Comparative study of an Eden model for the irreversible growth of spins and the equilibrium Ising model

Abstract

The Magnetic Eden Model (MEM) [N. Vandewalle et al., Phys. Rev. E. 50, R635 (1994)] with ferromagnetic interactions between nearest-neighbor spins is studied in (d + 1)−dimensional rectangular geometries for d = 1, 2. In the MEM, magnetic clusters are grown by adding spins at the boundaries of the clusters. The orientation of the added spins depends on both the energetic interaction with already deposited spins and the temperature, through a Boltzmann factor. A numerical Monte Carlo investigation of the MEM has been performed and the results of the simulations have been analyzed using finite-size scaling arguments. As in the case of the Ising model, the MEM in d = 1 is non-critical (only exhibits an ordered phase at T = 0). In d = 2 the MEM exhibits an order-disorder transition of second-order at a finite temperature. Such transition has been characterized in detail and the relevant critical exponents have been determined. These exponents are in agreement (within error bars) with those of the Ising model in 2 dimensions. Further similarities between both models have been found by evaluating the probability distribution of the order parameter, the magnetization and the susceptibility. Results obtained by means of extensive computer simulations allow us to put forward a conjecture which establishes a nontrivial correspondence between the MEM for the irreversible growth of spins and the equilibrium Ising model. This conjecture is certainly a theoretical challenge and its confirmation will contribute to the development of a framework for the study of irreversible growth processes.

Criticality and the onset of ordering in the standard Vicsek model

“…Considering statistical errors, the obtained exponents satisfy the hyperscaling relationship (dν − 2β = γ), which is firmly established for critical systems in equilibrium but less documented for the case of nonequilibrium critical systems (see e.g. [27]). This finding is a strong indication that the SVM undergoes a continuous nonequilibrium phase transition.…”

Magnetically controlled ballistic deposition. A model of polydisperse granular packing

“…However, the studies pertained to the growth of clusters starting from a point seed. In a recent set of investigations on the magnetic Eden model, Albano et al [20][21][22][23] have pointed out the interest of such models for examining deposition and ÿlm growth, thus starting from a substrate-sometimes with rather complex realistic rules. The studies are also related to nonequilibrium wetting questions [24], -and other deposition problems [25].…”

Exponential and Power Law Distribution of Mass Clusters in a (Magnetic-Like) Deposition Model of Elongated Grains in 2d Piles

“…in [72] from the orientation probability density of the dipole direction in the clusters that the ordered state found at low temperatures diminishes when the temperature increases, due to the disorder induced by the fractal geometry of the aggregates. The same was noticed in the magnetic binary Eden model [73][74][75] where the minimum in the fractal dimension corresponds to the onset of magnetization, even when there are finite size effects [76] One interpretation of the βJ parameter is a thermodynamic one, i.e 1/βJ = kT /J.…”