why phase transitions and points in phase diagrams important?
Phase transitions are qualitative changes (sharp boundaries between different phases) like boiling a liquid into gas, or a solid melting into liquid.
Also, its at phase transitions where you can classify systems. I'm just starting to learn about this but the way a phase transition occurs involves the notion of a critical exponent and this can be used to determine 'universality' classes. So very different systems can act exactly the same at a phase transition. Presumably this helps us organize and understand diverse phenomena. For example I saw a talk last fall suggesting that evolution was in the same class as the Bose-Einstein Condensate (if that surprised you it surprised most other folks there as well).
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