We will now turn to the investigation of the rigorous probabilistic formalism of the statistical mechanics of lattice spin systems, or lattice gases. The literature on this subject is well developed and the interested student can find in-depth material for further reading in, and the classical monographs by Ruelle. A nice short introduction with a particular aim in viewis also given in the first sections of the paper.

Spin systems and Gibbs measures

As mentioned in the last chapter, the idea of the spin system was born in about 1920 in an attempt to understand the phenomenon of ferromagnetism. At that time it was understood that ferromagnetism should be due to the alignment of the elementary magnetic moments (‘spins’) of the (iron) atoms, that persists even after an external field is turned off. The phenomenon is temperature dependent: if one heats the material, the coherent alignment is lost. It was understood that the magnetic moments should exert an ‘attractive’ (‘ferromagnetic’) interaction towards each other, which, however, is of short range. The question was then, how such a short-range interaction could sustain the observed very long-range coherent behaviour of the material, and why such an effect should depend on the temperature.