Coarsening in the q-State Potts Model and the Ising Model with Globally Conserved Magnetization

TL;DR

This paper investigates the nonequilibrium coarsening dynamics of the q-state Potts model and related Ising models with conserved magnetization, deriving exact solutions and approximations for correlation functions across dimensions.

Contribution

It provides exact solutions and a Gaussian closure approximation for the correlation functions in the Potts and Ising models with conserved magnetization, extending understanding of their scaling behavior.

Findings

Correlation functions follow dynamic scaling with length scale L(t) ~ t^{1/2}.

Exact solutions for the kinetic Potts model in 1D are presented.

The Gaussian closure approximation agrees well with numerical simulations.

Abstract

We study the nonequilibrium dynamics of the $q$-state Potts model following a quench from the high temperature disordered phase to zero temperature. The time dependent two-point correlation functions of the order parameter field satisfy dynamic scaling with a length scale $L(t)\sim t^{1/2}$. In particular, the autocorrelation function decays as $L(t)^{-\lambda(q)}$. We illustrate these properties by solving exactly the kinetic Potts model in $d=1$. We then analyze a Langevin equation of an appropriate field theory to compute these correlation functions for general $q$ and $d$. We establish a correspondence between the two-point correlations of the $q$-state Potts model and those of a kinetic Ising model evolving with a fixed magnetization $(2/q-1)$. The dynamics of this Ising model is solved exactly in the large q limit, and in the limit of a large number of components $n$ for the order parameter. For general $q$ and in any dimension, we introduce a Gaussian closure approximation and calculate within this approximation the scaling functions and the exponent $\lambda (q)$. These are in good agreement with the direct numerical simulations of the Potts model as well as the kinetic Ising model with fixed magnetization. We also discuss the existing and possible experimental realizations of these models.