I am currently revising the lecture notes in which different spin systems are analyzed, focussing on the occurrence (or absence) of phase transitions. Different techniques are applied to analyze the different models and - seemingly - also different notions of phase transitions. These notions are not mathematically precisely defined and the question is how to connect them.
I will briefly outline the analysis of three different models and their notions of phase transitions.
Absence of phase transitions via smoothness of the free energy: 1-D Ising Model. Here it is shown that the free energy per spin is a smooth (in general not analytic) function of the temperature and the magnetic field.
Phase transitions via magnetization: 1-dimensional chain of non-interacting spins in a magnetic field. Here it is shown that the average magnetization (averaged over all spins, not as in a expected value) converges a.s. to tanh(hb) in the thermodynamical limit where h is the magnetic field and b is the inverse temperature.
Phase transition via magnetization: Curie-Weiß-model without magnetic field. Here it is shown that for high temperatures the average magnetization converges to zero. For low temperatures however the magnetization is either positive or negative, each with proability 1/2
Phase transition via existence of Gibbs measure: 2D Ising model without magnetic field. Here it is shown that in the thermodynamical limit there exists a Gibbs measure such that above some critical temperature the expected direction of every spin is equal and non-zero. This means that the expected average magnetization must also be non-zero.
I am wondering how to connect these different concept and how these relate to the real-world intuition we have for phase transitions.