Wonders of the Universe

ENTROPY IN ACTION

In 1908 in the small town of Kolmanskop in southern Namibia, a railway worker by the name of Zacharias Lewala found a single diamond lying in the sand. He showed the precious stone to his manager – railway inspector August Stauch – who immediately realised its significance and set in motion a train of events that turned this desolate place into one of the most valuable diamond mines in the world. The colonial German government closed the entire area to outsiders; only German entrepreneurs were allowed to make their fortunes here. For 40 years, Kolmanskop was home to a thriving community as over a thousand people gathered, seeking to become millionaires by picking diamonds out of the desert. As the money rolled in, the residents built a town in the finest German tradition; grand houses stood beside a casino, a ballroom and the first X-ray station in the Southern Hemisphere. They led a champagne lifestyle in the desert, and created a little piece of opulent German architecture in the sand. Eventually, though, as with all cash cows, the diamonds could no longer be found and the town gradually lost its sparkle until it was abandoned in 1954. For half a century it has fallen into disrepair as the buildings are slowly reclaimed by the sands.

Today Kolmanskop is a ghost town, a place where our efforts to replace the geological grandeur of the desert with architectural grandeur of our own have been thwarted by the power of the winds.

Kolmanskop lies just outside the modern port town of Lüderitz, which sits in spectacular isolation on the southern Namibian coast. One of our guides told us that it takes a special kind of Namibian to set up home in Lüderitz – you have to really want to live there. The reason this place has a reputation for being particularly harsh, even by the standards of this part of the world, is the wind. This strip of the southern African coast is permanently assaulted by the untamed winds of the South Atlantic that whip up the fine-grained sands of the Namib Desert and hurl them unrelentingly into machinery, houses, camera equipment and eyes. I have never experienced anything like it. While filming, I found myself walking through the wind at Kolmanskop with my hands completely shielding my face. I didn’t do this for dramatic effect, I genuinely couldn’t look into the lacerating sand-laden wind. We also shot a scene showing a little sandcastle gradually blowing away; the camera we left in the desert for hours to film that had its lens sandblasted – the high-precision optics felt like sandpaper after a single spring afternoon in the vicinity of Lüderitz. If it wasn’t for the fact that it never rains here, and nothing rots, the ghost town of Kolmanskop would surely already have already disappeared back into the desert.

The opulent buildings of the once-glorious town of Kolmanskop are a shadow of their former selves as the desert sands blow across them and reclaim the landscape.

The little sandcastle slowly decaying in the desert wind vividly demonstrates the connection between decay, randomness and entropy. To understand why this is so, we’ll need a different and much more intuitive definition of entropy than that given by Clausius. Known as the statistical definition of entropy, it was developed by Ludwig Boltzmann in the 1870s.

A sandcastle is made of lots of little grains of sand, arranged into a distinctive shape – a castle. Let’s say there are a million sand grains in our little castle. We could take those million grains and, instead of carefully ordering them into a castle, we could just drop them onto the ground. They would then form a pile of sand. We would be surprised, to say the least, if we dropped our sand grains onto the floor and they assembled themselves into a castle, but why does this not happen? What is the difference between a pile of sand and a sandcastle? They both have the same number of sand grains, and both shapes are obviously possible arrangements of the grains. Boltzmann’s definition of entropy is essentially a mathematical description of the difference between a sandcastle and a sand pile. It says that the entropy of something is the number of ways in which you can rearrange its constituent parts and not notice that you’ve done so. For a sandcastle, the number of ways in which you can arrange the grains and still keep the highly-ordered shape of the castle is quite low, so it therefore has low entropy. For a sand pile, on the other hand, pretty much anything you do to it will still result in there being a pile of sand in the desert, indistinguishable from any other pile of sand. The sand pile therefore has a higher entropy than the sandcastle, simply because there are many more ways of arranging the grains of sand such that they form a pile of sand than arranging them into a castle. Boltzmann wrote this down in a simple equation, which is written on his gravestone:

S is the entropy, W is the number of ways in which you can arrange the component bits of something such that it is not changed, and k_B is a number known as Boltzmann’s constant. For the more mathematically adventurous, ln stands for ‘natural logarithm’. If you don’t know what that means, don’t worry; the equation simply relates to the number of ways in which you can arrange things to the entropy.

As long as each particular arrangement of the sand grains is equally likely, then if you start moving sand grains around at random they are overwhelmingly more likely to form a shapeless pile of sand than a sandcastle.

