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

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 $n\ge 2$ 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, $p_{\mathrm{eq}}\left(S\right)$, depends on the orientation or slope, $S$, of a planar interface separating those phases. The vacuum state displaces the populated state for $p>p_{\mathrm{eq}}\left(S\right)$, and the opposite applies for $p<p_{\mathrm{eq}}\left(S\right)$ for $0<S<\infty$. The special “combinatorial” rate choice $k_n=n\left(n-1\right)/12$ 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 $p_{\mathrm{eq}}\left(\max \right)=p_{\mathrm{eq}}(S=1)=0.09645$ and $p_{\mathrm{eq}}\left(\min \right)=p_{\mathrm{eq}}(S\to \infty )=0.08827$, values deviating less than 15% from KMC predictions. In the pair approximation, a perfect vertical interface is stationary for all $p<p_{\mathrm{eq}}(S=\infty )=0.08907$, a value exceeding $p_{\mathrm{eq}}(S\to \infty )$. One can regard an interface for large $S\to \infty$ as a vertical interface decorated with isolated kinks. For $p<p_{\mathrm{eq}}\left(S=\infty \right)$, the kink can move in either direction along this otherwise stationary interface depending upon $p$, but for $p=p_{\mathrm{eq}}\left(\min \right)$ the kink is also stationary.

Figure 1

Schematic: vertical interface (left) and an interface with slope–$S$ (right).

Figure 2

Pair-approximation analysis of the propagation of a vertical interface $\left(S=\infty \right)$. Periodic variation of $V_M$ with $t$ for: (a) $p=0.0900$; (b) $p=0.0892$ [which exceed $p_{\mathrm{eq}}(S=\infty )=0.08907$ so that the vacuum state displaces the populated state]. (c) Mean velocity, $V$, vs $p$.

Figure 3

$C_k\left(t\right)$ vs $t$ for four consecutive $k=k_0, k_0+1, k_0+1, k_0+3$ for a propagation vertical interface in the pair approximation: (a) $p=0.0895$; (b) $p=0.0950$.

Figure 4

$V$ vs $p$ for various $S$. Main figure: pair approximation. Inset: site approximation.

Figure 5

Motion of an isolated kink on an otherwise vertical interface. $C_{i,j}=0$ (o); $C_{i,j}>0$ (•). The schematic is idealized in that the actual $C_{i,j}$ does not suddenly become identically zero above the kink in the incomplete row along the step. See Fig. 8 for a realistic characterization.

Figure 6

Boundary conditions for 2D LDE analysis of the evolution of an isolated kink on a vertical interface.

Figure 7

Pair-approximation analysis of kink-propagation velocity, $V_{\mathrm{kink}}$, along the vertical interface vs $p$. The kink is stationary when $p=0.08827$ in the pair approximation.

Figure 8

Pair approximation prediction for $C_{i,j}$ in the vicinity of the stationary kink for $p=p_{\mathrm{eq}}(S\to \infty )$. A color scale is also included to illustrate the variation in concentration, $C_{i,j}$.