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# 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, ${p}_{\mathrm{eq}}$, of a control parameter, $p$, with metastability and hysteresis around ${p}_{\mathrm{eq}}$. 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 $p\to {p}_{\mathrm{eq}}$. 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. Herein, 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.

- Authors:

- National Chung Cheng Univ., Chiayi (Taiwan)
- Ames Lab., Ames, IA (United States)
- Ames Lab., Ames, IA (United States); Iowa State Univ., Ames, IA (United States)

- Publication Date:

- Research Org.:
- Ames Laboratory (AMES), Ames, IA (United States)

- Sponsoring Org.:
- Ministry of Science and Technology (MOST) of Taiwan; USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22). Chemical Sciences, Geosciences & Biosciences Division

- OSTI Identifier:
- 1602867

- Alternate Identifier(s):
- OSTI ID: 1602361

- Report Number(s):
- [IS-J-10048]

[Journal ID: ISSN 2470-0045; PLEEE8]

- Grant/Contract Number:
- [105-2115-M-194-011-MY2; AC02-07CH11358]

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Physical Review E

- Additional Journal Information:
- [ Journal Volume: 101; Journal Issue: 2]; Journal ID: ISSN 2470-0045

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING

### Citation Formats

```
Wang, Chi-Jen, Liu, Da-Jiang, and Evans, James W. Extended families of critical and stationary droplets for nonequilibrium phase transitions in spatially discrete bistable systems. United States: N. p., 2020.
Web. doi:10.1103/PhysRevE.101.022803.
```

```
Wang, Chi-Jen, Liu, Da-Jiang, & Evans, James W. Extended families of critical and stationary droplets for nonequilibrium phase transitions in spatially discrete bistable systems. United States. doi:10.1103/PhysRevE.101.022803.
```

```
Wang, Chi-Jen, Liu, Da-Jiang, and Evans, James W. Fri .
"Extended families of critical and stationary droplets for nonequilibrium phase transitions in spatially discrete bistable systems". United States. doi:10.1103/PhysRevE.101.022803.
```

```
@article{osti_1602867,
```

title = {Extended families of critical and stationary droplets for nonequilibrium phase transitions in spatially discrete bistable systems},

author = {Wang, Chi-Jen and Liu, Da-Jiang and Evans, James W.},

abstractNote = {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. Herein, 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.},

doi = {10.1103/PhysRevE.101.022803},

journal = {Physical Review E},

number = [2],

volume = [101],

place = {United States},

year = {2020},

month = {2}

}

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