Entropy

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What is Entropy?

Entropy is the element of chaos.

It sounds like one of the Empedoclesian four elements – earth, wind, fire, and water – or the Gallic four humours – phlegm, blood, and black and yellow bile. But it isn’t.

Integral to all modern branches of science, it describes how it’s easy to crumble a biscuit, but harder to put it back together. How, left un-fiddled-with, things tend to get messier and dirtier and shabbier.

Entropy, or disorder, is a scientific measure of things spreading out, like a drop of food colouring in a glass of water, gradually diffusing until the whole solution is an even, pale colour. When the particles in the glass of water randomly collide until the food colouring particles are evenly dispersed, they reach their highest entropy – the macrostate, or big picture, that can be made from the most microstates, or different ways of arranging particles.

Humans are living oppositions to entropy, a carefully structured and ordered system that cooperates to form a functioning organism. All our cells in their right places, and doing their correct jobs. This costs energy – because entropy always increases. Unless energy is put in to create order, particles disperse, spread out, and entropy increases. It costs a lot of energy to maintain order. It’s astonishing that we exist at all really.

Discovery of entropy.

Entropy has its roots in thermodynamics – the study of the movement of thermal energy. Thermodynamics works much the same way as the drop of food colouring in the water – heat spreads out. That’s why, if you touch a piece of metal, it feels hot (thermal energy is transferred to you) or cold (thermal energy is transferred away from you). It’s also why no machines are 100% efficient: some energy is always lost as heat, no matter how well you insulate.

It was this last observation that led Rudolf Clausius to develop the first definition of entropy in the 1850s. Entropy, he said, describes how some energy in any irreversible process is always lost to the surroundings via dissipation or friction.

Today, we use the term not only to describe heat transfer, but also the transfer of other kinds of energy, the transfer of matter, or information.

NASA/ WMAP Science Team
This diagram shows how scientists think the universe may have expanded from a single point into the vast space we see today. Image credit: Public domain

Cosmic microwave background.

The entropy of the universe is dominated by the cosmic microwave background, the radiative backdrop to the observable sky. Faint variations in this background radiation create an energy map from shortly after the big bang – an anisotropic landscape of fossilised energy fluctuations. The cosmic microwave background radiation tells the story of an early universe in thermal equilibrium, balanced at a maximum entropy. But something happened to that equilibrium. Material clustered together. Atoms formed. Planets formed. Galaxies.

But how can this be? If entropy is always increasing, the universe is becoming more chaotic, how can ordered, condensed systems like planets form? It doesn’t make sense.

Scientists think that it’s all due to another factor altogether: gravity. Gravity works very differently to thermal energy because very spread out things have a very low rather than high gravitational entropy. When matter sticks together, it releases gravitational potential energy to its surroundings, allowing more disorder. So the gravitational entropy of the universe has increased, although thermal entropy has decreased. This must mean that gravitational entropy dominated the early universe, that or there are other entropic factors at play.

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Dark matter entropy.

Does dark matter also have entropy? We don’t know. So far, the temperature fluctuations in the cosmic microwave background radiation can only be accurately modelled by assuming that it does. However, an alternative theory called entropic gravity, or emergent gravity does exist, which not only requires no dark matter contribution to entropy, but is in fact incompatible with dark matter theory entirely.

Entropic gravity is a theory that gravity is an emergent force rather than fundamental one, and arises from the statistical behaviour of a large bunch load of particles. The theory goes that macroscale (big picture) homogeneity arises from microlevel disorder or, in other words, that the even arrangement of dye molecules in water hides the chaotic interaction of lots and lots of small, tumbling particles.

The theory of entropic gravity is based on string theory, black hole physics, and quantum information theory, and the second law of thermodynamics, which states than entropy always increases. In the theory, because changing the arrangement of particles changes the amount of information needed to describe a system, it also changes the entropy. Matter affects the arrangement of particles, and so creates an entropic force – what we know as gravity. This theory is nearly impossible to test, however, because it predicts exactly the same gravitational behaviour as general relativity. And where it differs, at the extremes of physics, it’s not really practical to test it. Some have tried to test a theoretical equation for it developed by physicist Erik Verlinder. However, the equation was only intended to work in a narrow scenario, and many of the tests fall outside its scope.
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Unmismoobjetivo
Artist's logarithmic scale conception of the observable universe with the Solar System at the centre all the way to Big Bang's invisible plasma on the edge. Image credit: Unmismoobjetivo  

What is the entropy of the universe?

Even the inflation of the universe is just a theory, how it started and how it slowed remain unknown. Competing theories exist to explain how, such as the theory that, at extreme temperatures like those present at the start of the universe, light rays travelled faster than the speed of light today. This theory can be tested against the expansion theory as we get ever more accurate estimates for the spectral index of the universe.

This is important because a universe of limited size has a limited entropy – and we can measure it. A flat Newtonian universe is limitless, but an expanding universe has an event horizon: a maximum distance that an emitted signal could reach. The current distance to the event horizon has been calculated at 7.15 giga light years[1], and from this the maximum entropy of the universe, 122106.2kB. At this maximum entropy, there could be several possible universes with several difference balances between different kinds of opposing entropy – scientists still aren’t sure what the universe would look like. But how different would it be from now?

