Generic two-phase coexistence and non-equilibrium criticality in a type-2 Schloegl model for autocatalysis on a square lattice

Abstract

ABSTRACT Schloegl’s second model (a.k.a. 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 kn. KMC simulation reveals that these models exhibit a non-equilibrium discontinuous phase transition with generic two-phase co-existence: the p-value for equistability of coexisting populated and vacuum states, peq(S), depends on the orientation or slope, S, of a planar interface separating those phases. The vacuum state displaces the populated state for p > peq(S), and the opposite applies for p < peq(S) for 0<S<. The special “combinatorial” rate choice kn = 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 peq(max) = peq(S=1) = 0.09645 and peq(min) = peq(S) = 0.08827, values deviating less than 15% from KMC predictions. In the pair approximation, a perfect vertical interface is stationary for all p < peq(S=) = 0.08907, a value exceeding peq(S). One can regard an interface for large S as a vertical interface decorated with an isolated kink. For p < peq(S=), the kink can move in either direction along this otherwise stationary interface depending upon p, but for p = peq(min) the kink is also stationary.

A more general class of type-2 Schloegl models for particles on a square lattice with variable range interactions can be defined to involve: (i) spontaneous annihilation at rate p; (ii) autocatalytic creation at unoccupied sites (i,j) with n 2 particles within a specified neighborhood, N(i, j), of N sites at rate kn = = ; and also (iii) spontaneous creation at unoccupied sites at “small” rate 0. In some special cases, N just includes all symmetry-equivalent sites at a single specific distance d (measured in lattice constants) from the unoccupied site, e.g., d = 1 (nearest-neighbor sites) where N = 4, or d = 5 (or 13 or…) where N = 8. In other cases, N includes sites multiple distances from the unoccupied site, e.g., d = {1, 2}, where N = 8. KMC simulation reveals that these models exhibit a non-equilibrium discontinuous phase transition between high- and low-density states below a critical point, < c, with generic two-phase coexistence (2PC) at least for smaller N. In general, there is an approach towards mean-field behavior with increasing N (e.g., the regime of generic 2PC shrinks, and c approaches the mean-field value of 1/27), although there are exceptions. Additional insight into trends is provided by analysis of the exact master equations for the models via hierarchical truncation using suitably tailored pair approximations. For spatially heterogeneous states, these 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 < c. Pair approximation values of p = peq where interface is stationarity, and their interface orientation-dependence associated with generic 2PC, are in semi-quantitative agreement with KMC results. However, interpretation is complicated by subtle propagation failure behavior in the LDE.