When Pigs Fly

(UPDATED 7-19-08: Unfortunately, when I originally posted this, I missed a negative sign in the expression for entropy change. I hate when that happens! It might be instructive to tell you how I came to realize my mistake. I realized that the original expression, with both terms added, was monotonically increasing. That meant that the hotter the water gets, the greater the entropy change - making it very likely that the water would get even hotter than the oven, hotter without limit in fact. That, of course, is ridiculous. The expression below has been corrected, with one term added, because the entropy of the water increases, & the other subtracted, because the entropy of the oven decreases. This expression has a maximum when T_final = T_oven. That is as it should be, and all is well with physics.)

During spring term in thermodynamics class, we computed some thermodynamic probabilities. We found that the odds against a system spontaneously decreasing in entropy by 1 J/K were enormous. The number was so large that we couldn't think of anything that was comparable. I promised to compare it to other enormous numbers, but I never got around to it. Until now.

Imagine taking a cup of water at room temperature and placing it in an oven at 350^oF. We all know that the water is very unlikely to cool down, but just how unlikely is it?

We can calculate relative probabilities* by using entropy differences. For example, we can find out how much more likely it is for the water to warm up by 0.5^oC than it is for it to cool down by 0.5^oC. To do that, we find the entropy change in each case. Assuming that the oven is a large reservoir, and its temperature hardly changes, the entropy change is:

mc[ln(T_final/T_initial) - (T_final-T_initial)/T_oven]

where T_initial=293K, T_final = 293.5K or 292.5K, and T_oven = 450K, m is the mass of the water, and c is its specific heat.

This equation yields an entropy difference of about 1J/K.

This entropy difference is related to the ratio N₂/N₁, where N₁ is the number of different ways that the microscopic atoms and molecules can be, and still have it correspond to a temperature decrease of 0.5^oC (N₂ similar for a 0.5^oC increase).

Why do we care about this number? The probability of either of these events happening is proportional to that number - for the same reason that the odds of rolling a 2 rather than a 7 when rolling two fair six-sided dice is 6:1 against. You are six times more likely to roll a seven, because there 6 different ways that it can happen, but only one way to roll a two.

That means we can use the entropy difference to calculate the odds (here k is Boltzmann's constant):
1J/K = kln(N₂/N₁) = kln(odds)
Solving for the odds, we find that warming up by 0.5^oC is about 10^3X1022 times more likely than cooling down by 0.5^oC.

Wow! That's a big number. But just how big is it? Even a googol pales in comparison, it's "only" 10¹⁰², nowhere near 10^3X1022. How much bigger than a googol is this number? The ratio of this number to a googol is 10^3X1022/10¹⁰⁰ = 10^(3X1022-100) = 10^{2.99999999999999999999X1022}.

Let's compare this to other large numbers, like the age of the universe, about 4X10¹⁷seconds, or about 4X10³²femtoseconds. What about the volume of the observable universe? That's about 3X10⁸⁶ cubic centimeters, or 3X10¹¹⁰ cubic Angstroms. We have to divide the universe up into pieces as small as an atom to even get above a googol, but even these large numbers are nowhere near as large as the disparity in thermodynamic probabilities that we calculated above.

It is starting to look like this number is so huge (and the probability so small) that nothing compares to it.

But, don't despair. One way to get numbers as large as 10^3X1022 is to do something similar to our thermodynamic probability - take something unlikely for a microscopic particle and require that a macroscopic number of particles do this unlikely thing.** For example, imagine that all the atoms in a 2 g sample of radon-222 decayed in the same second. What are the odds of that? You guessed it, about 10^3X1022:1 against.

Just in case 2g of radon decaying all at once doesn't startle you, let me explain further. 2g of radon is about the amount of radon in every single basement on the whole planet. If 2g of radon decayed all in one second, it would release 4.7 gigawatts of power (almost enough to power 4 time-travel DeLoreans :) Ouch!

So, yes, there is a non-zero probability of macroscopic decreases of entropy, like room-temperature water cooling in a hot oven. But in my mind, it might as well be zero!


*_{Note: What I've calculated here is the relative probability of cooling down by 0.5^oC compared to warming up by 0.5^oC. It is not the probability of the cup of water cooling down in the oven. I expect that is much smaller yet, since you would have to compare all ways of cooling down to all ways of warming up. All ways of cooling down are very unlikely, but some ways of warming up are very likely, particularly those where the water boils and warms close to 350^oF.}
**_{Thanks for the idea, Nick}