# Nucleation of Spatio-Temporal Structures From Defect Turbulence in the Two-dimensional Complex Ginzburg-Landau Equation

## TL;DR

This study numerically examines the nucleation processes leading to the formation of spiral structures in the two-dimensional complex Ginzburg-Landau equation, revealing evidence for a discontinuous transition to a frozen state.

## Contribution

It provides the first detailed analysis of nucleation kinetics and finite-size effects in the transition to spiral structures in the 2D complex Ginzburg-Landau equation.

## Key findings

- Nucleation times increase with system size and suggest a discontinuous transition.
- Finite-size effects are accounted for using extrapolation and phenomenological models.
- Long tails in nucleation time distributions support a discontinuous transition scenario.

## Abstract

We numerically investigate nucleation processes in the transient dynamics of the two-dimensional complex Ginzburg-Landau equation towards its "frozen" state with quasi-stationary spiral structures. We study the transition kinetics from either the defect turbulence regime or random initial configurations to the frozen state with a well-defined low density of quasi-stationary topological defects. Nucleation events of spiral structures are monitored using the characteristic length between the emerging shock fronts. We study two distinct situations, namely either when the system is quenched far away from the transition limit or near it. In the former deeply quenched case, the average nucleation time for different system sizes is measured over many independent realizations. We employ an extrapolation method as well as a phenomenological formula to account for and eliminate finite-size effects. The non-zero (dimensionless) barrier for the nucleation of single spiral droplets in the extrapolated infinite system size limit suggests that the transition to the frozen state is discontinuous. We also investigate the nucleation of spirals for systems that are quenched close to but beyond the crossover limit, and of target waves which emerge if a specific spatial inhomogeneity is introduced. In either of these cases, we observe long, "fat" tails in the distribution of nucleation times, which also supports a discontinuous transition scenario.