Distribution of domain sizes in the zero temperature Glauber dynamics of the 1 D Potts model

TL;DR

This paper derives the exact distribution of domain sizes in the zero temperature Glauber dynamics of the 1D Potts model, revealing universal behavior and providing insights into domain wall reactions.

Contribution

It introduces an exact solution for domain size distribution in the 1D Potts model's zero temperature dynamics, connecting it to a soluble coagulation model.

Findings

Exact domain size distribution derived

Universal long-time behavior identified

Pair correlation functions calculated

Abstract

For the zero temperature Glauber dynamics of the $q$-state Potts model, we calculate the exact distribution of domain sizes by mapping the problem on an exactly soluble one-species coagulation model ($A+A\rightarrow A$). In the long time limit, this distribution is universal and from its (complicated) exact expression, we extract its behavior in various regimes. Our results are tested in a simulation and compared to the predictions of a simple approximation proposed recently. Considering the dynamics of domain walls as a reaction diffusion model $ A+A \to \ A $ with probability $ (q-2)/(q-1)$ and $A+A \to \emptyset $ with probability $1/(q-1)$, we calculate the pair correlation function in the long time regime.