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

Abstract

Soft-matter systems when driven out of equilibrium often give rise to structures that usually lie in between the macroscopic scale of the material and microscopic scale of its constituents. In this paper we review three such systems, the two-dimensional square-lattice Ising model, the Kuramoto model and the Rayleigh–Bénard convection system which when driven out of equilibrium give rise to emergent spatio-temporal order through self-organization. A common feature of these systems is that the entities that self-organize are coupled to one another in some way, either through local interactions or through a continuous media. Therefore, the general nature of non-equilibrium fluctuations of the intrinsic variables in these systems are found to follow similar trends as order emerges. Through this paper, we attempt to find connections between these systems, and systems in general which give rise to emergent order when driven out of equilibrium. This study, thus acts as a foundation for modeling a complex system as a two-state system, where the states: order and disorder can coexist as the system is driven away from equilibrium.

Coexisting Ordered States, Local Equilibrium-like Domains, and Broken Ergodicity in a Non-turbulent Rayleigh-Bénard Convection at Steady-state

“…Rather, we use this paper as an avenue to present experimental evidence in support of the local-equilibrium assumption in a prototypical driven system, the non-turbulent Rayleigh-Bénard convection at steady-state. This work builds on our previous studies where we have performed extensive thermal analysis of these convective cells, and have shown that the temperature manifold bifurcates into regions of local sources and sinks as macroscopic order emerges [28,29,30,31,32,33]. In this paper, we first show that the temperature distribution profiles of these localized domains that coexist together at a non-equilibrium steady-state exhibit room temperature equilibrium-like statistics.…”

Synchronization in phase-coupled oscillator with attractive–repulsive frequencies

“…In general, the presence of such randomness models a wide range of collective behaviors observed in diverse out-of-equilibrium systems, characterized by tilted or switching periodic potentials, affected by stochastic fluctuations of temperature, current, or magnetic field, indicates the presence of noise-induced phenomena, such as noise-enhanced stability and resonant activation with different features [32][33][34]. In fact, randomness is not only able to generate disorder, but it may also give rise to new ordered states and induce unexpected dynamical behaviors, which would not be found in the deterministic counterpart of the system such that a mixture of synchronized and unsynchronized oscillators in which some oscillators reach the common frequency and lock themselves in that state before the other, and the evolution of order is inversely related to the fluctuations of the intrinsic oscillators [35,36]. These significant studies show that the underlying frequency structure is critical since it controls the model synchrony transition phases and drastically alters the model's stability aspects and their dimensionality reduction, especially when they are altered by noise.…”

Pattern Formation in Thermal Convective Systems: Spatio-temporal Thermal Statistics, Emergent Flux, and Local Equilibrium

“…Therefore, a linear relationship between surface temperature standard deviation and emergent flux also implies a linear relationship between surface temperature standard deviation and thermodynamic force. We believe that this observation is non-trivial as this emergent thermodynamic force should comply with the second law of thermodynamics and decrease the potential of the system -as indicated by the largest decline during the time evolution of the surface temperature standard deviation, or equivalently lead to an increase in the entropy production during the dissipative process of pattern formation [5,17,[28][29][30].…”