Generic two-phase coexistence in a type-2 Schloegl model for autocatalysis on a square lattice: Analysis via heterogeneous master equations.

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Schloegl's second model (also known as the quadratic contact process) on a square lattice involves spontaneous annihilation of particles at lattice sites at rate p, and their autocatalytic creation at unoccupied sites with n≥2 occupied neighbors at rate k_{n}. Kinetic Monte Carlo (KMC) simulation reveals that these models exhibit a nonequilibrium discontinuous phase transition with generic two-phase coexistence: the p value for equistability of coexisting populated and vacuum states, p_{eq}(S), depends on the orientation or slope, S, of a planar interface separating those phases. The vacuum state displaces the populated state for p>p_{eq}(S), and the opposite applies for p<p_{eq}(S) for 0<S<∞. The special "combinatorial" rate choice k_{n}=n(n-1)/12 facilitates an appealing simplification of the exact master equations for the evolution of spatially heterogeneous states in the model, which aids analytic investigation of these equations via hierarchical truncation approximations. Truncation produces coupled sets of lattice differential equations which can describe orientation-dependent interface propagation and equistability. The pair approximation predicts that p_{eq}(max)=p_{eq}(S=1)=0.09645 and p_{eq}(min)=p_{eq}(S→∞)=0.08827, values deviating less than 15% from KMC predictions. In the pair approximation, a perfect vertical interface is stationary for all p<p_{eq}(S=∞)=0.08907, a value exceeding p_{eq}(S→∞). One can regard an interface for large S→∞ as a vertical interface decorated with isolated kinks. For p<p_{eq}(S=∞), the kink can move in either direction along this otherwise stationary interface depending upon p, but for p=p_{eq}(min) the kink is also stationary.