In some ways, a crowd at a stadium has two possible overall states. They can be a normal crowd, or they can do the wave. But first, some background:
A gas isn’t usually too hard to model at the statistical level. Treat is as a large collection of free particles and you’re pretty much set. For more accuracy you can take quantum effects such as the so-called second quantization into account, and you can model the interactions between atoms with an appropriate Hamiltonian and for all intents and purposes your model will be able to handle more or less any real-world situation with great precision.
Solids are harder, and liquids tend to be harder still. Still, their bulk statistical properties tend to be predictable at least in a qualitative way. Some solids are well described as a lattice of harmonic oscillators with phonons gently carrying energy around through the crystal structure. Metals are pretty well described as a gas of free conduction electrons. I’m not very conversant in how to handle liquids mathematically, but I’m reliably informed that though tricky it’s not all that egregious.
But how to describe that magical moment when a warming solid block melts into a puddle, or the drop of water flashes into vapor? That’s a phase transition, and they are a tremendous pain. It’s doable though in some cases, and we’ll explore the basics a little.
So what’s this got to do with people waving their arms at a football game? Well, a gas atom which happens to have fluctuated into a low energy state will quickly have that statistical fluctuation swamped because it’s constantly sharing energy with its nearest neighbors. The correlation length between what any given atom is doing and what it influences its neighbors to do is very small. Same thing for an atom in the lattice of a solid. A fluctuation is going to be decisively squashed by the massed statistical effects of its neighbors and their neighbors and so on throughout the solid.
Same thing with a person standing in a stadium. He might be sitting minding his own business, or he might be jumping around trying to start the wave with all his might. Either way, he won’t change the overall state of the stadium crowd. The correlation length that determines how far his influence spreads is small. Maybe he’ll be able to get one or two people near him to wave, but it doesn’t spread. The overall mood of the crowd – its “mean field”, as it were – is not going to allow the state of the stadium to change.
Unless the overall tenor of the crowd approaches a critical point, or the mean field of the atoms starts to creep toward an inflection point as the conditions (temperature, external magnetic field, whatever) change. As the critical point is approached, the correlation length rapidly becomes larger and larger until it’s actually infinite at the critical point itself. That one atom fluctuating can change the state of other atoms which changes the state of other atoms… and suddenly the solid is a liquid, and suddenly the rush of standing people waving their arms propagates strongly around the stadium.
And at that point the correlation length drops down to microscopic size again. One recalcitrant spectator can’t make the wave go away, and fluctuations in the atoms of the liquid aren’t going to freeze the whole thing back into a solid. Only a change in the external conditions can alter the mean field to the point where those fluctuations dominate and push the system past the critical point into another phase.
Not something that gets thought about a lot at football games, I’d wager, but interesting nonetheless.