Domain Statistics in Coarsening Systems

TL;DR

This paper investigates the statistical properties of domains in one-dimensional Ising and Potts models under zero-temperature Glauber dynamics, revealing nontrivial exponents and developing an approximation method.

Contribution

It introduces a new analysis of domain statistics, including exponents and distributions, and develops an independent interval approximation for these coarsening systems.

Findings

Survival probability exponent: $oxed{ ext{0.126}}$ for Ising.

Unreacted domain exponent: $oxed{ ext{1.27}}$ for Ising.

Approximate predictions match numerical simulations and exact solutions.

Abstract

We study the domain number and size distributions in the one-dimensional Ising and $q$-state Potts models subject to zero-temperature Glauber dynamics. The survival probability of a domain, $S(t)\sim t^{-\psi}$, and an unreacted domain, $Q_1(t)\sim t^{-\delta}$, are characterized by two independent nonrtrivial exponents. For the Ising case, we find $\psi=0.126$ and $\delta=1.27$ using numerical simulations. We develop an independent interval approximation (IIA) that predicts the qualitative behavior of the domain distribution and provides good estimates for the exponents. Exact results for the domain distribution are also obtained in several solvable cases.