# An Overview of Emergent Order in Far-from-Equilibrium Driven Systems: From Kuramoto Oscillators to Rayleigh–Bénard Convection

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## Abstract

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## 1. Introduction

#### 1.1. Ising Model

#### 1.2. Kuramoto Model

#### 1.3. Rayleigh–Bénard Convection

## 2. Results

## 3. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Figure shows phase transition in a two-dimensional square lattice Ising model. The magnetization in the system ($S/\overline{S}$) is plotted as a function of the inverse temperature ($\beta $). Vertical dotted line denotes ${\beta}_{c}\approx 0.44$ on the abscissa. (

**b**) Figure shows magnetization as a function of time for three cases: $h=0$ (dashed), $h=Asin\omega t$ (solid) and $h=\mathrm{constant}$ (dotted). (

**c**) Figure shows magnetization as a function of inverse temperature for the previous three cases. (

**d**) Figure shows magnetization (in black) and standard deviation of magnetization (${\sigma}_{S/\overline{S}}$, in red) as a function of simulation time-steps.

**Figure 2.**(

**a**) Figure shows the scaled standard deviation ($\sigma /{\sigma}_{max}$) of the angular frequency as a function of time (log-scale) in a two-dimensional Kuramoto system on a lattice for different coupling strengths ($\kappa $). (

**b**–

**d**) Figures show scaled standard deviation of the temperature as a function of time (log-scale) for different fluid samples in a Rayleigh–Bénard convection system. Note that the Rayleigh number ($Ra$) changes from non-turbulent to turbulent. (

**e**–

**h**) Figures show the evolution of the order parameter (R) as a function of time (log-scale) for the four systems. Note that time is in seconds.

**Figure 3.**(

**a**) Figure shows the probability density functions (log-scale) for the scaled angular frequency fluctuation ($\delta {\omega}^{\star}$). The initial randomized state data is fit with a Gaussian (in black) and the final state data is fit with a Lorentzian (in red). (

**b**) Figure shows the probability density functions (log-scale) for the scaled thermal fluctuation ($\delta {T}^{\star}$) for two different fluid samples at room temperature along with respective Gaussian fits. (

**c**) Figure shows the probability density functions (log-scale) for the scaled thermal fluctuation for two different fluid samples at steady-state along with kernel density estimates (KDE). Note that in the final state the two samples correspond to two separate Rayleigh numbers. (

**d**) Figure shows the probability density functions (log-scale) for the scaled thermal fluctuation for two different fluid samples at steady-state along with respective Gaussian (in black) and Lorentzian (in red) tails. The absence of sufficient data points prevent us from fitting the final state data of the $Ra=1790$ sample with a Lorentzian function.

**Figure 4.**Figure shows scaled standard deviation ($\sigma /{\sigma}_{max}$) of the angular frequency and lattice entropy ($S/{S}_{max}$) as a function of time in a two-dimensional Kuramoto system on a lattice.

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**MDPI and ACS Style**

Chatterjee, A.; Mears, N.; Yadati, Y.; Iannacchione, G.S.
An Overview of Emergent Order in Far-from-Equilibrium Driven Systems: From Kuramoto Oscillators to Rayleigh–Bénard Convection. *Entropy* **2020**, *22*, 561.
https://doi.org/10.3390/e22050561

**AMA Style**

Chatterjee A, Mears N, Yadati Y, Iannacchione GS.
An Overview of Emergent Order in Far-from-Equilibrium Driven Systems: From Kuramoto Oscillators to Rayleigh–Bénard Convection. *Entropy*. 2020; 22(5):561.
https://doi.org/10.3390/e22050561

**Chicago/Turabian Style**

Chatterjee, Atanu, Nicholas Mears, Yash Yadati, and Germano S. Iannacchione.
2020. "An Overview of Emergent Order in Far-from-Equilibrium Driven Systems: From Kuramoto Oscillators to Rayleigh–Bénard Convection" *Entropy* 22, no. 5: 561.
https://doi.org/10.3390/e22050561