Mapping the phase transitions of the ZGB model with inert sites via nonequilibrium refinement methods

In this paper, we revisit the ZGB model and explore the effects of the presence of inert sites on the catalytic surface. The continuous and discontinuous phase transitions of the model are studied via time-dependent Monte Carlo simulations. In our study, we are concerned with building a refinement procedure, based on a simple concept known as coefficient of determination $r$, in order to find the possible phase transition points given by the two parameters of the model: the adsorption rates of carbon monoxide, $y$, and the density of inert sites, $\rho_{is}$. First, we obtain $10^6$ values of $r$ by sweeping the whole set of possible values of the parameters with an increment $10^{-3}$, i.e., $0 \leq y \leq 1$ and $0 \leq \rho_{is} \leq 1$ with $\Delta y=\Delta\rho_{is}=10^{-3}$. Then, with the possible phase transition points in hand, we turn our attention to some fixed values of $\rho_{is}$ and perform a more detailed refinement considering larger lattices and increasing the increment $\Delta y$ by one order of magnitude to estimate the critical points with higher precision. Finally, we estimate the static critical exponents $\beta$, $\nu_{\parallel}$, and $\nu_{\perp}$, as well as the dynamic critical exponents $z$ and $\theta$.