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

TL;DR

This study compares the Magnetic Eden Model's irreversible spin growth with the equilibrium Ising model, revealing similar critical behavior and proposing a conjecture linking the two, advancing understanding of growth processes in statistical physics.

Contribution

It introduces a detailed numerical analysis of the MEM and establishes a conjecture connecting its behavior to the well-understood Ising model.

Findings

MEM exhibits a second-order phase transition in 2D

Critical exponents match those of the 2D Ising model

Similar probability distributions of order parameters

Abstract

The Magnetic Eden Model (MEM) 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.