Here’s an amusing question. Let’s say that I took all the atoms in the observable universe and shuffled their positions by independently randomly and uniformly choosing positions for them through the volume of the observable universe. What is the probability that I would get any molecules?
It turns out not to be hard to answer this if we just want to get a very rough upper bound. The above-linked Wikipedia article on the observable universe gives us some useful data:
Volume of observable universe (V): about 4 × 1080 m3
Number of atoms in observable universe (N): about 1080
Next, all the bonds in molecules that I can find references to are under about 4 × 10−10 m, and anyway the most relevant one is hydrogen-hydrogen, which is much smaller. Thus, if we keep a sphere of volume of v = (4/3)π(4 × 10−10)3 m3 ≈ 3 × 10−28 around each atom empty of other atoms, we can suppose there are no molecules. What’s the probability of doing that? It’s
- p = ((V − v)/V)((V − 2v)/(V − v))⋯((V − Nv)/(V − (N − 1)v)).
Lots of stuff cancels out and we get:
- p = (V − Nv)/V = 1 − Nv/V.
Thus, the probability that we won’t succeed in clearing such a space around each atom is:
- 1 − p = Nv/V ≈ 10−28.
So, it’s extremely unlikely that a random rearrangement of the atoms in the observable universe would result in a single molecule.
Does this have any interesting philosophical consequences? I don’t know. I wanted to do this calculation to have a better intuitive picture of how incredibly unlikely it would be to get the observable universe by chance to be remotely like what we have—having at least one molecule is my “remotely like what we have” condition.
Of course, nobody thinks our current observable universe was produced directly by chance in its current state. But if something like Liouville’s theorem is applicable to the observable universe, and my above estimate gets close to the probability in the relevant phase space, then the probability of an initial state that results in at least one molecule in 13.8 billion years is going to be the same as the probability of getting at least one molecule directly by the chance arrangement. But I know little about this kind of physics stuff.
