A central concept in Shannon’s information theory is something that he called entropy, which in this context is a measure of how statistical patterns in a source of messages affect the amount of information that the messages can convey. If certain patterns of bits are more likely than others, then their presence conveys less information, because the uncertainty is reduced by a smaller amount. In English, for example, the letter ‘E’ is much more common than the letter ‘Q’. So receiving an ‘E’ tells you less than receiving a ‘Q’. Given a choice between ‘E’ and ‘Q’, your best bet is that you’re going to receive an ‘E’. And you learn the most when your expectations are proved wrong. Shannon’s entropy smooths out these statistical biases and provides a ‘fair’ measure of information content.
In retrospect, it was a pity that he used the name ‘entropy’, because there is a longstanding concept in physics with the same name, normally interpreted as ‘disorder’. Its opposite, ‘order’, is usually identified with complexity. The context here is the branch of physics known as thermodynamics, which is a specific simplified model of a gas. In thermodynamics, the molecules of a gas are modelled as ‘hard spheres’, tiny billiard balls. Occasionally balls collide, and when they do, they bounce off each other as if they are perfectly elastic. The Laws of Thermodynamics state that a large collection of such spheres will obey certain statistical regularities. In such a system, there are two forms of energy: mechanical energy and heat energy. The First Law states that the total energy of the system never changes. Heat energy can be transformed into mechanical energy, as it is in, say, a steam engine; conversely, mechanical energy can be transformed into heat. But the sum of the two is always the same. The Second Law states, in more precise terms (which we explain in a moment), that heat cannot be transferred from a cool body to a hotter one. And the Third Law states that there is a specific temperature below which the gas cannot go – ‘absolute zero’, which is around -273 degrees Celsius.
The most difficult – and the most interesting – of these laws is the Second. In more detail, it involves a quantity that is again called ‘entropy’, which is usually interpreted as ‘disorder’. If the gas in a room is concentrated in one corner, for instance, this is a more ordered (that is, less disordered!) state than one in which it is distributed uniformly throughout the room. So when the gas is uniformly distributed, its entropy is higher than when it is all in one corner. One formulation of the Second Law is that the amount of entropy in the universe always increases as time passes. Another way to say this is that the universe always becomes less ordered, or equivalently less complex, as time passes. According to this interpretation, the highly complex world of living creatures will inevitably become less complex, until the universe eventually runs out of steam and turns into a thin, lukewarm soup.
This property gives rise to one explanation for the ‘arrow of time’, the curious fact that it is easy to scramble an egg but impossible to unscramble one. Time flows in the direction of increasing entropy. So scrambling an egg makes the egg more disordered – that is, increases its entropy – which is in accordance with the Second Law. Unscrambling the egg makes it less disordered, and decreases energy, which conflicts with the Second Law. An egg is not a gas, mind you, but thermodynamics can be extended to solids and liquids, too.
At this point we encounter one of the big paradoxes of physics, a source of considerable confusion for a century or so. A different set of physical laws, Newton’s laws of motion, predicts that scrambling an egg and unscrambling it are equally plausible physical events. More precisely, if any dynamic behaviour that is consistent with Newton’s laws is run backwards in time, then the result is also consistent with Newton’s laws. In short, Newton’s laws are ‘time-reversible’.
However, a thermodynamic gas is really just a mechanical system built from lots of tiny spheres. In this model, heat energy is just a special type of mechanical energy, in which the spheres vibrate but do not move en masse. So we can compare Newton’s laws with the laws of thermodynamics. The First Law of Thermodynamics is simply a restatement of energy conservation in Newtonian mechanics, so the First Law does not contradict Newton’s laws. Neither does the Third Law: absolute zero is just the temperature at which the spheres cease vibrating. The amount of vibration can never be less than zero.
Unfortunately, the Second Law of Thermodynamics behaves very differently. It contradicts Newton’s laws. Specifically, it contradicts the property of time-reversibility. Our universe has a definite direction for its ‘arrow of time’, but a universe obeying Newton’s laws has two distinct arrows of time, one the opposite of the other. In our universe, scrambling eggs is easy and unscrambling them seems impossible. Therefore, according to Newton’s laws, in a time-reversal of our universe, unscrambling eggs is easy but scrambling them is impossible. But Newton’s laws are the same in both universes, so they cannot prescribe a definite arrow of time.
Many suggestions have been made to resolve this discrepancy. The best mathematical one is that thermodynamics is an approximation, involving a ‘coarse-graining’ of the universe in which details on very fine scales are smeared out and ignored. In effect, the universe is divided into tiny boxes, each containing (say) several thousand gas molecules. The detailed motion inside such a box is ignored, and only the average state of its molecules is considered.
It’s a bit like a picture on a computer screen. If you look at it from a distance, you can see cows and trees and all kinds of structure. But if you look sufficiently closely at a tree, all you see is one uniformly green square, or pixel. A real tree would still have detailed structure at this scale – leaves and twigs, say – but in the picture all this detail is smeared out into the same shade of green.
