Models in statistical mechanics

Models are the main instrument of statistical mechanics that allow to "translate" the physical configuration of a system in a mathematical language, so that it can be thoroughly studied and its behaviour predicted. The role of models in statistical mechanics, and more in general the role of models in the whole scientific method, is of course a very intriguing and philosophically demanding one; of course we will not cover such a vast and difficult issue, and we will only limit ourselves to consider the two diametrically opposing points of view about how models are used.

The "traditional" point of view has been to describe as much faithfully as possible a system, including all its details; in case the theory is unable to explain the experimental results, then some parameters of the system are fine-tuned, or additional parameters are included. On the other hand a more modern point of view, born in the framework of statistical mechanics and motivated by the study of phase transitions and critical phenomena, is that of describing a system with the most minimal and simple possible model, introducing few parameters or properties eventually motivated by symmetry or very general arguments. In such systems in fact it is not necessary to introduce deep levels of details in order to understand their phenomenology, and the study of such minimal models has often led to answers that turned out to be even more universal that those that were looked for when the model was introduced. The use of this more minimalistic approach has been ultimately justified with the introduction of the concepts of universality (see Critical exponents and universality) and Renormalization Group (see chapter The Renormalization Group).

We are now going to introduce some very important models in statistical mechanics: their study will allow us to learn a lot about the statistical mechanics of phase transitions.