Finite-time effects on a first-order irreversible phase transition

TL;DR

The paper analyzes finite-time effects in the FOIPT of the ZGB model by applying a slowly varying CO pressure with a driving time scale $t_s$. Using a dynamic protocol and the coefficient of determination to locate a dynamical transition, it demonstrates spinodal-like scaling and extracts a dynamic critical point at $p_c=0.5266(1)$ and $\theta_c=0.087(1)$. The study shows robust data collapses with scaling forms that include power-law behavior and logarithmic corrections for the susceptibilities, linking finite-time scaling to a dynamic spinodal scenario. These findings reveal new states and provide a precise framework for identifying dynamic transitions in far-from-equilibrium irreversible phase transitions, distinguishing them from the equilibrium-like evaporation/condensation transitions observed in other ensembles.

Abstract

The first-order irreversible phase transition (FOIPT) of the ZGB model [Ziff, Gulari, Barshad, Phys. Rev. Lett. \textbf{56} (1986) 2553] for the catalytic oxidation of carbon monoxide is studied numerically in the presence of a slowly time-dependent, spatially uniform carbon monoxide pressure, with standard constant pressure simulations. This method allows us to observe finite-time effects close to the FOIPT, as well as evidence that a dynamic phase transition occurs. The location of this transition is measured very precisely and compared with previous results in the literature.

Figures (9)

• Figure 1: Results of Monte Carlo simulation obtained with the dynamic protocol given by Eq. (\ref{['Eq.protocol']}) for a lattice of side $L=512$ and using different times $t_s$ as indicated. (a) Plots of $\langle \theta_{CO} \rangle$ as a function of the control parameter $p_{CO}(t)$. Open black circles correspond to stationary results of Monte Carlo simulation obtained using the standard ensemble. (b) Plots of the susceptibility $\chi_{CO}$ versus $p_{CO}(t)$. In both graph the vertical dashed lines correspond to the first-order IPT point $p_{2}=0.525615$.
• Figure 2: Idem Figure 1 for (a) the density of empty sites $\langle \theta_{vac} \rangle$ and (b) susceptibility of empty sites $\chi_{vac}$.
• Figure 3: Plots of $\langle\theta_{CO}\rangle$ as a function of the control parameter $p_{CO}$. Color lines are results of Monte Carlo simulations obtained with the dynamic protocol for time scale $t_s=10k$. Different curves correspond to the indicated initial pressure $p_{CO}^{ini}(t)$. Open circles correspond to stationary results obtained by using the standard ensemble (dashed line is used to guide the eyes). Vertical dashed line corresponds to the first-order IPT point $p_{2}=0.525615$.
• Figure 4: (a) Plots of the coverage $\theta_{CO}$ and (b) susceptibility $\chi_{CO}$ versus the control parameter $p_{CO}$. The data were obtained with the dynamic protocol for $t_s=10$k and different sizes $L$ as indicated. Size independence is observed for $L\geq 64$. (c) Plots of the coverage $\theta_{CO}$ and (d) susceptibility $\chi_{CO}$ versus the control parameter $p_{CO}$. The data were obtained with the dynamic protocol for $t_s=2560$k and different sizes $L$ as indicated. Size independence is observed for $L\geq 256$. Vertical dashed lines indicate the first order IPT point $p_{2}=0.525615$.
• Figure 5: Determination of the critical pressure $p_c$ using the $CO$-coverage. (a) Effective transition pressure $p^{eff}_{CO}$ corresponding to the maximum susceptibility of $CO$, showed in Fig. \ref{['fig.1']}, as function of the time scale $t_s$. Horizontal line corresponds to the first-order IPT point $p_2$. (b) Plot of the coefficient of determination $r$ calculated for several $CO$-pressures $p^*$. The best fit is obtained for $p^*=0.5266$ as indicated. (c) Log-log plots of the difference $p^{eff}_{CO}-p^*$ versus $t_s$. The continuous line is the best fit obtained for $p^*=0.5266$. Symbols are greater than statistical errors.