In this approximation, once ‘order’ has disappeared below the level of the coarse-graining, it can never come back. Once a pixel has been smeared, you can’t unsmear it. In the real universe, though, it sometimes can, because in the real universe the detailed motion inside the boxes is still going on, and a smeared-out average ignores that detail. So the model and the reality are different. Moreover, this modelling assumption treats forward and backward time asymmetrically. In forward time, once a molecule goes into a box, it can’t escape. In contrast, in a time-reversal of this model it can escape from a box but it can never get in if it wasn’t already inside that box to begin with.
This explanation makes it clear that the Second Law of Thermodynamics is not a genuine property of the universe, but merely a property of an approximate mathematical description. Whether the approximation is helpful or not thus depends on the context in which it is invoked, not on the content of the Second Law of Thermodynamics. And the approximation involved destroys any relation with Newton’s laws, which are inextricably linked to that fine detail.
Now, as we said, Shannon used the same word ‘entropy’ for his measure of the structure introduced by statistical patterns in an information source. He did so because the mathematical formula for Shannon’s entropy looks exactly the same as the formula for the thermodynamic concept. Except for a minus sign. So thermodynamic entropy looks like negative Shannon entropy: that is, thermodynamic entropy can be interpreted as ‘missing information’. Many papers and books have been written exploiting this relationship – attributing the arrow of time to a gradual loss of information from the universe, for instance. After all, when you replace all that fine detail inside a box by a smeared-out average, you lose information about the fine detail. And once it’s lost, you can’t get it back. Bingo: time flows in the direction of information-loss.
However, the proposed relationship here is bogus. Yes, the formulas look the same … but they apply in very different, unrelated, contexts. In Einstein’s famous formula relating mass and energy, the symbol c represents the speed of light. In Pythagoras’s Theorem, the same letter represents one side of a right triangle. The letters are the same, but nobody expects to get sensible conclusions by identifying one side of a right triangle with the speed of light. T
he alleged relationship between thermodynamic entropy and negative information isn’t quite that silly, of course. Not quite.
As we’ve said, science is not a fixed body of ‘facts’, and there are disagreements. The relation between Shannon’s entropy and thermodynamic entropy is one of them. Whether it is meaningful to view thermodynamic entropy as negative information has been a controversial issue for many years. The scientific disagreements rumble on, even today, and published, peer-reviewed papers by competent scientists flatly contradict each other.
What seems to have happened here is a confusion between a formal mathematical setting in which ‘laws’ of information and entropy can be stated, a series of physical intuitions about heuristic interpretations of those concepts, and a failure to understand the role of context. Much is made of the resemblance between the formulas for entropy in information theory and thermodynamics, but little attention is paid to the context in which those formulas apply. This habit has led to some very sloppy thinking about some important issues in physics.
One important difference is that in thermodynamics, entropy is a quantity associated with a state of the gas, whereas in information theory it is defined for an information source: a system that generates entire collections of states (‘messages’). Roughly speaking, a source is a phase space for successive bits of a message, and a message is a trajectory, a path, in that phase space. In contrast, a thermodynamic configuration of molecules is a point in phase space. A specific configuration of gas molecules has a thermodynamic entropy, but a specific message does not have a Shannon entropy. This fact alone should serve as a warning. And even in information theory, the information ‘in’ a message is not negative information-theoretic entropy. Indeed the entropy of the source remains unchanged, no matter how many messages it generates.
There is another puzzle associated with entropy in our universe. Astronomical observations do not fit well with the Second Law. On cosmological scales, our universe seems to have become more complex with the passage of time, not less complex. The matter in the universe started out in the Big Bang with a very smooth distribution, and has become more and more clumpy – more and more complex – with the passage of time. The entropy of the universe seems to have decreased considerably, not increased. Matter is now segregated on a huge range of scales: into rocks, asteroids, planets, stars, galaxies, galactic clusters, galactic superclusters and so on. Using the same metaphor as in thermodynamics, the distribution of matter in the universe seems to be becoming increasingly ordered. This is puzzling since the Second Law tells us that a thermodynamic system should become increasingly disordered.
The cause of this clumping seems to be well established: it is gravity. A second time-reversibility paradox now rears its head. Einstein’s field equations for gravitational systems are time-reversible. This means that if any solution of Einstein’s field equations is time-reversed, it becomes an equally valid solution. Our own universe, run backwards in this manner, becomes a gravitational system that gets less and less clumpy as time passes – so getting less clumpy is just as valid, physically, as getting more clumpy. Our universe, though, does only one of these things: more clumpy.
Paul Davies’s view here is that ‘as with all arrows of time, there is a puzzle about where the asymmetry comes in … The asymmetry must therefore be traced to initial conditions’. What he means here is that even with time-reversible laws, you can get different behaviour by starting the system in a different way. If you start with an egg and stir it with a fork, then it scrambles. If you start with the scrambled egg, and very very carefully give each tiny particle of egg exactly the right push along precisely the opposite trajectory, then it will unscramble. The difference lies entirely in the initial state, not in the laws. Notice that ‘stir with a fork’ is a very general kind of initial condition: lots of different ways to stir will scramble the egg. In contrast, the initial condition for unscrambling an egg is extremely delicate and special.
