to biofilms, and to which (if any) universality class biofilms belong, remains unclear.…”

THE critical exponent of the tree lattice generating function in the eden model

“…The cluster growth in the Eden model results from randomly adding particles (sites, bonds) on the cluster boundary with equal probability [1]- [6]. This process is especially simple on the Bethe lattice [2], [3], where it is assumed that each new bond is added in a new dimension, and the lattice therefore has an infinite dimension.…”

“…We encounter difficulties in constructing a tree on a hypercubic lattice of dimension d following the same rules if we prohibit intersections of branches (repeated site occupations). The branch interaction caused by this restriction, called the excluded volume interaction [7], does not allow finding a rigorous analytic solution of the problem, and approximate methods [1], [3] or computer modeling [4], [6] are therefore used. The excluded volume interaction affects tree properties and makes them dependent on variants of the tree construction.…”