What is Entropy? Entropy is a scientific thermodynamic measure (of the universe, system, or control volume) that can increase with time because it is mostly a consequence of rearranging energy and distributing it (often more uniformly) while the system attempts to reach equilibrium or a steady state. While it is most commonly described as a “measure of disorder or randomness,” engineers more precisely define it as a measure of how many different ways a system can rearrange itself within its constraints. Entropy is given the symbol S, with units of joules per kelvin (J/K). Or the lowercase “s,” which is the entropy per unit volume or per mole. The discovery of entropy as a thermodynamic quantity came about with the understanding that led to the articulation of the Second Law of thermodynamics: the unavailability of a part of a system’s thermal energy for conversion into work, often interpreted as the degree of disorder or randomness in the system, which is natural. Because energy conservation is a consequence of local time-symmetry conditions, and energy itself could be a property of ripples in space-time, one definition of entropy is that it is merely a logarithmic measure of the rate at which information can be accurately transmitted in a particular message or language. When multiplied by the Boltzmann constant, this definition too yields dimensions of entropy. The Boltzmann constant (k or kB) is a fundamental physical constant that links the average kinetic energy of particles in a gas to the temperature of that system. Its exactly defined value is 1.380649 × 10⁻²³ J K⁻ (the same as 1.380649 × 10-23 m2 kg s-2 K-1). When multiplied by Avogadro’s number, it yields the universal molar gas constant R, which has an exactly defined value of 8.3144 J mol-1 K-1.
Classical thermodynamics recognizes distinct pathways of work and heat for a change of state but stops short of describing the influence of the rate of work and heat production (also known as the useful power or heating rate, respectively) and consequently does not easily address the entropy generation rate. The manifestations of work and heat rates are always practically apparent, but the manifestation of the entropy generation rate is less so.
What is Entropy Generation Rate (EGR) and why is it important? When energy is transferred, rearranged, or used for an objective, new entropy is always created. This new entropy production is called entropy generation. It arises from gradients of temperature, pressure, chemical potential, electrical charge, or surfaces. Such gradients can be caused by temperature or pressure differences, chemical species reacting, ionization, voltage differences, or phase-change reactions and similar processes, i.e., driven by a thermodynamic force or imbalances. Whenever there is a gradient, new entropy is generated. The rate at which new entropy is generated (produced) is called the entropy generation rate (EGR). Unlike energy, entropy is not a conserved quantity, so this newly produced entropy adds to the overall entropy of the universe. More importantly, while dissipating the gradient, the entropy generation can produce beautiful patterns. This is the basis of observable patterns in a self-organizing system. Patterns have repeatable spacings and so often influence engineering properties, such as resilience. For example, grain patterns influence the measured engineering hardness in crystallized metallic solids. A lower grain diameter commonly indicates a higher hardness. A higher rate of entropy production yields a finer pattern. This entropy generation invariably results in a loss of traditional thermodynamic efficiency (in extracting work, e.g., electrical energy). Still, it can be used in several creative ways for energy storage and other useful purposes (such as resilience to cracks and dislocations due to pattern spacing). Pattern formation is linked to the rate of entropy generation, especially in steady-state open systems. Weather selection and cloud-type formation are related to the rate of tropospheric entropy generation. A steady-state entropy generation rate can also be applied to bird flight formation assessments where entropy generation leads to stored work potential. Patterns are related to the entropy generation rate.
The importance of the rate of entropy generation (EGR) lies in two key practical aspects, namely, (i) it is related to the loss of efficiency in power-producing systems and (ii) it is related to pattern spacings and thus to contextual properties like resilience and connectivity. By analyzing EGR for a process, engineers can better design systems to minimize high-quality energy waste and predict long-term connectivity for environmental and mechanical stability. Source (link).
Most high-efficiency applications preserve energy quality.
The distribution possibilities of newly produced entropy and the overall energy into compartments of the whole depend on the process. Rapid processes typically give rise to gradients (this is why conventional thermodynamics assumes slow change). A flux and a conjugate force generate entropy. For example, a flux of heat (energy) is driven by a temperature gradient (the conjugate force) and thus creates new entropy (this is an example of Onsager’s relationship). In the picture below, new entropy is produced by the thermal energy (heat) that flows down a temperature gradient in a metal rod held between two reservoirs at temperatures T1 and T0. This is a simple example that shows the entropy generation rate is maximized for several processes- see the article.
