Spinodal-like scaling behavior after a temperature quench across the first-order phase transition in three-dimensional $q$-state Potts models

We study the out-of-equilibrium spinodal-like behavior of three-dimensional (3D) $q$-state Potts models (for $q\ge 3$), observed when the temperature is quenched across the first-order transition (FOT) point ${\beta }_{\mathrm{f}\mathrm{o}}=T_{\mathrm{f}\mathrm{o}}^{-1}$. We consider a standard quench protocol, in which high-temperature configurations, thermalized at ${\beta }_i<{\beta }_{\mathrm{f}\mathrm{o}}$, are driven across the FOT by a purely relaxational dynamics at $\beta >{\beta }_{\mathrm{f}\mathrm{o}}$. We focus on the emergence of spinodal-like behaviors in the thermodynamic limit, associated with the dynamic phase change. We argue that, if the nucleation of smooth droplets is the relevant mechanism of the post-quench phase change, for sufficiently small ${\beta }_{\mathrm{f}\mathrm{o}}-{\beta }_i>0$, the time-dependent energy density should scale in terms of $\rho =(\ln t{)}^{3\mathrm{/}2}\delta$, where $\delta =\beta \mathrm{/}{\beta }_{\mathrm{f}\mathrm{o}}-1$, with a discontinuity at a particular value $\rho ={\rho }_s>0$. This implies the emergence of a spinodal-like behavior, whose time scale $\tau$ increases exponentially as $\ln \tau \approx ({\rho }_s\mathrm{/}\delta {)}^{2\mathrm{/}3}$ in the limit $\delta \to 0^{+}$. We present a numerical analysis of the quench protocol in the 3D $q=6$ Potts model, which supports the above spinodal-like scenario.