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

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.