I What is the effect of ionization at low densities?

[Swamp Thing]

John Baez says here http://math.ucr.edu/home/baez/end.html

... everything except the black holes will have a tendency to "sublimate" or "ionize", gradually losing atoms or even electrons and protons, despite the low temperature. ... If you take a box of hydrogen and keep making the box bigger while keeping its temperature fixed, it will eventually ionize. This happens no matter how low the temperature is, as long as it's not exactly absolute zero — which is forbidden by the 3rd law of thermodynamics, anyway.

This may seem odd, but the reason is simple: in thermal equilibrium any sort of stuff minimizes its free energy, E - TS: the energy minus the temperature times the entropy. This means there is a competition between wanting to minimize its energy and wanting to maximize its entropy. Maximizing entropy becomes more important at higher temperatures; minimizing energy becomes more important at lower temperatures — but both effects matter as long as the temperature isn't zero or infinite.

What is the technical term for this process? Is there some online stuff that explains this in more detail?

 

[TeethWhitener]

I think he’s probably referring to the somewhat surprising fact that the statistical mechanical partition function of the hydrogen atom is divergent, which implies that you’ll never find a proton bound to an electron. The reason is a little contrived: in an infinite universe with only one proton and one electron, there is essentially no chance that the proton or electron will ever interact; in other words, entropy wins (there are far more states where the proton and electron are unbound than states where they are bound).

However, the divergence of the partition function is extraordinarily slow: even putting a proton and an electron into an empty box the size of the observable universe—not infinite but ridiculously big—predicts that the two will be bound.

Edit: here’s a paper with more info:
https://www-liphy.ujf-grenoble.fr/pagesperso/bahram/Phys_Stat/Biblio/Miranda_Hydrogen_2001.pdf