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How much has the entropy of the universe increased?

Scientists have also tried to estimate how much the entropy of the universe has increased overall. This gives us an idea how quickly it’s changing.

The number they get is a factor of 1015 which seems big until you recall the value of the overall entropy of the universe ~1090kB. The increase in entropy of the universe is almost entirely due to the formation of black holes. The entropy contribution from the cosmic microwave background radiation is excluded because even though it dominates the entropy of the universe[2], the amount hasn’t changed very much. There is also an entropy increase from starlight, but this has probably only increased the entropy of the universe by a factor of 1.02 (dark matter could significantly alter this figure. Or not. We don’t know)[3][4]. Forming planets and stars has also decreased entropy – but only by a small amount. Just 10% of matter is stars and 0.01% planets. In fact, most visible matter hasn’t changed very much.

“It would seem that all structured matter in the universe is but a thin scum floating on an enormous ocean of chaos.”

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The hole in the center is within the schwarzschild radius - the “event horizon” beyond which light cannot escape. The larger the schwarzschild radius the lower the density of the black hole. Image credit: Public domain

The entropy of black holes.

Since the 1990s, people have wondered whether the universe really needs three dimensions. Isn’t two enough? As it turns out, describing the universe in two dimensions could resolve some incompatibilities between relativity and quantum mechanics, including the information paradox that nothing can escape a black hole, BUT matter can never be destroyed. This is because, in two dimensions, things do not “enter” a black hole, merely pass beyond the edge and stay there, locked to the middle by gravity. This hologram hypothesis holds that the three dimensional nature of black holes could just be an illusion. In agreement with the hypothesis, scientists including Stephen Hawking have suggested that the entropy of black hole could be proportional to its area, and not its volume. This idea is consolidated in the Bekenstein-Hawking formula for black hole entropy, the best mathematical description we have of black hole entropy. However, two dimensional black holes and their implications on other areas of physics remain a hypothesis.

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Spacetime curvature. Image credit: Licensed under CC BY-SA 3.0 via Wikimedia Commons 

Time.

If two dimensions doesn’t sound like enough, how about four? Time – often named as the fourth dimension – is sometimes defined in terms of entropy, as the direction in which entropy increases. That is, we are moving from a position of low entropy (low disorder or high order) at the start of the universe, and towards a higher and higher entropy state – the possible heat death of the universe. According to this description, time time moves irreversibly forward in the direction of change, ensuring that all events don’t occur simultaneously.

But now imagine two particles colliding and bouncing apart. And again in reverse – two particles colliding and bouncing apart. The physical direction is the same, but it looks in time the same forwards as backwards. This means at a particle level there is no entropy – it’s not more chaotic one way or the other way. However, the big picture looks a bit different – things are generally spreading out. So, according to our entropy definition of time, it doesn’t really exist at a particle or quantum level.

Materials and Structure.

Particles may not behave like time exists when they’re moving, but when they are trapped in solid materials, their level of order – or entropyis important. In fact, how ordered or disordered solid materials are directly effects their material properties, things like magnetism, conductivity, flexibility, and thermal expansion.

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Orbital disorder

Disorder in materials can begin at an orbital level – where the electrons in an atom sit.

Whether an orbital is full, empty or, in particular, partially filled, can lead to long-range orbital disorder, especially in compounds containing transition metals. The more partially filled orbitals, the higher the orbital entropy because the more configurations, or different ways of arranging the orbitals. What’s more, bonds are formed when orbitals overlap, so orbital disorder changes the strengths and lengths of bonds. One example is the Jahn-Teller distortion, where six bonds around a metal centre form one long pair and two short pairs.

Despite orbital disorder, the same configurations are seen over and over again, because of something called “coupling” – when atoms influence their neighbours. If coupling is weak, the material shows no long-range order, but if coupling is strong, materials exhibit long-range order and low entropy. Materials scientists are interested in how strong and weak coupling happen.

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An electron moves in an atomic “orbital” and its position is described by a probability distribution. Here, shading density shows that distribution. Image credit: Via Wikipedia Commons RJHall Bryce and Paint Shop Pro 

Correlated disorder.

If the configuration is neither completely ordered nor completely disordered (somewhere between highest and lowest entropies), we call the structure correlated disorder. Correlated disorder can profoundly influence material properties, and scientists are interested in how and why it happens.

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Examples of chains, layers and open frameworks. There are so many good ways to bond. Image credit: Public domain

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There are many ways of arranging spins in a tetrahedron, but none of them minimise entropy. Image credit: Rowena Fletcher-Wood

Geometric frustration.

Correlated disorder can come about because of geometric frustration – a situation where a structure can’t lower its energy (and so become more stable) because of the arrangement of atoms in space. A simple example is a tetrahedron (triangular based pyramid) of atoms with spin. If one corner of the tetrahedron is a spin up, all adjacent corners would like to be spin down so that the spins can couple and lower the energy of the structure. However, all three adjacent spins are themselves adjacent, and want to couple. The geometric arrangement of atoms makes it impossible for all of them to couple simultaneously, and a higher energy configuration must be adopted.

