# Discontinuity of free energy at phase transitions

I am learning about the two dimensional Ising model at the moment, and how it exhibits phase transitions. At the very beginning of the course we were told that:

Thermodynamically, a phase transition occurs when there is a singularity in the free energy.

Is there any particular intuitive physical reason for this? How about non-thermodynamical systems such as real-space percolation?

I know that a phase transition is defined by an order parameter, that has to be zero on one side of the phase transition and non-zero on the other. However, the transition can be continuous, why must the free-energy diverge then?

• Nov 25 '17 at 16:20