I'm currently trying to understand the physics of phase transitions and I'm having a hard time doing that. First of all, the discussions on the topic seem to be confusing and there is no methodical approach to study such systems. I will list a couple questions concerning this topic, and hope someone could clarify them to me.
The wikipedia definition of a critical point is:the end point of a phase equilibrium curve. This seems to be qualitatively ok to me, but is there a more rigorous, mathematical definition? I know this is related to second order derivatives of a convenient free energy but in most books the critical temperature, critical pressure etc shows up from nowhere and after that some connections between the second order derivatives are made. For example, take the Ising model in $d$ dimensions with the mean field approximation. The critical temperature is obtained from the behavior of the (Taylor expansion) of the free energy and no connection to the heat capacity (the appropriate second order derivative of the free energy to be taken account) is made. Besides, how the discontinuity of the second order derivative of the free energy imply the wikipedia definition?
Is it possible for a system to have more than one critical point?
Since childhood we learn in school about basic phase transitions such as liquid-solid, gas-liquid etc. It is funny that these phase transitions we are familiar with seem to be all of first order. Is it accurate? Is there a reason for that? The only reason I could think of is that the critical point of a given system is considerably high and we are not familiar with such systems.
What does the system become if we go beyond the critical point?
If two systems have the same critical point, are their critical exponents necessarily the same?