Essays

Maths

The thermodynamics of tidying a room

It is a basic rule of the universe that any room will, without any external organising influence, tend to a state of disorder. Take a tidy room and add a high energy particle, say a small child, and the room will get messier and messier. Eventually the room will reach a state of total disorder, nothing will be where it should be, and the child will be exhausted. This can be likened to the heat death of the universe.

Thermodynamics concerns itself with a concept called entropy, roughly a measure of the disorder of a system or the messiness of a room. A nicely stacked set of letters block in strict alphabetical order has low entropy, a set of blocks scattered randomly about the room has high entropy. The universe is slowly undergoing a process of increasing entropy, from a very ordered state at the big bang to a state where every atom is moving about in a random direction - the heat death of the universe.

We can see this process of increasing entropy in many natural phenomena: if you leave a bowl of water in a room atoms in the air will knock the water molecules out of the bowl and into the air. Over time this process of diffusion will result in the water molecules being randomly spread about the air and the bowl will be dry. Other natural phenomena like the weathering of mountain, the rusting of metal and other processes of decay are all good examples of increasing entropy.

These processes are all summed up in the infamous and rather tricky second law of thermodynamics: the entropy of a closed system will increase with time.

For the example of a tidy room we can see a similar process. Rather than atom bouncing water molecules about consider people coming into a room, picking up an object and putting it down in a random place. Repeat this a hundred times and a once tidy room will soon become messy. Chaos can be the only result. This leads us to the first law of tidy-room-dynamics: left unchecked a room will tend to a state of total disorder. This alas is a natural process, rooms do not stay tidy all by themselves, another process is needed to maintain perfect room tidiness and the process is life.

What is life, the thermodynamics of a cleaner

Scherodinger, he of quantum cat fame, wrote an influential book in the 1940's called "What is life?" In it the explored the mystery of life from a physicist perspective. In particular with regards to the ideas of thermodynamics and entropy.

At first glance life does not seem to follow the rules of entropy, rather than an increase in disorder we actually see an increase in order. Over life's two billion year history we have seen more and more complex organisms evolve - from single celled amoeba to large complex multi-celled creatures to large scale structures with many creatures working together for the common good, say a termite mound.

How does this tie with thermodynamics? Scherodinger viewed life as a process of converting ordered energy into a small region of very high order, the creature itself, and a large amount of waste heat. This waste heat has very high entropy - its essentially molecules bouncing about in random directions and you can't get useful energy from these molecules. When you take the total entropy of the system, the low entropy of the organism, and the high entropy of the waste heat you actually end up with more total entropy than you started with. The second law of thermodynamics has not been broken and Scherodinger and his cat can sleep soundly.

There are many examples of creating a little local order from high energy. Plants take highly ordered energy from the sun, use it in photosynthesis to grow the plant and create waste heat in the process; animals eat the plants and again produce waste heat, other animal eat these animals creating more order and more waste heat. Overall the whole of life on earth can be seen as a process taking the energy from the sun, creating a richly complex biosphere and a lot of waste heat speeding out into the solar system.

On a less grand scale, we can see the same process with tidying a room. To maintain a little local order in a tidy room take a good breakfast and a lot of coffee (i.e. lots of energy) plus an example of living-organism, typically a mother or, rarely, a father. This creature can then uses the energy to create a little oasis of order and a lot of waste heat in the process. So we now have our second law of tidy-room-dynamics: creating order out of chaos takes a lots of time and energy.

One of the hardest working of all room tidiers is Maxwell's daemon. Maxwell wanted to separate red molecules from blue molecules putting red molecules on the left and blue ones on the right, so he created a little daemon who could control a gate between two halves of a room. (technically he just though the daemon up and it is a companion of the cat in the strange thought-world of physics) Initially the room has red and blue molecules on both sides of the room bouncing about at random. Whenever a red molecule approached the gate from the right the daemon opens the gate letting it through to the left, if a red molecule approaches from the left the daemon keeps the gate shut so it bounces back to the left. The same happens for blue molecules but they are only allowed to travel from left to right. Over time this hard working daemon manages to get most of the red molecules on the left and blue one on the right and all the opening and shutting of the gate takes quite a bit of energy. Certain technical problems, such as the fact that cloths don't tend to fly about by themselves, mean this is not a practical approach to tidying a room.

Room tidying strategies: free will and information theory

After a certain age, we acquire free will - there is no longer a parent saying "go and tidy your room". Free will can be a blessing or a curse but it gives us the choice of whether we wish to tidy our rooms or not. There are several different approaches we can take to this existential dilemma:

The tidy room strategy can be broken down into two main sub-stratergies:

To distinguish between these two we need to venture into the world of information theory. Compare these three line

In the first we have a lot of order, but it is a little boring. The third is completely random and also rather boring. But second hits a sweat spot between order and disorder. In terms of entropy its the second which has the lowest entropy. To get a grip on such lines Claud Shannon introduced the idea of information theory, and the entropy of a system is likened to the simplest way a system could be described - high entropy = simple descriptions. The first line can be described simply as "26 a's" the third as "26 random digits", both contain little information.The second can't really be described any simpler than the sequence "I wandered lonely, as a cloud" so has high information. If you also want to incorporate the cultural and emotional significance of the phrase the information content could be much higher.

Applying this idea to various types of tidy room and we see the cataloguer has a much lower entropy than the minimalists. Indeed a library with a well catalogued system of books is pretty good from a tidy-room-dynamics point of view.

If we want want to consider the whole system, and the total entropy of the system there is one better option. Consider that, rather than spend energy tidying my room, I have spent my time writing this essay - 8000 highly ordered letters with a highish information content, and rather low entropy. Taking the total entropy, quite high entropy of fairly untidy room plus the low entropy of the essay the theory of tidy-room-dynamics shows I have reached the peek of perfection.

Notes

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