Dynamics of the order parameter in symmetry breaking phase transitions

Second-order phase transitions can be investigated numerically, by solving partial differential equations for the evolution of the order parameter in space and time, such as the Langevin equation. We demonstrate that the ordinary differential equations provide surprisingly substantial insights into the dynamics of the phase transition. This includes the essence of the adiabatic-impulse scenario and the scaling of the freeze-out time, which are crucial to the Kibble-Zurek mechanism for topological defect formation.This finding enables the exploration of the Kibble-Zurek scaling using ordinary differential equations over a large range of quench timescales, which is otherwise difficult to achieve with numerical simulations of the full partial differential equations.