Schloegl's second model for autocatalysis on hypercubic lattices: Dimension dependence of generic two-phase coexistence

Schloegl's second model on a ($d$ ≥ 2)-dimensional hypercubic lattice involves: (i) spontaneous annihilation of particles with rate $p$ and (ii) autocatalytic creation of particles at vacant sites at a rate proportional to the number of suitable pairs of neighboring particles. This model provides a prototype for nonequilibrium discontinuous phase transitions. However, it also exhibits nontrivial generic two-phase coexistence: Stable populated and vacuum states coexist for a finite range, $p_f\left(d\right)<p<p_e\left(d\right),$ spanned by the orientation-dependent stationary points for planar interfaces separating these states. Analysis of interface dynamics from kinetic Monte Carlo simulation and from discrete reaction-diffusion equations (dRDEs) obtained from truncation of the exact master equation, reveals that $p_{e\left(f\right)}\sim 0.2113765+c_{e\left(f\right)}/d$ as $d\to \infty ,$ where $\Delta c=c_e-c_f\approx 0.014$. A metastable populated state persists above $p_e$($d$) up to a spinodal $p=p_s\left(d\right)$, which has a well-defined limit $p_s(d\to \infty )=\frac{1}{4}$. The dRDEs display artificial propagation failure, absent in the stochastic model due to fluctuations. This feature is amplified for increasing $d$, thus complicating our analysis.

Figure 1

Steady-state $C$ versus $p$ for (a) $d$ = 2; (b) $d$ → ∞ MF behavior. $p_e$ ($p_f$) = upper (lower) 2PC boundaries; $p_s$ = spinodal. Inset to (a), dependence of equistability $p_{eq}$ on interface orientation. Inset to (b), profile of a hyperskew interface for $d$ → ∞ at $p_e$ = $p_f$  = 0.211 3765.

Figure 2

KMC results for $p_{eq}$ versus 1/$d$ for hyperskew (filled square) and vertical (open square) interfaces where the exact result is also shown for $d$ → ∞.

Figure 3

Schematic for $d$ = 3 of (a) a rare horizontal and (b) a hyperskew step on a vertical interface.