Scaling properties of the relaxation time near the mean-field spinodal

TL;DR

This paper investigates the relaxation dynamics near the spinodal point in a mean-field spin model, deriving a finite-size scaling function and validating it through numerical and simulation methods.

Contribution

It introduces a unified finite-size scaling function for relaxation near the spinodal point, including metastable, spinodal, and unstable regions, using a master equation and Fokker-Planck analysis.

Findings

Derived a finite-size scaling function near the spinodal point.

Confirmed scaling behavior through numerical solutions and Monte Carlo simulations.

Analyzed the relaxation process using the van Kampen Omega expansion.

Abstract

We study the relaxation processes of the infinitely long-range interaction model (the Husimi-Temperley model) near the spinodal point. We propose a unified finite-size scaling function near the spinodal point, including the metastable region, the spinodal point, and the unstable region. We explicitly adopt the Glauber dynamics, derive a master equation for the probability distribution of the total magnetization, and perform the so-called van Kampen Omega expansion (an expansion in terms of the inverse of the systems size), which leads to a Fokker-Planck equation. We analyze the scaling properties of the Fokker-Planck equation and confirm the obtained scaling plot by direct numerical solution of the original master equation, and by kinetic Monte Carlo simulation of the stochastic decay process.