The many faces of far-from-equilibrium thermodynamics: Deterministic chaos, randomness, or emergent order?
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ntroduction A system, isolated from its surrounding, will continue to be in a state of equilibrium unless driven by an external steady flow of energy. Statistically, a state of equilibrium implies a state of randomness, and randomness implies symmetry. Therefore, from a microscopic sense, all states in a statistical ensemble are equally likely, and the system explores all possibilities before collapsing into a single-state point in the phase space, characterized by macroscopic thermodynamic variables such as pressure, temperature, and volume. The physics of such equilibrium systems forms the basis of classical thermodynamics and statistical mechanics.1–5 However, if we observe our surroundings, there are numerous systems that are open and are constantly being fed with energy. Examples of such actively driven systems include self-assembly in biological systems; the reaction-diffusion process in chemical and ecological sciences; thermal-convective phenomena in fluid dynamics, geophysical and atmospheric sciences; and fracture propagation in materials science.6–14 The unifying theme across all of these examples, from nanoscale to macroscale, is the staggering complexity that emerges spontaneously. As expected, equilibrium thermodynamics is insufficient in explaining the underlying dynamics. Typically, far-from-equilibrium thermodynamics is treated as a natural extension of equilibrium thermodynamics. Such an approach is based on the local equilibrium hypothesis, according to which a system can be viewed as a
collection of subsystems where the rules of equilibrium thermodynamics hold true.15,16 However, in reality, a simple theoretical Carnot engine (C), which exchanges heat between two reservoirs maintained at different temperatures and generates work, becomes incredibly difficult to visualize in practice (see Figure 1a).17,18 Even in order to maintain the heat baths at a constant temperature, a steady heat influx is mandatory. Thus, a practical Carnot engine (C′) no longer remains as efficient as a theoretical Carnot engine, and its efficiency is now expressed as a function of steady-state nonequilibrium temperature of the baths and subsequent far-from-equilibrium corrections (see Figure 1b). The problem we face is twofold—the absence of definition of the thermodynamic variables in a far-from-equilibrium scenario, and the ability to fit the ideas of emergence of order and complexity into a theoretical framework. While a system at equilibrium is completely random, a system when driven out-of-equilibrium is extremely sensitive to the magnitude of the driving perturbation. For example, when water is heated over a flame, it takes some time to create a steadystate thermal gradient. Once the gradient is established, a convection current is set up that drives the hotter molecules to the top and colder molecules to the bottom, in a cycle. This onset of convection is denoted by the critical value of the dimensionless constant, the Rayleigh number (Ra). In further increasing the gradient, the convective motion becomes chaotic
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