My thinking... (1) On temperature

This was originally written in 2012, in Polish language, for my friend who did not have a good knowledge of mathematical sciences but really wanted to learn more about them. Since we also shared an enthusiasm for a use of words to paint things inside an imagination, I decided for an unusual approach: to write about scientific topics with no equations or illustrations whatsoever. Other readers might consider it an unfortunate choice, but I always liked to try doing things differently, even if against the odds.

Sometimes a sudden surge - of fascination or disbelief, I don't know exactly - makes me think about trivial details that I don't pay attention to for most of my life. These are not questions that one would expect from an adult, they sound more like inquiries of a kid just getting to know the world. For example: why is the tea I just made with a boiling water now cooling?

I am no longer a child, I know this and that about the world, so I immediately answer, looking at the light mist floating over the cup: first of all, by evaporation.

After all, a steam forms because most vigorous particles of water break out of the tether and fly away - and since these are the fastest ones, the cup is left with slowpokes and the average speed of particles drops down. Isn't temperature simply a measure of average liveliness of molecules?

But something is not right, an inquisitive kid in me is not satisfied. He turns his nose up, although it is not yet clear at what. One could expect that he would start asking about the temperature of a single particle (after all, average of a single value can also be taken) - but no, this time he tricks me into considering more subtle issues. He makes me imagine throwing the cup at a very high speed. Would its temperature rise? Because the average speed of particles would certainly grow. And what if all the molecules were flying side by side, in a consistent formation and at the same speed as that average? Would such rigid block running through the space have the same temperature?

I don't allow him to back me into a corner and I start to explain everything patiently. Particle velocity is actually a vector - it has a direction. As we add all these vectors to each other, we get a resultant representing how fast and in which direction my tea moves as a whole. Then we can subtract this from each vector and what remains is the chaotic motion of particles within the cup that does not translate into movement of the whole. This time we may not be interested in their directions, just take the average of the numerical values alone, and this will be a measure of temperature. Is that good enough?

I made this up on the spot, but it seemed quite sensible - so I was ready to claim the problem solved. But suddenly, in a moment of sincerity, I admitted to myself that I am just playing with the definitions I read somewhere, and I don't understand what temperature actually is. That I only have a foggy idea of how these average speeds translate into me being able to warm my hands with this cup of tea. And that I don't know what it really means that the temperatures of some two things are equal. After all, the particles are of various shapes and sizes, some are easier to accelerate, others more difficult. What is it all about?

I became convinced that I am touching some deeper knowledge, that this old story about particle velocity is only a simplification, a description of a special case and that temperature is something much more fundamental. And it became clear to me that it would bug me until I understood what it means.

In order to explore this problem, I had to reconcile myself with statistics. For a long time I was reluctant to look towards this area, it seemed to me too contrived, even frivolous. Because it looks like an attempt to predict unpredictable events; a science which, by its very nature, is never entirely certain of what it concludes.

On the other hand, it is close to how we deal with an uncertainty of what may happen in everyday life. Taking a step on a pavement block, I don't expect that it will collapse under me - but this may happen once in blue moon, for whatever reason. So while I am convinced that this is not a threat - if it happens, I'll be at most a little surprised. Such circumstance can usually be summed up with the words of Kurt Vonnegut: "so it goes". The possibility that something will go not as we expected is inscribed in the nature of the world - which is too complex for us to take everything into account.

Statistics takes delight in large numbers. Only in such company can it feel confident and show its usefulness. When I'm going to throw a dice once, the knowledge of the exact probability of rolling a six is not all that useful to me. It will either land six up or not. But if I do it a thousand times and I would like to roll a six every time, then even without precise calculations I realize that among all possible sequences this one is less than a drop in the ocean - and, although, of course, it may happen, I will not count on it. One can quite reliably predict that something is not going to happen if it is just one scenario among myriad equally probable. And the larger disproportion, the better for the predictions.

Therefore, when I want to think about things like the behavior of molecules that make up my tea, I cannot do without statistics. There are so many particles there that every other approach is doomed to fail - even the most powerful computer will not be able to predict where each one of them is going to fly. While a statistician looking at such large numbers not only won't be frightened, but will be enthusiastic.

So when I started to explore the subject of what temperature is, it was inevitable to meet the statistical mechanics - a science that deals with such swarms of particles. The first to go was a simple example - a little bit of gas enclosed in a container.

The gas molecules used to be presented as small balls, bouncing off one another - a bit like balls in lottery draws. I thought it's quite a good analogy, so I imagined them exactly like that, thrashing around inside a box, hitting walls and each other. They keep forming new and new constellations, and when there are more than just a couple of them, it is next to impossible to predict their subsequent layout. It looks like they might set up themselves in any possible configuration at any moment.

How many of these potential arrangements are there? Certainly a lot. It is not without importance how precisely may the positions of the particles be measured - the finer the differences are noted, the more variants can be distinguished. But the number is huge anyway - with the addition of every single molecule it grows as many times as there are places that particle can visit.

This means that the number of possible configurations (so-called "microstates") is going to grow not only when new particles arrive, but also when they get more space for themselves. Following this trail, I imagined that my hypothetical gas container was in fact one of the two identical chambers in a larger box, separated by a tight partition. The second chamber is completely empty. What happens when the barrier is removed?

