coffee and time

A demon in your coffee

Americano, Maxwell's demon
So innocent looking. But could an imaginary demon lurking within help us to understand more fully a major theory in physics?

Is there a way of preparing an Americano that can reveal a particularly knotty problem in physics with implications for information theory?

The question arises out of a field of physics, developed through the nineteenth century, that deals with energy and temperature: thermodynamics. It is the theory that describes how a hot coffee, left in a cold room, will eventually cool to the temperature of the (ever so slightly warmer) room. And though this may seem a trivial example, the theory is immensely powerful with applications from steam engines to superconductors. But it is back with the cooling coffee that we may find a demon, and it is worth finding out a bit more about him.

There are four laws of thermodynamics (the original three and then what is known as the ‘zeroth’ law). But it is the second that concerns us here. It can be phrased in a number of different ways but essentially says that there is no process for which the only result is the transfer of heat from a cold object to a hot one. To think about our coffee, the coffee will cool down to the same temperature as the room, but as the law describes, the room cannot get colder by giving its heat to the coffee cup (so the coffee gets hotter)!

It is in fact, one of the few places in physics where there is a ‘direction’ to time. For most of the laws of physics, time could run in the opposite direction without changing the effect, but not so for this one. The second law of thermodynamics is a definite provider for an arrow of time.

coffee clock, Rosslyn coffee
The clock at Rosslyn Coffee in the City of London. But the image alludes to a fundamental truth: the way that coffee cools is one of the few areas of physics for which it matters which way time ‘flows’.

But that is a digression. We ought to return to the demon in the coffee. The second law of thermodynamics seems to be based on our common sense (though perhaps that is because our common sense is formed within the laws of physics that determine the second law of thermodynamics). But with confidence in our common sense to understand the second law of thermodynamics, let’s do a thought experiment in which we make a strange type of Americano. Imagine a cup of coffee with an impermeable partition cutting through it. Into one half of the cup we pull a lovely, single origin, espresso. The crema rising onto the surface with some brilliant tiger striping on show. Into the other half of the cup we pour some water, initially at the same temperature as the coffee. We drill a small hole in the partition and watch what happens. Of course we know what happens. Ever so slowly, the coffee starts to get into the water and the water into the coffee until we are left with a balanced Americano on both sides with both sides at the same temperature.

Great, but now let us introduce the demon. Actually, he’s called “Maxwell’s Demon” because it was Maxwell who first proposed him (in ~1871), but we can call him anything we like. Perhaps he’s not a he at all. Our demon sits next to the small hole we have made in the partition and watches as the molecules travel towards the hole from the water’s side and the side holding the coffee. This demon is a bit of a trouble maker and so any fast moving molecules (hot) from the water he allows to get into the coffee and any slow moving molecules (cold) from the coffee he allows to get into the water. He does not allow slow molecules from the water into the coffee or fast molecules from the coffee into the water. Just to add to the mix, any coffee solubles he returns to the coffee allowing only water molecules through the hole in the partition.

If our demon exists, we would end up with a lot of very fast molecules on the coffee side (which will therefore be hotter) while the water would hold slower molecules (and be colder). We’d have a very hot espresso on one side of the partition and some luke warm water on the other. It’s not only a terrible Americano but a violation of the second law of thermodynamics! Which is worse?

Although he was proposed as a thought experiment, it is a problem with serious implications for the second law of thermodynamics (which otherwise seems to be a very good model of how things work). Because while we may not seriously consider an actual demon in the coffee, what stops some mechanical tool that we make from violating the second law, if the demon, in principle, could exist? Could the second law be wrong? Could there be a way of getting heat into our coffee from a cold room?

3D hot chocolate art on an iced chocolate, Mace, Mace KL, dogs in a chocolate
Art on a hot chocolate at Mace in KL. Well, what is your mental image of Maxwell’s demon?

The consensus has been that even were the demon to exist, ultimately he is powerless against the second law which does not get overturned by his presence. Because even if we could end up with a super hot espresso on one side of the barrier and cold water on the other side, this is not the whole system; the whole system includes the demon. And the second law applies to the whole system not the system minus the demon. So when we consider the energy (and entropy) of the demon in doing the work necessary to decide which molecules to let through and which to filter out, we find that work is done on the system (by the demon) and the entropy, the disorder if you like, of the whole system has increased (which is another way of phrasing the second law). Calm is restored, we get our Americano back, the laws of physics as we understand them are retained.

But Maxwell’s demon has not been completely exorcised yet, or at least, he is proving to be quite helpful. Because it turns out that there are methods for which the energy cost for the demon is minimal and the argument above no longer works. It seems we are back to square one. But even in that situation, it was realised that the demon has to record, make a note of, which molecules are fast and which are slow, which are coffee and which are water. It has led to an understanding that information has to be part of our consideration of thermodynamics. And as our ability to manipulate nanostructures and individual atoms improves, so experiments are able to explore how information ties into thermodynamics and why Maxwell’s demon still has not undone the second law yet. But it is here that we encounter another demon, the one that is found in the details, so if you are interested you can read more about it here.