Are you wondering what patterns could be related to solid-state heat transfer? In metals, or at least when solids are near their melting points, heat transfer can cause noticeable atomic diffusion. Atoms migrate from hotter (higher-stress) regions to cooler regions along the boundaries between crystalline grains, which themselves respond to thermal conditions. This bulk movement of atoms is a microscopic analog of fluid convection. In some solids, electron motion and related patterns can be driven by thermal gradients, and vice versa.
The entropy generation rate is given the symbol.
which can only be zero or positive. The units are joules per kelvin per second (J/K · s).
Entropy generation rate (Sgen rate) by heat transfer between two temperatures connected by a metal rod of length L. The direct power (rate of work) extracted, Pmax (J/s), is zero in this configuration.
Why is the concept of Maximum Entropy Generation Rate (MEPR) important? The Maximum Entropy Generation Rate (MEPR) principle – also sometimes referred to as the Maximum Entropy Production Principle (MEPP) – is important because it likely predicts how complex systems will behave (resilience) or self-organize in nature. The maximum entropy generation rate appears to predict S-shaped transformations over time. At equilibrium, gradients are dissipated. If you think about it- open systems at steady state produce entropy that is at its maximum possible amount (rate). It stands to reason that small (infinitesimal) changes in the system will have to produce the maximum rate during transition. Large changes will be an accumulation or a pathway for the maximum entropy generation rate and hence are the pathway for self-organization. see the article and cited references. Source (link).
Applications related to pattern evolution: Tropospheric weather development, Equilibrium and non-equilibrium clouds, solidification patterns (crystallization), transformation kinetics, chemical reactions, metal deformation, sintering, surface patterns in nature, bird organization, self-organization of chemical and sociological groups, heat pumps, energy efficiency calculations, or flow-flow resistance of a fluid in a pipe, i.e., wherever it takes a finite amount of time to complete a process. There is a quantum mechanical analogy to entropy generation that is not discussed here.
Keeping the entropy generation rate maximized seems to be a natural outcome of spontaneous processes.
A Statistical Basis. Although an S-curve based on a symmetrical (normal) distribution of rate can be used to represent a self-organizing process, one should be aware that skewness in a distribution can also yield an S-curve-like transformation (albeit with a tail), as shown.
By definition, a normal distribution has skewness 0. A skewness of 2 corresponds to an exponential distribution. Normalized for the standard normal distribution with mean μ, std. dev. σ, and z=(x-μ)/σ) are shown below.
PDF(z) (Normal Distribution) = e^(-z^2/2)
CDF(z) (Normal Distribution) = 1/2 [1+erf (z/√2)]. This is a classic S-Curve
For an Exponential (λ=1) (so μ=1and σ=1). The domain is x ≥ 0, which implies z ≥ -1. For an exponential distribution (λ=1) (μ=1 and σ=1 and z=(x-1/λ)/(1/λ)) as shown below.
PDF(x) (Exponential Distribution) = λe^(-λx), x≥0
CDF(x) (Exponential Distribution) = 1-e^(-λx), x≥0
(λe^(-λx))/[(1-e^(-λx))^2] = (λe^(-(z+1)))/(1-e^(-(z+1)) )^2 for z≥-1
The PDF (e.g., rate of change of temperature vs. time) and CDF (S-shaped curves, e.g., for temperature) for each distribution are shown for the normal and exponential distributions. The PDF/(CDF)^2 approximates the entropy generation rate. The cumulative integral (which also looks like an S-curve) is shown below. The total (cumulative) entropy produced (generated) by a skewed distribution process is lower than from a normal distribution-behaved process. It is good to bear in mind that lack of order or predictability, and gradual decline into disorder, is one definition of entropy.
Why are the Gaussian and exponential distributions important to compare for EGR? Amongst many things, it shows how anisotropies can generate entropy and a natural attempt to establish the rate of entropy generation, whether in small systems or galaxies, therefore fundamentally answering the question of the creation of resilient order while pursuing a pathway towards disorder so as to maximize the rate of entropy generation.
One of our AI friends, after reading the source article above and the key references, opined that the real question is: What kind of order best dissipates the available gradients produced by entropy generation? The real mystery may not be why entropy increases. The deeper question may be why entropy generation so often produces beautiful, persistent, and resilient structures (patterns) and connectivity along the way.
