Dynamic Off-Equilibrium Transition in Systems Slowly Driven across Thermal First-Order Phase Transitions

We study the off-equilibrium behavior of systems with short-range interactions, slowly driven across a thermal first-order transition, where the equilibrium dynamics is exponentially slow. We consider a dynamics that starts in the high-$T$ phase at time $t=t_i<0$ and ends at $t=t_f>0$ in the low-$T$ phase, with a time-dependent temperature $T(t)/T_c\approx 1-t/t_s$, where $t_s$ is the protocol time scale. A general off-equilibrium scaling (OS) behavior emerges in the limit of large $t_s$. We check it at the first-order transition of the two-dimensional $q$-state Potts model with $q=20$ and 10. The numerical results show evidence of a dynamic transition, where the OS functions show a spinodal-like singularity. Therefore, the general mean-field picture valid for systems with long-range interactions is qualitatively recovered, provided the time dependence is appropriately (logarithmically) rescaled.

Figure 1

MC data of $E_r$ for $q=20$ at $t=0$ versus $s_1=t_s/(L^{2+\alpha }{e}^{\sigma L})$, using the optimal value $\alpha =2$ [57]. For $s_1\to \infty$, the data converge to ${\mathcal{E}}_{\mathrm{eq}}(0)=1/(1+q)$ (dashed line), since equilibrium is approached for $t_s\gg \tau (L)$. The inset shows the approach to the large-$L$ limit at fixed $s_1$.

Figure 2

MC data of $E_r(t)$ for $q=20$ for some values of $t_s$, in the $L\to \infty$ limit. The large-$L$ convergence (within errors) is checked by increasing $L$ at fixed $t_s$, analogously to the case shown in the inset. The full lines show the equilibrium energy density at $\beta ={\beta }_c(1+t/t_s)$.

Figure 3

The infinite-volume $E_r$ and $I_G$ (inset) for $q=20$ versus $s_2$. The dotted lines show the conjectured singular large-$t_s$ limit; see the text.

Figure 4

Scaling of the infinite-volume $E_r$ and $I_G$ (inset) around $s_2^{*}$.

Figure 5

Snapshots of a system of size $L=512$ for ${\tilde{s}}_2=(s_2-s_2^{*})t_s^{1/3}=-20$, 0, 20, 40, 100, 1500 (from left to right, top to bottom). Here $q=20$ and $t_s=640000$. We use different colors for each value of $s_x$. The range of ${\tilde{s}}_2$ covers the region where $E_r$ significantly changes; see Fig. 4.

Figure 6

Scaling of the infinite-volume $E_r$ for the reversed protocol, in which the dynamics starts from an ordered configuration at ${\beta }_i>{\beta }_c$ and $\beta (t)={\beta }_c(1-t/t_s)$. The optimal collapse is obtained for $s_2^{*}\approx 0.85$ and $\theta \approx 1/4$ [the accuracy of the data does not exclude the value $\theta \approx 1/3$ obtained for protocol (1)]; see [51].