# Investigating the long-time behavior of the two-dimensional Ising model by leveraging the Java Virtual Machine (JVM)

# Summary

Novel computational methods are developed to allow for very long time simulations of the two-dimensional Ising model with 10 billion Monte Carlo updates in each simulation. Using these methods, the time-dependent behavior of quenching from random initial states is analyzed to determine the quenching behavior. Simulations are run across a range of parameters, including the lateral size of the grid, *l*, and the pair interaction strength, *J*.

In some cases, the simulation trajectory converges to a configuration with a predominantly up spin or a predominantly down spin. There does not appear to be a simple relationship between the parameters of grid lateral size and interaction strength with the behavior of whether or not the simulation converges to a predominant state. In general, smaller lateral grid sizes results are more likely to converge to a single predominant state, which can be explained by the smaller number of spins that need to be aligned. Moderate interaction strengths are found to maximize the fraction of simulated systems that converge to a predominant state with higher interaction strengths leading the system to become arrested in a mixed state configuration.

# Background and Introduction

The Ising model is among the simplest models of cooperative behavior in physics and it useful for modeling such phenomena as ferromagnetism. Simpler physics models assume each individual entity operates independently of other entities within the systems. As cooperative phenomena are ubiquitous in nature, it is useful to understand the properties of such a simple model of cooperative interactions. Further, the Ising model has been related to Machine Learning models (see discussion in Quora: How is the Ising model related to machine learning?) and I postulate that it will be increasingly more important as researchers further work to explore and understand the structure of machine learning models.

In the present work, we consider the two-dimension Ising model, which consists of a periodic grid of cells whereby each cell interacts with its four nearest neighbors. Each cell can be in one of two states: up or…