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 $\delta \beta \left(t\right)\equiv \beta \left(t\right)-{\beta }_{\mathrm{fo}}\sim t/t_s$, where ${\beta }_{\mathrm{fo}}$ is the inverse temperature at the transition point, $t$ is the time, and $t_s$ is a timescale. The dynamics starts at $t_i<0$ from an ensemble of disordered configurations equilibrated at an inverse temperature $\beta \left(t_i\right)<{\beta }_{\mathrm{fo}}$ and ends at positive values of $t$, corresponding to $\beta \left(t\right)>{\beta }_{\mathrm{fo}}$ in the ordered phase (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 $\sigma \equiv t{(lnt)}^{\kappa }/t_s$. The exponent $\kappa$ turns out to be consistent with $\kappa =3/2$ (with good accuracy), which is the value obtained by assuming that the initial nucleation of ordered regions is the relevant mechanism providing the largest timescale. This 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 $\delta \beta \left(t\right)=\delta {\beta }_{*}>0$, which decreases as $\delta {\beta }_{*}\sim {(ln{t}_s)}^{-\kappa }$ in the large-$t_s$ limit.

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• Nonequilibrium statistical mechanics
• Phase transitions
• Lattice models in statistical physics
• Monte Carlo methods
• Scaling methods