Schloegl's second model for autocatalysis with particle diffusion: Lattice-gas realization exhibiting generic two-phase coexistence

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.

Phase transitions in the quadratic contact process on complex networks

The quadratic contact process (QCP) is a natural extension of the well-studied linear contact process where infected (1) individuals infect susceptible (0) neighbors at rate λ and infected individuals recover (1→0) at rate 1. In the QCP, a combination of two 1's is required to effect a 0→1 change. We extend the study of the QCP, which so far has been limited to lattices, to complex networks. We define two versions of the QCP: vertex-centered (VQCP) and edge-centered (EQCP) with birth events 1-0-1→1-1-1 and 1-1-0→1-1-1, respectively, where "-" represents an edge. We investigate the effects of network topology by considering the QCP on random regular, Erdős-Rényi, and power-law random graphs. We perform mean-field calculations as well as simulations to find the steady-state fraction of occupied vertices as a function of the birth rate. We find that on the random regular and Erdős-Rényi graphs, there is a discontinuous phase transition with a region of bistability, whereas on the heavy-tailed power-law graph, the transition is continuous. The critical birth rate is found to be positive in the former but zero in the latter.

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, 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)∼0.2113765+ce(f)/d as d→∞, where Δc=ce-cf≈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→∞)=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.

Metastability in Schloegl's second model for autocatalysis: Lattice-gas realization with particle diffusion

We analyze metastability associated with a discontinuous nonequilibrium phase transition in a stochastic lattice-gas realization of Schloegl's second model for autocatalysis. This model realization involves spontaneous annihilation, autocatalytic creation, and diffusion of particles on a square lattice, where creation at empty sites requires an adjacent diagonal pair of particles. This model, also known as the quadratic contact process, exhibits discontinuous transition between a populated active state and a particle-free vacuum or "poisoned" state, as well as generic two-phase coexistence. The poisoned state exists for all particle annihilation rates p>0 and hop rates h≥0 and is an absorbing state in the sense of Markovian processes. The active or reactive steady state exists only for p below a critical value, p{e}=p{e}(h) , but a metastable extension appears for a range of higher p up to an effective upper spinodal point, p{s+}=p{s+}(h) (i.e., p{s+}>p{e} ). For selected h , we assess the location of p{s+}(h) by characterizing both the poisoning kinetics and the propagation of interfaces separating vacuum and active states as a function of p .