More than a googol things to count

So this week I said that there weren’t a googol of things in this universe to count.  That might not be quite true. It really depends on what you mean by a thing. If it’s something tangible, that you can hold in your hand (like a grain of sand), then that is true. But we can construct intangible things that far exceed a googol in number.

Imagine you are watching a cross-country race with 100 competitors. How many different finishing orders are there? For example, runner number 23 might cross the line first, followed by number 55, then 38, 93, 17 all the way down to 43 who trails in last. Or maybe runner number 2 takes the tape, just beating number 95 and 52 and so on.

It’s an easy calculation; there are 100 possibilities for first place, then given the winner there are 99 possibilities for second (only 99 since one runner has already come in first), then 98 for third place, etc etc etc all the way down to 2 for 99th place (only two left to cross the line at this stage) and just one for last place. That gives us the number of possibilites as 100 x 99 x 98 x 97 … x 3 x 2 x 1 which is a very large number indeed (about 10 to the power of 157 I reckon – you would-be mathematicians work out how I did that).

Who cares?  Well, in a branch of physics called statistical mechanics, when we look at behaviour of say a room full of molecules, this kind of calculation becomes very important. One definition of entropy (as in 2nd law of thermodynamics) is a constant times the natural logarithm of the number of ways of arranging our system. (The logarithm function allows us to handle otherwise unmanageably  large numbers). The second law of thermodynamics then says that the number of ways of arranging something has to increase.

Like in my wine bottle incident; there are relatively few ways of assembling lots of glass ‘molecules’ to make a wine bottle; many more ways (bigger than a googol for sure) of assembling them into small pieces scattered all over the kitchen floor. So while I can expect a bottle to turn to pieces, it would be wishful thinking to expect the pieces to combined to give a bottle.

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