A class of type-2 Schloegl models is considered for particles on a square lattice with variable-range cooperativity. These models involve: (i) spontaneous particle annihilation at rate $p$; (ii) autocatalytic particle creation at unoccupied sites (i, j) with $n\ge 2$ particles within a specified neighborhood, ${\mathrm{\Omega }}_N(\mathrm{i},\mathrm{j})$, of $N$ sites at rate $k_n=\frac{\left(\genfrac{}{}{0pt}{}{n}{2}\right)}{\left(\genfrac{}{}{0pt}{}{N}{2}\right)}=\frac{n\left(n-1\right)}{N\left(N-1\right)}$; and (iii) possible spontaneous particle creation at unoccupied sites at “small” rate $\varepsilon \ge 0$. In some cases, ${\mathrm{\Omega }}_N$ just includes all symmetry-equivalent sites at a single specific distance $d$ (in units of lattice constants) from the unoccupied site, e.g., $d=1$ (nearest-neighbor sites) where $N=4$, or $d=\surd 5$ (or $\surd 13$ or…) where $N=8$. In other cases, ${\mathrm{\Omega }}_N$ includes sites multiple distances from the unoccupied site, e.g., $d=\{1,\surd 2\}$, where $N=8$. Kinetic Monte Carlo (KMC) simulation reveals that these models exhibit a nonequilibrium discontinuous phase transition between high- and low-density states below a critical point, $\varepsilon <{\varepsilon }_{\mathrm{c}}$, with generic two-phase coexistence (2PC) at least for smaller $N$. With some exceptions, there is an approach toward mean-field behavior with increasing $N$ (so the regime of generic 2PC shrinks, and ${\varepsilon }_{\mathrm{c}}$ approaches the mean-field value of 1/27). Additional insight into trends is provided by analysis of the exact master equations for the models via hierarchical truncation. These truncations utilize suitably tailored pair approximations which reflect the dominant nonequilibrium spatial correlations. These correlations in turn are shown to reflect the details of the autocatalytic particle creation process. For spatially heterogeneous states, the truncations produce coupled sets of lattice differential equations (LDE) which can describe orientation-dependent propagation of an interface between high- and low-density steady states for $\varepsilon <{\varepsilon }_{\mathrm{c}}$. Pair approximation values of $p=p_{\mathrm{eq}}$ where the interface is stationary, and its orientation-dependence, are in semiquantitative agreement with KMC results. This comparison accounts for propagation failure in the LDE which complicates interpretation.

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• Critical phenomena
• Nonequilibrium statistical mechanics
• Phase separation
• Phase transitions