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

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≥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.

Phase transitions in Schloegl's second model for autocatalysis on a Bethe lattice

Schloegl's second model (also known as the quadratic contact process) on a lattice involves spontaneous particle annihilation at rate p and autocatalytic particle creation at empty sites with n≥2 occupied neighbors. The particle creation rate for exactly n occupied neighbors is selected here as n(n-1)/[z(z-1)] for lattice coordination number z. We analyze this model on a Bethe lattice. Precise behavior for stochastic models on regular periodic infinite lattices is usually surmised from kinetic Monte Carlo simulation on a finite lattice with periodic boundary conditions. However, the persistence of boundary effects for a Bethe lattice complicates this process, e.g., by inducing spatially heterogenous states. This motivates the exploration of various boundary conditions and unconventional simulation ensembles on the Bethe lattice to predict behavior for infinite size. We focus on z=3, and predict a discontinuous transition to the vacuum state on the infinite lattice when p exceeds a threshold value of around 0.053.

Extended families of critical and stationary droplets for nonequilibrium phase transitions in spatially discrete bistable systems

Bistable nonequilibrium systems are realized in catalytic reaction-diffusion processes, biological transport and regulation, spatial epidemics, etc. Behavior in spatially continuous formulations, described at the mean-field level by reaction-diffusion type equations (RDEs), often mimics that of classic equilibrium van der Waals type systems. When accounting for noise, similarities include a discontinuous phase transition at some value, p_{eq}, of a control parameter, p, with metastability and hysteresis around p_{eq}. For each p, there is a unique critical droplet of the more stable phase embedded in the less stable or metastable phase which is stationary (neither shrinking nor growing), and with size diverging as p→p_{eq}. Spatially discrete analogs of these mean-field formulations, described by lattice differential equations (LDEs), are more appropriate for some applications, but have received less attention. It is recognized that LDEs can exhibit richer behavior than RDEs, specifically propagation failure for planar interphases separating distinct phases. We show that this feature, together with an orientation dependence of planar interface propagation also deriving from spatial discreteness, results in the occurrence of entire families of stationary droplets. The extent of these families increases approaching the transition and can be infinite if propagation failure is realized. In addition, there can exist a regime of generic two-phase coexistence where arbitrarily large droplets of either phase always shrink. Such rich behavior is qualitatively distinct from that for classic nucleation in equilibrium and spatially continuous nonequilibrium systems.

Discontinuous Phase Transitions in Nonlocal Schloegl Models for Autocatalysis: Loss and Reemergence of a Nonequilibrium Gibbs Phase Rule

We consider Schloegl models (or contact processes) where particles on a square grid annihilate at a rate p and are created at a rate of k_{n}=n(n-1)/[N(N-1)] at empty sites with n particles in a neighborhood Ω_{N} of size N. Simulation reveals a discontinuous transition between populated and vacuum states, but equistable p=p_{eq} determined by the stationarity of planar interfaces between these states depends on the interface orientation and on Ω_{N}. The behavior for large Ω_{N} follows from continuum equations. These also depend on the interface orientation and on Ω_{N} shape, but a unique p_{eq}=0.211 376 320 4 emerges imposing a Gibbs phase rule.

Minimal mechanism leading to discontinuous phase transitions for short-range systems with absorbing states

Motivated by recent findings, we discuss the existence of a direct and robust mechanism providing discontinuous absorbing transitions in short-range systems with single species, with no extra symmetries or conservation laws. We consider variants of the contact process, in which at least two adjacent particles (instead of one, as commonly assumed) are required to create a new species. Many interaction rules are analyzed, including distinct cluster annihilations and a modified version of the original pair contact process. Through detailed time-dependent numerical simulations, we find that for our modified models, the phase transitions are of first order, hence contrasting with their corresponding usual formulations in the literature, which are of second order. By calculating the order-parameter distributions, the obtained bimodal shapes as well as the finite-scale analysis reinforce coexisting phases and thus a discontinuous transition. These findings strongly suggest that the above particle creation requirements constitute a minimum and fundamental mechanism determining the phase coexistence in short-range contact processes.