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Catastrophe theory

The reader might think that this theory should be discussed at least a bit at this juncture. Here only a certain elementary relevance to statistical physics is mentioned.

 If the number of parameters controlling the phase diagram is less than or equal to 4, then for a scalar order parameter model, the catastrophe theory can tell that generically the form of the quasi free energy (or the Landau-Ginzburg free energy) that can explain what happens near the phase transition point is either the $\phi^4$ model discussed here, or the model with a $\phi^6$ term added to it. Thus, Thom’s classification theorem is worth studying. 

 However, attempts to devise free energy functions that could reproduce even critical indices by mean field theory are meaningless. See the lecture notes: Informal Notes on Renormalization and Phase Transitions.