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Threshold versions of Schloegl's model on a lattice, which involve autocatalytic creation and spontaneous annihilation of particles, can provide a simple prototype for discontinuous non-equilibrium phase transitions. These models are equivalent to so-called threshold contact processes. A discontinuous transition between populated and vacuum states can occur selecting a threshold of N ≥ 2 for the minimum number, N, of neighboring particles enabling autocatalytic creation at an empty site. Fundamental open questions remain given the lack of a thermodynamic framework for analysis. For a square lattice with N = 2, we show that phase coexistence occurs not at a unique value but for a finite range of particle annihilation rate (the natural control parameter). This generic two-phase coexistence also persists when perturbing the model to allow spontaneous particle creation. Such behavior contrasts both the Gibbs phase rule for thermodynamic systems and also previous analysis for this model. We find metastability near the transition corresponding to a non-zero effective line tension, also contrasting previously suggested critical behavior. Mean-field type analysis, extended to treat spatially heterogeneous states, further elucidates model behavior. [ABSTRACT FROM AUTHOR]

We analyze a discontinuous nonequilibrium phase transition between an active (or reactive) state and a poisoned (or extinguished) state occurring in a stochastic lattice-gas realization of Schloegl’s second model for autocatalysis. This realization, also known as the quadratic contact process, involves spontaneous annihilation, autocatalytic creation, and diffusion of particles on a square lattice, where creation at empty sites requires a suitable nearby pair of particles. The poisoned state exists for all annihilation rates p>0 and is an absorbing particle-free “vacuum” state. The populated active steady state exists only for p below a critical value, pe. If pf denotes the critical value below which a finite population can survive, then we show that pf

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, pf(d) < p < pe(d), 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 pe(f) ∼ 0.211 3765 + ce(f),/d as d →∞, where Δc = ce -- Cf &thkap; 0.014. A metastable populated state persists above pe(d) up to a spinodal p = ps(d), which has a well-defined limit ps(d →∞) = ¼. 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. [ABSTRACT FROM AUTHOR]

A two-dimensional atomistic realization of Schlögl’s second model for autocatalysis is implemented and studied on a square lattice as a prototypical nonequilibrium model with first-order transition. The model has no explicit symmetry and its phase transition can be viewed as the nonequilibrium counterpart of liquid-vapor phase separations. We show some familiar concepts from study of equilibrium systems need to be modified. Most importantly, phase coexistence can be a generic feature of the model, occurring over a finite region of the parameter space. The first-order transition becomes continuous as a temperature-like variable increases. The associated critical behavior is studied through Monte Carlo simulations and shown to be in the two-dimensional Ising universality class. However, some common expectations regarding finite-size corrections and fractal properties of geometric clusters for equilibrium systems seems to be inapplicable. [ABSTRACT FROM AUTHOR]

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 kn=n(n-1)/[N(N-1)] at empty sites with n particles in a neighborhood ΩN of size N. Simulation reveals a discontinuous transition between populated and vacuum states, but equistable p=peq determined by the stationarity of planar interfaces between these states depends on the interface orientation and on ΩN. The behavior for large ΩN follows from continuum equations. These also depend on the interface orientation and on ΩN shape, but a unique peq=0.211 376 320 4 emerges imposing a Gibbs phase rule. [ABSTRACT FROM AUTHOR]