  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.
