## Entropy is Not Disorder

The second law of thermodynamics — the law of entropy — is a fascinating thing.  It’s the law that makes the past different from the future; it’s the law that predicts an effective end to the Universe, yet it’s the law that makes life possible.  It’s also a deeply mysterious law.  It took well over a century for true meaning of entropy to be understood (in fact arguments on the subject still rage today), and we still don’t understand, on a cosmological level, exactly why it was so low in the past.

One of the things that’s often said about entropy is that it means “disorder”.  This post is about that idea.  It’s worth discussing for two reasons: firstly, it’s wrong.  It’s close to the truth, in the same sort of way that a spectrum is close to a rainbow, but not the same.  Secondly, the real truth is much more interesting.

Imagine filling a glass jar with water and cooking oil.  Give it a good shake: the oil is now interspersed with the water.  It’s an opaque, homogeneous mess. Very disordered. But now, leave the jar on the table for a few minutes.  In the parlance of thermodynamics, this is a closed system (more or less — we can safely ignore the various ways in which it isn’t), so its entropy must increase over time.  But as we watch, the oil separates from the water, forming a nice, orderly layer on top of it.  The system, to the eye at least, appears to have become less disordered even as its entropy increased.

Now, I know what some of you will say at this point.  You’ll say that although the emulsion that formed when we shook the oil appeared to be less ordered than the separated state, it was actually more ordered.  You’ll say it had to be so because the emulsion had a lower entropy.  If you want, you’re free to do this — to define “disorder” to mean “entropy” — and then it will be true that entropy is disorder.  But I hope you’ll still read the rest of this post, and eventually perhaps you’ll agree that this definition of disorder doesn’t really match our intuitions very well.

Another motivating example: how is it that plants are able to convert apparently disordered air, water and sunlight into the precisely ordered machinery of cells, mitochondria, chloroplasts, etc.?  I remember asking this question to my parents as a child (my parents are scientists), and I was told it’s because the photons coming in from the sun are very ordered, even though white light seems to be about as homogeneous and scrambled as anything.  It’s this invisible “order” which apparently gets converted into the order of a plant’s vascular architecture.  On the other hand, the photons that the plants give off to space (which are at a much lower temperature, and longer wavelength) are very disordered.

This idea – that some light is ordered and other light is disordered, and that this order is comparable to the very tangible order of biology – never seemed quite right to me.  If one replaces “disorder” by “entropy”, however, it becomes much more reasonable.  Entropy, after all, is simply a physical concept that you can put a number to.  High frequency photons do have less entropy than low frequency ones (or, more accurately, high frequency light has a lower entropy than the same amount of low frequency light because it contains fewer photons).  Plants do have low entropy — you can calculate it from their chemical make-up — and their entropy does stay low because of the incoming low-entropy light that they capture.

So where did the idea that entropy is disorder come from?  It dates back to the early years of statistical mechanics.  At this time, the concept of entropy was already well known.  It had been conceived of in the 1820s by Carnot, and later developed by Clausius, in order to build more efficient steam engines.  In Clausius’ time nobody knew what heat and temperature were, exactly, except that they were things you could measure, and entropy was defined in terms of them.  But around the end of the 19th century, people (and in particular Ludwig Boltzmann) began to seriously propose that atoms existed, and that heat is the energy of their vibrations.  The central idea of this kinetic theory was Boltzmann’s famous equation $S = k \log W$ — Enropy is proportional to the logarithm of the number of possible microstates of a system.  This was a radically new idea, and one that would take a very long time to be fully understood. (I’ll cover some of the subtleties of its meaning in future posts.)

As people grasped their way towards an understanding of this new concept, one of the things they did was to work through the mechanics of the simplest system to which it can be applied: the ideal gas.  Let us imagine we have a chamber of an ideal gas (truly ideal gasses are always imaginary), and we’ve heated part of it up.  According to the thermodynamics of Clausius, the cold parts of the gas will then warm up and the hot parts cool down.  As this happens the entropy increases, by definition.

Now the handy thing about the ideal gas, from a statistical mechanics point of view, is that there aren’t any forces acting between the molecules, except when they collide.  As a result, when the gas reaches thermal equilibrium — that is, when its entropy is the highest — the motion of every particle is completely uncorrelated with the motion of every other molecule.  This fits rather nicely to our intuitive notion of disorder, and it was this result that sank into the scientific and popular unconscious as “entropy is disorder”.

The problem is, it doesn’t generalise beyond the ideal gas.  In fact, ideal gasses are the only type of system in which it does hold.  In any other system there are forces between the molecules, and this means their motions are correlated when they’re in thermal equilibrium.  Hence we can find, on the microscopic scale, the orderly lattice of a crystal.  This is an equilibrium structure, meaning that its entropy is as high as it can get.  That’s why it forms — if you grind it up you’ll destroy that order but decrease the entropy.  Leave it long enough and it will eventually reform.  Similarly, there are a lot of intramolecular forces at the interface between the oil and water we considered above, and it’s those forces that cause the two liquids to separate.  It seems wrong to say that these intramolecular forces mean that a growing crystal lattice becomes more disordered even as it becomes more structured. In informal language, order and structure are basically the same thing.  It seems to me much better to say that a crystal lattice is both highly ordered and has a high entropy; and that entropy is only equivalent to disorder in the case of an ideal gas.

I’m not saying that entropy is completely unconnected to order.  If “order” means “correlations between the motion of the atoms” then any low-entropy system you’re ever likely to find will also be a highly ordered one.  It’s just that the reverse isn’t necessarily true.

All of this is not to say that the universe can escape the second law, allowing the kind of order we see around us to last forever.  Equilibrium structure is different from non-equilibrium structure.  Life, for example, is ordered in a very low-entropy way, and it needs a continual input of low-entropy food or photons in order to persist.  Crystals are highly ordered, but they’re ordered in a particularly inert kind of way.  Atoms vibrate and shuffle around the lattice, but they do so in a more-or-less random way, and this kind of equilibrium order does not allow the same kind of complex behaviour that we observe in life.

On a cosmological time scale, the universe will still suffer a “heat death”, where all the free energy has been used up and life can no longer persist.  This will not be because the universe has become infinitely disordered, but simply because it has reached its highest possible entropy.  There might still be order – and that order will persist, pretty much unchanging, forever – but it will be of the inert, crystal lattice kind.

Similarly, we can think about the as yet unsolved mystery of why the entropy was so low in the cosmological past.  The question is not “why did the Universe begin in a very ordered state”, it is simply “why did it begin in a low entropy state?”.  I have a feeling that we’re more likely to be able to answer that question if we don’t assume that entropy is the same as disorder.  (There is also, to my mind, a deep mystery about what it could even mean to say the Universe had a low entropy even when no-one was around to observe it — but that’s a question for another time.)

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