# Approaching equilibrium and the distribution of clusters

## TL;DR

This paper studies how Ising models approach equilibrium by analyzing cluster distributions and nucleation times, revealing early suppression of nucleation rates and structural differences in nucleating droplets.

## Contribution

It introduces a detailed analysis of cluster distributions and nucleation dynamics during the approach to equilibrium in Ising models and related systems.

## Key findings

- Nucleation rate is suppressed at early times despite stable global variables.
- Number of clusters of critical size becomes time-independent when nucleation rate stabilizes.
- Structural differences exist between pre- and post-metastable nucleating droplets.

## Abstract

We investigate the approach to stable and metastable equilibrium in Ising models using a cluster representation. The distribution of nucleation times is determined using the Metropolis algorithm and the corresponding $\phi^{4}$ model using Langevin dynamics. We find that the nucleation rate is suppressed at early times even after global variables such as the magnetization and energy have apparently reached their time independent values. The mean number of clusters whose size is comparable to the size of the nucleating droplet becomes time independent at about the same time that the nucleation rate reaches its constant value. We also find subtle structural differences between the nucleating droplets formed before and after apparent metastable equilibrium has been established.