Out-of-equilibrium spinodal-like scaling behaviors at the thermal first-order transitions of three-dimensional q-state Potts models

Abstract

We study the out-of-equilibrium spinodal-like dynamics of three-dimensional $q$-state Potts systems driven across their thermal first-order transition in the thermodynamic limit, by a relaxational (heat-bath) dynamics. During the evolution, the inverse temperature $\beta$ increases linearly with time, as $\delta\beta(t)\equiv \beta(t)- \beta_{\rm fo} \sim t/t_s$, where $\beta_{\rm fo}$ is the inverse temperature at the transition point, $t$ is the time and $t_s$ is a time scale. The dynamics starts at $t_i< 0$ from an ensemble of disordered configurations equilibrated at inverse temperature $\beta(t_i)<\beta_{\rm fo}$ and ends at positive values of $t$, when the system is ordered (this is analogous to a standard Kibble-Zurek protocol). The time-dependent energy density shows an out-of-equilibrium scaling behavior in the large-$t_s$ limit, in terms of the scaling variable $t(\ln t)^\kappa/t_s$. The corresponding exponent turns out to be consistent with $\kappa=3/2$ (with a good accuracy), which is the value obtained by assuming that the initial nucleation of ordered regions provides the relevant mechanism for the passage from one phase to the other. The scaling behavior implies a spinodal-like phenomenon close to the transition point: the passage from the disordered to the ordered phase, composed of large ordered regions of different color, occurs at $\delta\beta(t)=\delta\beta_*>0$, where $\delta\beta_*$ decreases as $1/(\ln t_s)^{3/2}$ in the large-$t_s$ limit.