Dynamical fluctuations in critical regime and across the 1st order phase transition

In this proceeding, we study the dynamical evolution of the sigma field within the framework of Langevin dynamics. We find that, as the system evolves in the critical regime, the magnitudes and signs of the cumulants of sigma field, $C_{3}$ and $C_{4}$, can be dramatically different from the equilibrated ones due to the memory effects near $T_c$. For the dynamical evolution across the 1st order phase transition boundary, the supercooling effect leads the sigma field to be widely distributed in the thermodynamical potential, which largely enhances the cumulants $C_3, \ C_4$, correspondingly.