Discontinuity of free energy at phase transitions

Question

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?

Answer 1

I think the question comes from a misunderstanding of the cited sentence.

Indeed, free energy, as well as any other thermodynamic potential (or fundamental equation) cannot diverge at finite values of the state variables and cannot have discontinuities. The reason is directly related to the stability condition of equilibrium in thermodynamic systems which implies that every fundamental equation (and then every thermodynamic potential) must have a well definite convexity with respect to each of its variables. A theorem of convex analysis states that convex functions must be continuous and must have left and right derivatives almost everywhere in their domain.

As a consequence, free energy could exhibit singularities but divergences or discontinuities are allowed only starting with the first derivatives (first order phase transitions). All remaining singularities must be related to higher order derivatives (they are named continuous phase transitions; however continuity here is referred to the first derivatives).