A Method to Study Relaxation of Metastable Phases: Macroscopic Mean-Field Dynamics

TL;DR

This paper introduces two macroscopic dynamics models to describe the decay of metastable phases in many-particle systems, validated through applications to the Ising model showing good agreement with theory and simulations.

Contribution

It presents novel macroscopic dynamics formulations that accurately capture metastable decay, linking microscopic and macroscopic descriptions via a projection-operator formalism.

Findings

Good agreement with droplet theory and Monte Carlo simulations

Quantitative exponential dependence of lifetime on inverse field

Observation of oscillatory behavior at low temperatures

Abstract

We propose two different macroscopic dynamics to describe the decay of metastable phases in many-particle systems with local interactions. These dynamics depend on the macroscopic order parameter $m$ through the restricted free energy $F(m)$ and are designed to give the correct equilibrium distribution for $m$. The connection between macroscopic dynamics and the underlying microscopic dynamic are considered in the context of a projection- operator formalism. Application to the square-lattice nearest-neighbor Ising ferromagnet gives good agreement with droplet theory and Monte Carlo simulations of the underlying microscopic dynamic. This includes quantitative agreement for the exponential dependence of the lifetime on the inverse of the applied field $H$, and the observation of distinct field regions in which the derivative of the lifetime with respect to $1/H$ depends differently on $H$. In addition, at very low temperatures we observe oscillatory behavior of this derivative with respect to $H$, due to the discreteness of the lattice and in agreement with rigorous results. Similarities and differences between this work and earlier works on finite Ising models in the fixed-magnetization ensemble are discussed.