When we study the theory of phase transitions and critical phenomena, we are taught that phase transitions are characterized by some mathematical properties. It works basically as follows. For a given system, we consider an adequate thermodynamic potential and study its derivatives, which represent other thermodynamic quantities. If the first derivative of this potential is not continuous, we say that a first order phase transition occurs; if, instead, the first derivative is again differentiable and the second derivative is not continuous or is divergent, we say that a second order phase transition occurs.
I'd like to better understand these classifications. Are they purely mathematical criteria for spotting phase transitions or does they say something about the physical process underlying it? More specifically, are these different 'types' of phase transition, in some sense, manifested differently in the real world?
