Wonders of the Universe - Brian Cox [92]
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 kB 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.
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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.
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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