What is a Phase Transition?

Dear Diary,

I recently was wandering the woods and thought of a good analogy for a coherence length.  Condensed matter is rife with coherence lengths while something like particle physics deals with energy scales.  Really, these two things are the same since there's a relation like

Energy ~ 1 / Length


Consider a big stick stuck in the ground with the sun shining on.  This setup can only be in two phases:  day or night, but without looking at anything else, we can determine if it is day or night (or how much day there is) by looking at the stick's shadow. The length of the shadow is the coherence length for "day phase".

Now, consider what happens as set-up goes from day to night.  The shadow grows and grows and grows until there is a point where the shadow has infinite length but just for one instant.  You blink and you miss it.  What do you miss if you blink?  You missed the phase transition from day to night!  The coherence length went off to infinity.

Now, this happens all the time in physics when there's a phase transition.  So, the coherence length for solids goes to infinity when we transition to a liquid and so on.