In a way this is an attractive option. Our clumping universe is like an unscrambling egg: its increasing complexity is a consequence of very special initial conditions. Most ‘ordinary’ initial conditions would lead to a universe that isn’t clumped – just as any reasonable kind of stirring leads to a scrambled egg. And observations strongly suggest that the universe’s initial conditions at the time of the Big Bang were extremely smooth, whereas any ‘ordinary’ state of a gravitational system presumably should be clumped. So, in agreement with the suggestion just outlined, it seems that the initial conditions of the universe must have been very special – an attractive proposition for those who believe that our universe is highly unusual, and ditto for our place within it.
From the Second Law to God in one easy step.
Roger Penrose has even quantified how special this initial state is, by comparing the thermodynamic entropy of the initial state with that of a hypothetical but plausible final state in which the universe has become a system of Black Holes. This final state shows an extreme degree of clumpiness – though not the ultimate degree, which would be a single giant Black Hole. The result is that the entropy of the initial state is about 10-30 times that of the hypothetical final state, making it extremely special. So special, in fact, that Penrose was led to introduce a new time-asymmetric law that forces the early universe to be exceptionally smooth.
Oh, how our stories mislead us … There is another, much more reasonable, explanation. The key point is simple: gravitation is very different from thermodynamics. In a gas of buzzing molecules, the uniform state – equal density everywhere – is stable. Confine all the gas into one small part of a room, let it go, and within a split second it’s back to a uniform state. Gravity is exactly the opposite: uniform systems of gravitating bodies are unstable. Differences smaller than any specific level of coarse-graining not only can ‘bubble up’ into macroscopic differences as time passes, but do.
Here lies the big difference between gravity and thermodynamics. The thermodynamic model that best fits our universe is one in which differences dissipate by disappearing below the level of coarse-graining as time marches forwards. The gravitic model that best fits our universe is one in which differences amplify by bubbling up from below the level of coarse-graining as time marches forwards. The relation of these two scientific domains to coarse-graining is exactly opposite when the same arrow of time is used for both.
We can now give a completely different, and far more reasonable, explanation for the ‘entropy gap’ between the early and late universes, as observed by Penrose and credited by him to astonishingly unlikely initial conditions. It is actually an artefact of coarse-graining. Gravitational clumping bubbles up from a level of coarse-graining to which thermodynamic entropy is, by definition, insensitive. Therefore virtually any initial distribution of matter in the universe would lead to clumping. There’s no need for something extraordinarily special.
The physical differences between gravitating systems and thermodynamic ones are straightforward: gravity is a long-range attractive force, whereas elastic collisions are short-range and repulsive. With such different force laws, it is hardly surprising that the behaviour should be so different. As an extreme case, imagine systems where ‘gravity’ is so short range that it has no effect unless particles collide, but then they stick together forever. Increasing clumpiness is obvious for such a force law.
The real universe is both gravitational and thermodynamic. In some contexts, the thermodynamic model is more appropriate and thermodynamics provides a good model. In other contexts, a gravitational model is more appropriate. There are yet other contexts: molecular chemistry involves different types of forces again. It is a mistake to shoehorn all natural phenomena into the thermodynamic approximation or the gravitic approximation. It is especially dubious to expect both thermodynamic and gravitic approximations to work in the same context, when the way they respond to coarse-graining is diametrically opposite.
See? It’s simple. Not magical at all …
Pe
rhaps it’s a good idea to sum up our thinking here.
The ‘laws’ of thermodynamics, especially the celebrated Second Law, are statistically valid models of nature in a particular set of contexts. They are not universally valid truths about the universe, as the clumping of gravity demonstrates. It even seems plausible that a suitable measure of gravitational complexity, like thermodynamic entropy but different, might one day be defined – call it ‘gravtropy’, say. Then we might be able to deduce, mathematically, a ‘second law of gravitics’, stating that the gravtropy of a gravitic system increases with time. For example, gravtropy might perhaps be the fractal dimension (‘degree of intricacy’) of the system.
Even though coarse-graining works in opposite ways for these two types of system, both ‘second laws’ – thermodynamic and gravitic – would correspond rather well to our own universe. The reason is that both laws are formulated to correspond to what we actually observe in our own universe. Nevertheless, despite this apparent concurrence, the two laws would apply to drastically different physical systems: one to gases, the other to systems of particles moving under gravity.
With these two examples of the misuse of information-theoretic and associated thermodynamic principles behind us, we can turn to the intriguing suggestion that the universe is made from information.
Ridcully suspected that Ponder Stibbons would invoke ‘quantum’ to explain anything really bizarre, like the disappearance of the Shell Midden People. The quantum world is bizarre, and this kind of invocation is always tempting. In an attempt to make sense of the quantum universe, several physicists have suggested founding all quantum phenomena (that is, everything) on the concept of information. John Archibald Wheeler coined the phrase ‘It from Bit’ to capture this idea. Briefly, every quantum object is characterised by a finite number of states. The spin of an electron, for instance, can either be up or down, a binary choice. The state of the universe is therefore a huge list of ups and downs and more sophisticated quantities of the same general kind: a very long binary message.
The Science of Discworld II Page 21