Like a lot of people I am watching a fair bit of the olympics. What the athletes can do is amazing, they are astonishingly fit and skilled. I am also working on how crystals start off life. You may think that there is no connection between the Olympics and crystallisation. But there is, the connection is extremes.
The olympic athletes are all at extremes of what the human body can do, the fastest, the strongest, the best hand-eye coordination, and so on. See here for a BBC page on the weights and heights of the Olympic athletes, the lightest is an astonishing 30 kg, while the heaviest is a very hefty 218 kg!
Now, the most common use of stats is to look at averages, which for the human body would be the strength of an average man, the speed of an average woman, etc. But this is obviously not the bit of stats you need for olympic athletes. The athletes are very far from average, they are extremes. Helpfully, there is an entire branch of stats devoted not to extremes. Rather unimaginatively, it is called extreme value statistics.
These stats can get a bit complicated but there is a simple and powerful result that applies in many circumstances. This is that as the population size, N, increases the size of the most extreme value, e.g., height of tallest, weight of the heaviest, etc., increases only logarithmically with N. For example if H is the height of the tallest man in a country with a population of N people, we expect that H obeys
H = A + B × ln [ N ]
where A and B are constants. In words, the maximum height is expected to increase only logarithmically with population size. A logarithmic increase is very slow; if N is multiplied by 1,000, ln [ N ] only increases by about 7. So even though China’s population is about 2,000 times larger than that of Luxemburg, the tallest Chinese man is unlikely to be much taller than the tallest man from Luxemburg.
This may (in part) explain why although China is winning many medals, other countries like the UK and even smaller countries, are still getting a look in. If the natural talent of the most talented athlete in a particular discipline only increases logarithmically with population size then this is a slow increase. So if a small country has a smarter training regime even though their athlete has a bit less natural talent, then better training would put them in the gold medal position.
Extreme value stats also come up in crystallisation. In practice, when water freezes, ice crystals almost always start to form on microscopic bits of dirt. Totally pure water probably freezes around —40°C. We can only make ice cubes in our freezers which are a lot warmer than that, because of tiny bits of dirt in tapwater. These are far too small to be seen by the naked eye but they are there.
In one cc of water there are a lot of tiny dirt particles, and there is a big variation from one particle to another in how easily an ice crystal can form on the particle. One of these thousands of dirt particle will, just by chance, be best for this, it is the extreme case here. This extreme dirt particle then determines at what temperature the water will freeze. So here again, the logarithmic law is useful.
