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 the 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-assemble are coupled to one another in some way, either through local interactions or through continuous media. Therefore, the general nature of non-equilibrium fluctuations of the intrinsic variables in these systems is found to follow similar trends as order emerges. Through this paper, we attempt to look for connections between among these systems and systems in general which give rise to emergent order when driven out-of-equilibrium.

Let's look at a combustion engine, which in the most literal sense is designed to exploit the Second law of Thermodynamics. Take a glass of petrol, it is in an ordered form of energy by itself but when the petrol is ignited in a combustion engine, the volume increases (not to mention the heat and sound dumped onto the environment) and the once ordered energy becomes disordered.  The combustion engine is special in that it can harness the dissipating energy and tap into the flow of heat and use it to create a small pocket of order to run its pistons in the midst of all that disorder.  Combustion engines are not the only ones, even life itself does this.  When you receive energy in the form of food, your body processes the ordered energy and converts it into more disordered energy but it uses the proceeds of it to power itself.  All the while the entropy of the universe is constantly increasing. In fact, we see this all around us, from crack propagation in materials to phase and glass transitions, from inter-cellular transportation at the nano-scale that form the basis of life to the weather systems that dictate the movement of the water, carbon, and nitrogen cycles of our planet.  There is a unifying theme that comes across from this and that is, there is a high level of complexity that emerges spontaneously in these systems. Classical equilibrium thermodynamics cannot answer this and is therefore rendered insufficient in explaining the underlying dynamics that allow for such complexity. In this report, we study one such system, the steady-state Rayleigh-B ́enard Convection.  An important observation we were able to see through the current study was how temperature took on a time-independent trait yet maintained a spatially varying character.  Our study employs a unique technique by using an Infrared camera to extrapolate thermal profiles and present them statistically.  Ona further notes, the report also discusses the different approaches that can be adapted to extend the current study to turbulence, complex networks, and information theory.

A challenge in fundamental physics and especially in thermodynamics is to understand emergent order in far-from-equilibrium systems. While at equilibrium, temperature plays the role of a key thermodynamic variable whose uniformity in space and time defines the equilibrium state the system is in, this is not the case in a far-from-equilibrium driven system. When energy flows through a finite system at a steady-state, temperature takes on a time-independent but spatially varying character. In this study, the convection patterns of a Rayleigh-Bénard fluid cell at steady-state is used as a prototype system where the temperature profile and fluctuations are measured spatio-temporally. The thermal data is obtained by performing high-resolution real-time infrared calorimetry on the convection system as it is first driven out-of-equilibrium when the power is applied, achieves steady-state, and then as it gradually relaxes back to room temperature equilibrium when the power is removed. Our study provides new experimental data on the non-trivial nature of thermal fluctuations when stable complex convective structures emerge. The thermal analysis of these convective cells at steady-state further yields local equilibrium-like statistics. In conclusion, these results correlate the spatial ordering of the convective cells with the evolution of the system’s temperature manifold.

In this paper, we present a detailed description of the statistical and computational techniques that were employed to study a driven far-from-equilibrium steady-state Rayleigh-Bénard system in the non-turbulent regime. In our previous work on the Rayleigh-Bénard convection system we try to answer two key open problems that are of great interest in contemporary physics:(i) how does an out-of-equilibrium steady-state differ from an equilibrium state and (ii) how do we explain the spontaneous emergence of stable structures and simultaneously interpret the physical notion of temperature when out-of-equilibrium. We believe that this paper will offer a useful repository of the technical details for a first-principles study of a similar kind. In addition, we are also hopeful that our work will spur considerable interest in the community which will lead to the development of more sophisticated and novel techniques to study far-from-equilibrium behavior.

Systems that are out-of-equilibrium and exhibit complex patterns are difficult to characterize. These systems are thermodynamically open, thereby constantly exchanging heat and matter with the surrounding. In order to be able to systematically and rigorously characterize such a system we present an experimental study of an out-of-equilibrium thermodynamic system, the Rayleigh-Bénard convection cells. When a thin layer of liquid is evenly heated from the bottom the liquid tends to self-organize into patterns of hexagonal cells or a series of rolls. The cells or rolls are an outcome of an upward flow of the hot layer of the liquid from the bottom, and a downward flow of the cool layer of the liquid from the top along with the competing effects of viscosity and thermal gradient. The goal of this presentation is to measure the flow of energy through the system and be able to quantify it, along with steady state measurements of the temperature fluctuation, entropy and internal work calculation by infrared imaging as a function of fluid viscosity and thickness over a wide range of heat input. The resulting thermal patterns will be interpreted in terms of a balance between internal entropy and work.