That may seem a bit complicated, and not entirely illuminating yet, but here is the key point: as long as each particular arrangement of the sand grains is equally likely, then if you start moving sand grains around at random they are overwhelmingly more likely to form a shapeless pile of sand than a sandcastle. This is because most of the arrangements you create at random look like a formless pile, and very few look like a sandcastle.

This is common sense, of course, but now think about what this looks like at a microscopic level – the level of individual sand grains. There is nothing at all in the laws of nature to stop the wind blowing a grain of sand off one of the turrets of our castle and then picking up another grain from the desert and blowing it back onto the turret again, leaving our castle perfectly unchanged. Nothing at all, that is, other than pure chance. It is much more likely that the grains of sand blown off the castle are not replaced with others from the desert, and so our castle gradually disintegrates, which is to say it gradually changes into a formless sand pile. In Boltzmann’s language, this is simply the statement that the entropy of the castle will increase over time; the castle will become more and more like a sand pile. Why? Because there are many more ways of arranging the grains of sand into a pile than there are into a castle, so if you just randomly blow grains around they will tend to form piles more often than castles. Here is the deep reason that entropy always increases: it’s simply more likely that it will! Notice that there is nothing in the laws of nature that prevent it from decreasing; it’s possible that the wind will build a sandcastle, but the chances are akin to tossing a coin billions of times and each one coming up heads. It’s simply not going to happen.

Boltzmann’s statistical definition of entropy is the key to understanding Eddington’s arrow of time. This is such a key concept with such profound consequences that it is worth repeating it once more in a slightly different way. If there are a million different ways of arranging a handful of sand grains, with 999,999 of the ways producing disordered sand piles but only 1 producing a beautifully ordered castle, then if you keep throwing the sand grains up in the air they will usually land in the form of a disordered pile. So, over time, if there is a force like the wind that acts to rearrange things, things will get more messy or disordered simply because there are more ways of being disordered than ordered. This means that there is a difference between the past and the future: the past was more ordered and the future will be less ordered, because this is the most likely way for things to play out. This is what Eddington meant by his statement that the future is more random than the past, and his description of the arrow of time as the thing that points in the direction of increasing randomness. And this is why entropy always increases.

Watching my carefully constructed sandcastle gradually disintegrate in the strong desert winds perfectly demonstrates how entropy increases, and the idea that the past is always more ordered than the future.

For the purposes of our story, this is sufficient; if you take a university physics degree, this is what you will learn about entropy and the arrow of time. But there is still a great deal of debate and research surrounding entropy, and it centres on something we have dodged slightly. We have only spoken about entropy differences; the past had a lower entropy than the future; ordered things become disordered as time ticks by, but one might legitimately ask where all the order in the Universe came from in the first place. In the case of our sandcastle, it’s obvious – I made it – but how did I get here? I’m very ordered. How did Earth get here? It’s very ordered too. And how did the Milky Way appear if it is composed of billions of ordered worlds orbiting around billions of ordered stars? There must have been some reason why the Universe began in such a highly ordered state, such that it can gradually fall to bits. The answer is that we don’t know why the Universe began with sufficient order in the bank to allow planets, stars and galaxies to appear. We understand how gravity can create local order in the form of solar systems and stars, but this must be at the expense of creating more disorder somewhere else. So there must have been a lot of order to begin with. In other words, the Universe was born in a highly ordered state, and there should be a reason for that. It is unlikely to have been chance, because by definition a highly ordered state is less likely to pop into existence than a less ordered one; a sandcastle is less likely to be formed by the desert winds than a pile of sand. Since the Universe is far less ordered today than it was 13.75 billion years ago, this means it is far more likely that our universe popped into existence a billionth of a second ago, fully formed with planets, stars, galaxies and people, than it is that the Universe popped into existence at the Big Bang in a highly ordered state. There is clearly something fascinating about the entropy of the early Universe that we have yet to understand

The sands of time are slowly and literally overrunning Kolmanskop, dismantling the highly crafted town and returning it to dust once more.

The arrow of time has been playing out dramatically in Kolmanskop since the mining facility was abandoned in 1954. In every building you can see the gradual transition from order to disorder; every room that was once full of structure is slowly being returned to a less-ordered state. This is the march of the arrow of time on Earth, but it is nothing compared to the grand journey that time’s arrow forces our universe to make.