Each molecule suddenly has twice as many places to go compared to a moment before. And as a result, the number of possible states doubles as many times as there are particles. If the previous number was huge, this new one appears monstrous. Although each of the previous states is still viable, their number is insignificant compared to all the new possibilities. Therefore, the particles will spread along both chambers and it is almost certain that they are not going to get crammed in a single one anymore.

How many possible microstates something has, is an important attribute and it even has been given a name - it is an entropy. In my example, the gas filling the whole container had a greater entropy than the gas accumulated in one half - therefore the latter is almost impossible. But for such predictions to make sense, the disproportion between the numbers of states must be really high. Therefore, entropy is calculated in such a way that it adequately represents the scale of these quantities - how many times one is larger than the other.

For example, I could have an entropy unit corresponding to an increase of the number of states one thousand times. And if I added a particle to my container and it could be in any of a million different places, it would increase entropy by two units - because the number of microstates would increase one million times. When calculating possible configurations of a complex system I need to multiply the numbers of states corresponding to individual pieces - but when I translate it into entropy, each element simply adds several units. The entropy of any thing is therefore the sum of entropy of all its components, which is a very useful property. From a mathematical point of view, the entropy can be viewed as a logarithm of the number of microstates, in computer science it would be a number of bits required to encode that number... but I do not need to delve into it.

When thinking about a gas in a container, I omitted at least one important aspect, namely that each of the particles has a certain energy - the greater energy, the faster it moves. Molecules exchange energy by colliding with each other, so its distribution is constantly changing. Every thing, like my tea, has therefore an entropy related to how the energy contained in it is distributed among individual particles. Some get a little more, others a bit less - it is obvious at first glance that there are lots of possible variants.

Fortunately for me, there is such a thing as the smallest amount of energy that particles can exchange among themselves (so-called quantum), so the number of possible distributions should be quantifiable. In particular, if the whole object has very little energy - let's say a few quanta - then the ways of distributing this energy among molecules are not so many. However, the more energy to divide, the more possibilities how to do it - and this number increases with each added quantum.

To see exactly what it looks like, I chose the simplest example: two particles and one quantum of energy. There are only two possibilities: either everything will go to the first one or to the other, because it is impossible to divide this portion. But if I add another quantum, the possibilities are already three, as either each particle is going to get one, or one of them is going to grab everything. After the next quantum is added, the number of variants increases to four and so on. Indeed, with every added portion of energy, entropy grows - but gradually it is growing less and less. Because in order for the number of possible states to grow a thousand-fold, the number of quanta needs to increase quite similarly - thus more and more energy is needed to add another unit of entropy.

The key term here is the measure of how much entropy of a certain thing is going to increase, when you add a little energy (e.g. a quantum) to it. It baffled me that I don't have a proper name for this. Out of necessity I decided that I would be calling it "greediness". In mathematics it would be a derivative of entropy with respect to energy, but this is way too wordy.

It is high time to go back to the contents of my mug and finally explain what the temperature is all about. When I heated water in a kettle, I supplied it with a lot of energy - and it now resides within my tea. Hence it has a lot of entropy, but its greediness is quite small. The number of possible configurations already is so huge that adding or subtracting a little energy has no major impact on it.

By contrast, the cup walls contain significantly less energy - and thus less entropy. It is a solid body, so the particles do not move freely there, since they are fused together. But they can contain energy all the same - in their case it translates to how much they wiggle in place. So, a cup also has some greediness, and at the moment a greater one that the tea has - if it receives some energy, its entropy is going to increase considerably.

So what happens? Tea comes into contact with the cup, so it is possible to transfer energy between them, one way or the other. If the walls give up some of their energy, their entropy will decrease sharply (as greediness is a double-edged weapon), while for tea it may not make much difference. The total entropy of a cup together with tea, if I could calculate it, would have been reduced - because in one part it decreases more strongly than it increases in the other. Hence, this is a highly unlikely option - just like in my previous example it was impossible for gas to return to one half of the container. The configurations implementing such a scenario are too few compared to the number of all possible ones.

However, if tea gives some of its energy to the walls, its entropy will decrease slightly while for the cup it will increase substantially. In this case, total entropy is going to grow, which means that there are so many microstates corresponding to such an event that it is probably going to happen.

So the energy is going to flow to the cup, because the laws of statistics say so, and it is going to continue doing so as long as the greediness of the cup is greater than that of the tea. But the greediness is decreasing as it becomes satisfied (that is, with an increase in energy) and there will finally come a moment when the cup and the tea have the same level of it. What does it mean? That the temperatures have evened out.

I was surprised myself when it turned out that what I called greediness is nothing other than an inverse of temperature. But when I got there, everything became clear. When there is more energy, the greediness decreases - and so the temperature rises. Particles actually start to move faster - as long as they have enough freedom, otherwise they have to content themselves with vibrating in place. And when it is possible to transfer energy between two objects, it is always going to flow to the one with a higher greediness, i.e. a lower temperature.

It is a statistical rule, but bordering on certainty. Even the smallest pollen contains more particles than a mind can grasp, so the entropy of anything that can be encountered every day is gigantic. This makes phenomena such as a flow of energy from a colder to a warmer object so unlikely in the macroscopic world that their prohibition seems like a hard law, impossible to break. It is most likely that nobody will ever see something like this. But if it somehow happened anyway, the only appropriate comment might be: and so it goes.