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We study the non-equilibrium dynamics of second-order phase transitions in asimplified Ginzburg-Landau model using the Fokker-Planck formalism. Inparticular, we focus on deriving the Kibble-Zurek scaling laws that dictate thedependence of spatial correlations on the quench rate. In the limiting cases ofoverdamped and underdamped dynamics, the Fokker-Planck method confirms thetheoretical predictions of the Kibble-Zurek scaling theory. The developedframework is computationally efficient, enables the prediction of finite-sizescaling functions and is applicable to microscopic models as well as theirhydrodynamic approximations. We demonstrate this extended range ofapplicability by analyzing the non-equilibrium linear to zigzag structuralphase transition in ion Coulomb crystals confined in a trap with periodicboundary conditions.

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