Alternative method to characterize continuous and discontinuous phase transitions in surface reaction models

In this paper we revisited the Ziff-Gulari-Barshad model to study its phase transitions and critical exponents through time-dependent Monte Carlo simulations. We use a method proposed recently to locate the nonequilibrium second-order phase transitions and that has been successfully used in systems with defined Hamiltonians and with absorbing states. This method, which is based on optimization of the coefficient of determination of the order parameter, was able to characterize the continuous phase transition of the model, as well as its upper spinodal point, a pseudocritical point located near the discontinuous phase transition. The static critical exponents $\beta , {\nu }_{\parallel }$, and ${\nu }_{\perp }$, as well as the dynamic critical exponents $\theta$ and $z$ for the continuous transition point, were also estimated and are in excellent agreement with results found in literature.

Figure 1

Coefficient of determination $r$ as function of $y$. The maximum occurs at the expected critical point and in a region related to the discontinuous phase transition, which is also observed in our study. Both regions deserve our attention and are explored in this paper.

Figure 2

Determination of continuous phase transition point using the density of $\text{CO}$ molecules through the curve $r\times y$. (a): Localization of the continuous phase transition point for $L=320$. (b) Extrapolation of $y\times 1/L$ for different lattice sizes used in this paper.

Figure 3

Determination of continuous phase transition point using the density of empty sites through the curve $r\times y$. (a) Localization of the continuous phase transition point for $L=320$. (b) Limit procedure used to determine $y_1$ when $1/L\to 0$.

Figure 4

Determination of the upper spinodal point. (a) Using the density of $\text{CO}$ molecules; (b) using the density of empty sites. The plots show the curves $r\times y$ for different lattice sizes. The inset shows the plot only with our larger lattice $L=320$ with error bars indicating the pronounced (first) peak, which corresponds to the upper spinodal point.

Figure 5

(a) Time evolution of $\rho \left(t\right)$ when the initial lattice is completely empty. (b) Limit procedure $L\to \infty$ to obtain $\beta /{\nu }_{\parallel }$ in the thermodynamic limit.

Figure 6

(a) Time evolution of $\rho \left(t\right)$ when the initial lattice is completely filled with $\text{O}$ atoms but a unique random site which remains empty. (b) Limit procedure $L\to \infty$ to obtain the dynamic exponent $\theta$ in the thermodynamic limit.

Figure 7

(a) Time evolution of $F_2\left(t\right)$ in $\text{log}-\text{log}$ scale for $L=320$. The error bars are smaller than the symbols. (b) Limit procedure $L\to \infty$ to obtain the dynamic exponent $d/z$ in the thermodynamic limit.

Figure 8

(a) Time evolution of $D\left(t\right)$ in $\text{log}-\text{log}$ scale for $L=320$. (b) Limit procedure $L\to \infty$ to obtain the static exponent $1/{\nu }_{\parallel }$ in the thermodynamic limit.