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.

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What about the ends of

U-shapedstadiums? Do they act as some sort of capacitor or buffer--killing small waves that don't have enough potential?The statistical mechanics of liquids is pretty difficult. A readable introduction is in David L. Goodstein's _States of Matter_, http://www.amazon.com/States-Matter-Phoenix-Editions-Phoneix/dp/0486495….

What makes liquids difficult is the fact that the zeroth-order approximation for a liquid is a box of hard spheres, and there's no closed-form solution for the free energy (or the structure). Indeed, the famous Metropolis Monte Carlo paper (http://en.wikipedia.org/wiki/Equation_of_state_calculation_by_fast_comp…) is entirely about using computer simulations to solve that problem.

Two words: "critical opalescence"

They should act as sharp impedance mismatches in the waveguide, causing reflections.

A gas? The crowd can't move around.

In any case, didn't you just finish studying for a Stat Mech exam? Your first choice for modeling this system should have been an 1-D Ising model on a circular ring with nearest neighbor interactions (maybe even next nearest) and some thermal noise (which simulates the mood of the crowd).

That, in fact, would have made an excellent exam question for your Stat Mech class.

When I say this post I was sure you were going to reference the nature paper about the wave.

http://angel.elte.hu/wave/

Check that out. This group has studied, simulated and published about the wave. Its fun and interesting.

It is really great.

About the sports crowd waves.. An interesting project comes into mind. It's not exactly physics but rather art.

If you line the play field with an array of programmable lights pointing at the spectators, you could use the lights to "drive" the crowd wave in interesting ways. For example you could speed it up until the wave rounds the stadium in just a few seconds, or you could change its direction, or create several waves, etc..

Just imagine what it would look like in a huge stadium full of people.

Or perhaps you could use the system to make people shout in synchrony and create interesting acoustic effects? :D