# Fokker-Planck formalism approach to Kibble-Zurek scaling laws and non-equilibrium dynamics

## TL;DR

This paper employs the Fokker-Planck formalism to analyze non-equilibrium dynamics and Kibble-Zurek scaling laws in phase transitions, validating theoretical predictions and extending applicability to microscopic models and ion Coulomb crystals.

## Contribution

It introduces a Fokker-Planck based framework for deriving and validating Kibble-Zurek scaling laws in various dynamical regimes of phase transitions.

## Key findings

- Fokker-Planck method confirms Kibble-Zurek predictions in overdamped and underdamped limits.
- Framework predicts finite-size scaling functions.
- Applied to ion Coulomb crystals, demonstrating broader applicability.

## Abstract

We study the non-equilibrium dynamics of second-order phase transitions in a simplified Ginzburg-Landau model using the Fokker-Planck formalism. In particular, we focus on deriving the Kibble-Zurek scaling laws that dictate the dependence of spatial correlations on the quench rate. In the limiting cases of overdamped and underdamped dynamics, the Fokker-Planck method confirms the theoretical predictions of the Kibble-Zurek scaling theory. The developed framework is computationally efficient, enables the prediction of finite-size scaling functions and is applicable to microscopic models as well as their hydrodynamic approximations. We demonstrate this extended range of applicability by analyzing the non-equilibrium linear to zigzag structural phase transition in ion Coulomb crystals confined in a trap with periodic boundary conditions.