When there’s more than one possible geometrically frustrated arrangement or “ground state”, the structure has configurational entropy. Many ground states have been theorised but not observed.

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Ice, made of tetrahedrally-coordinated water molecules, is geometrically frustrated. In a three dimensional lattice, tetrahedrons with two two slightly negative lone pair corners and two slightly positive hydrogen corners can’t perfectly link up to lower the energy, and so have residual entropy. The lowest entropy arrangement, “spin ice” has not yet been observed, although its equivalent structure has been seen in similar minerals.
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Topological phases.

At zero Kelvin, -273oC, absolute zero, solids form with imperfect ordering. This leaves them with residual entropy – because there are lots of ways they can be imperfectly ordered. They also don’t have the energy to transition into another topological phase (or state). This can profoundly affect properties, and scientists are interested in the quantum entanglement effects that lead to different topological phases. In fact, the mere existence of topological phases contradicts existing natural symmetry laws, yet we keep discovering “new” (topological) phases of matter that could be immediately used in technology such as quantum computing.

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Material Properties.

Some materials even have a “glass transition”, a reversible transition between a hard, brittle state and molten, rubbery state with very little difference in entropy. These two states have the same thermodynamic quantities such as heat capacity, suggesting that the state change is kinetic. However, this theory has a flaw in it: because the entropy difference gets smaller and smaller at lower temperatures… meaning that if you could supercool the liquid, the entropy difference would eventually become negative! This is unreasonable, because it would mean the liquid was more organised than the solid, which is opposite to their definitions, and is known as the Kauzmann Paradox[5].

Solutions include the idea that heat capacity changes, and that there’s a third intermediate state, but scientists don’t know and can’t agree.
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Thermodynamics is very successful at describing the way energy interacts with itself and matter - but every now and then a paradox will pop up, or someone will spot an event which no one has an explanation for yet. Image credit: Public domain

Some materials exhibit negative thermal expansion and/or negative compressibility – contracting upon heating and expanding upon cooling, or expanding under pressure and contracting when pressure is reduced. Scientists now think that one cause for this may be entropy, after discovering that some smaller volume materials are more disordered than larger volume materials[6].

Negative thermal expansion materials are important in engineering photonics, electronics and structures, such as for the formation of composite materials like tooth filling that have the same expansion properties as tooth enamel.
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Information entropy.

These days, entropy has a wider application in the field of information theory. Originally developed to measure how the loss of information down phone lines varies statistically, information entropy is used as a description of how much information there is in an event. The more information, the more entropy.

It can also be explained in terms of uncertainty. The greater the uncertainty around an event or measurement, the more information is needed. Very uncertain things can only be described in terms of probability distributions – guessing which outcomes are more statistically likely. This means that decreasing the uncertainty around an event or measurement is the same as decreasing the amount of information needed to describe it. We can also see intuitively that decreasing the uncertainty means decreasing the disorder, or entropy.

Information entropy is used by scientists, especially computer scientists, in cryptography and data transfer, but it’s also important in biology, chemistry, physics and other forms of practical science filled with events and measurements we just don’t know the answers to yet.

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Information entropy was originally developed to describe the statistical loss of information through phone lines. Image credit: via Wikipedia Commons 

Entropy, chaos, the element of disorder is all around us. In our stars, in our materials, and in – but not as much as we might think – our bodies. There are various definitions for different applications, but there are still lots of things we don’t know about entropy or how it acts. In fact, entropy is entwined with various fundamental principles that we rely on in science, and understanding it better could change our perspective of time, fundamental particles, or the dimensionality of the universe.


This article was written by the Things We Don’t Know editorial team, with contributions from Jon Cheyne, Freya Leask, Johanna Blee, Rowena Fletcher-Wood, and Alice Wayne.

This article was first published on and was last updated on 2020-07-30.

References
why don’t all references have links?

1. Egan, Chas A., and Charles H. Lineweaver. "A larger estimate of the entropy of the universe." The Astrophysical Journal 710.2 (2010): 1825 doi: 10.1088/0004-637X/710/2/1825/meta
2. Bousso, R., Harnik, R., (2010) Entropic landscape Physical Review D 82(12) doi: 10.1103/PhysRevD.82.123523
3. Hawking, S,W., (1974) Black hole explosions? Nature 248(5443):30-31 doi: 10.1038/248030a0
4. Bekenstein, J,D., (1972) Black holes and the second law Lettere Al Nuovo Cimento Series 2 4(15):737-740 doi: 10.1007/BF02757029
5. Speedy, Robin J. Kauzmann's paradox and the glass transition. Biophysical chemistry 105.2-3 (2003): 411-420 doi: 10.1016/S0301-4622(03)00105-4.
6. Liu, Z., Wang, Y., Shang, S., (2011) Origin of negative thermal expansion phenomenon in solids Scripta Materialia 65(8):664-667 doi: 10.1016/j.scriptamat.2011.07.001

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