1.8. First-order phase transitions In the second-order phase transitions, the state with the lowest value of the free energy changes continuously when the system crosses its critical point. On the contrary, in the first-order phase transition, the order parameter changes discontinuously.

a. Study the behaviour of the minima of the free energy F(x) = a(T)x2 +x4 by varying the temperature T as a(T) = (T −Tc) and determine if we are in presence of a first- or second-order phase transition.

b. Analyze the same questions for the free energy given by F(x) = (x2 −1)2 (x2 +a(T))
with the same expression for a(T).