Relaxation behavior near the first-order phase transition line

Using the Metropolis algorithm, we simulate the relaxation process of the three-dimensional kinetic Ising model. Starting from a random initial configuration, we first present the average equilibration time across the entire phase boundary. It is observed that the average equilibration time increases significantly as the temperature decreases far from the critical temperature $T_{\rm c}$. The average equilibration time along the first-order phase transition (1st-PT) line exhibits an ultra-slow relaxation. We also investigate the dynamic scaling behavior with system sizes, and find that dynamic scaling holds not only near $T_{\rm c}$, but also at $T\ll T_{\rm c}$. The dynamic exponent at $T\ll T_{\rm c}$ is larger than that near $T_{\rm c}$. Additionally, we analyze the dynamic scaling of the average autocorrelation time and find that it depends on system size only near $T_{\rm c}$, while it becomes size-independent both above and below $T_{\rm c}$. The extremely slow relaxation dynamics observed near the 1st-PT is attributed to the complex structure of the free energy.