Coarsening in a 1-D system of Orienting Arrowheads: Persistence with $A+B \rightarrow$ 0

TL;DR

This paper investigates the coarsening dynamics and persistence properties in a 1D arrowhead system with reorientation, revealing novel annihilation kinetics and persistence exponents distinct from classical models.

Contribution

It introduces a new kinetic model with $A+B ightarrow 0$ domain wall annihilation, showing different decay laws and persistence exponents compared to Glauber-Ising dynamics.

Findings

Survival probability of walls decays exponentially over time.

Number of walls decreases as $t^{-1/2}$, persistence as $t^{- heta}$ with $ heta oughly 1/4.

Persistence exponent varies with diffusion rate of $B$ walls.

Abstract

We demonstrate the large scale effects of the interplay between shape and hard core interactions in a system with left- and right-pointing arrowheads ~$\textless ~~ \textgreater$~ on a line, with reorientation dynamics. This interplay leads to the formation of two types of domain wall, diffusive ($A$) and static ($B$). The correlation length in the equilibrium state diverges exponentially with increasing arrowhead density, with an ordered state of like orientations arising in the limit. In time, the approach to the ordered state is described by a coarsening process governed by the kinetics of domain wall annihilation $A+B\rightarrow 0$, quite different from $A+A \rightarrow 0$ kinetics pertinent to the Glauber-Ising model. The survival probability of a finite set of walls is shown to decay exponentially in time, in contrast to the power law decay known for $A+A \rightarrow 0$. In the thermodynamic limit with a finite density of walls, coarsening as a function of time $t$ is studied by simulation. While the number of walls falls as $t^{-\frac{1}{2}}$, the fraction of persistent arrowheads decays as $t^{-\theta}$ where $\theta$ is close to $\frac{1}{4}$, quite different from the Ising value. The global persistence too has $\theta=\frac{1}{4}$, as follows from a heuristic argument. In a generalization where the $B$ walls diffuse slowly, $\theta$ varies continuously, increasing with increasing diffusion constant.