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

Abstract

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, peq, of a control parameter, p, with metastability and hysteresis around peq. 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→peq. 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.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 ...

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

“…This behavior will be described in more detail in a separate publication focused on such more general models for N = 4, 8,.... In addition, analysis of LDEs for similar models indicates that interface propagation is also possible for S = 1, but only for an extremely narrow "negligible" range of p ~10 -6 [26].…”

“…Ref. [26]). This behavior will be described in more detail in a separate publication focused on such more general models for N = 4, 8,....…”

“…Examples for related models are provided in previous work. Note that these are far more complex that LDEs typically considered in the applied mathematics literature since: (i) spatial coupling deriving from exact master equations for the models [24][25][26]38] is more complex than standard discrete Laplacian choice [20][21][22][23]; and (ii) the mathematical literature considers only mean-field-type models where the variables are just concentrations, Ci,j , versus pair approximation (or higher-level) treatments where equations for the Ci,j are coupled to those for pair probabilities (or multi-site probabilities) or related quantities. effective in describing the location of these boundaries for small , noting that peq(max)  0.0965 (versus 0.0849 from KMC) and peq(min)  0.0883 (versus 0.0798 from KMC) as   0 (see Table 3.1).…”

“…Hierarchical truncation using mean-field site approximation or the Kirkwood-type [19] pair approximation produces a closed set of rate equations for homogeneous states describing kinetics. For spatially heterogeneous states, truncation produces a set of coupled lattice differential equations (LDE) [20][21][22][23], although with unconventional spatial coupling [24][25][26], where these can describe interface propagation.…”