On the dynamical behavior of the ABC model

TL;DR

This paper studies the ABC model's dynamics on a ring, revealing a phase transition and analyzing how relaxation time scales with system size at different temperatures.

Contribution

It provides a detailed analysis of the relaxation time behavior of the ABC model, highlighting the impact of temperature on convergence rates.

Findings

Relaxation time scales as N^2 at high temperature.

Relaxation time scales at least as N^3 at low temperature.

The model exhibits a second order phase transition.

Abstract

We consider the ABC dynamics, with equal density of the three species, on the discrete ring with $N$ sites. In this case, the process is reversible with respect to a Gibbs measure with a mean field interaction that undergoes a second order phase transition. We analyze the relaxation time of the dynamics and show that at high temperature it grows at most as $N^2$ while it grows at least as $N^3$ at low temperature.