## Wednesday, October 20, 2021

### If we reshuffled the atoms in the observable universe, how likely is it we would get any molecules?

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.

Alexander R Pruss said...

What if we want intelligent life remotely like we know it? Well, here's a prerequisite for life remotely like we know it: somewhere there is a cubic meter of space with at least a kilogram of matter (think: human brain) concentrated in it. How likely is that? My back-of-envelope calculation is something like 10^(-10^27).

IanS said...

I’m not seeing what this calculation is relevant to.

Cosmologists have a broadly accepted story of how the universe evolved from the very early stages of the big bang to the present. As I understand it, they think that if the universe were re-run with the same physics and the same initial parameters, but slightly different thermal fluctuations in the early state of the big bang, the resulting universe would be broadly similar to ours. In particular, the proportion of atoms to molecules would be much as it is in our universe.

If you think that such a universe is improbable, it must be because you think that either the laws of physics or the early state of the big bang are improbable. In fact, that is just what cosmologists do think: the hot, dense, smooth early state of the big bang was low entropy (= improbable). It is this that calls for explanation.

IanS said...

On the calculation itself: (a) Molecules are not just atoms close together. They are formed in reactions that typically release either photons or kinetic energy. Calculations of thermodynamic equilibrium have to take account of this. Broadly, lower temperatures favour molecules, higher temperatures favour individual atoms. (b) Don’t overlook gravity. It is typically negligible at small scales but crucial at cosmological scales. Gravity makes things lumpy. It is because of gravity that the universe can have a low average density combined with huge local densities (e.g. at the centres of stars). Stars are formed by gravitational collapse of relatively low-density clouds.

Alexander R Pruss said...

Ian:

No disagreement, of course.